Lectures on

Dynamical Systems,Structural Stabilityand their Applications

KOTIK K. LEEDept. of Electrical and Computer Engineering

University of Colorado

`World Scientific1 Singapore New Jersey London Hong Kong

Published by

World Scientific Publishing Co. Pte. Ltd.

P 0 Box 128, Farrer Road, Singapore 9128

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LECTURES ON DYNAMICAL SYSTEMS, STRUCTURALSTABILITY AND THEIR APPLICATIONS

Copyright ®1992 by World Scientific Publishing Co. Pte. Ltd.

All rights reserved. This book, or parts thereof, may not be reproduced in any formor by any means, electronic or mechanical, including photocopying, recording or anyinformation storage and retrieval system now known or to be invented, withoutwritten permission from the Publisher.

ISBN 9971-50-965-2

Printed in Singapore by JBW Printers and Binders Re. Ltd.

Dedicated to

Peter G. Bergmann and Heinz Helfenstein,

who taught me physics and mathematics,

and to Lydia,

together they taught me humanity.

V

PREFACE

In this past decade, we have witnessed the enormous

growth of an interdisciplinary field of study, namely,

dynamical systems. Yet, dynamical systems, as a subfield of

mathematics, has been established since late last century by

Poincare [1881], and reinforced by Liapunov [1892]. Parts of

such a surge are due to many major advances in differential

topology, the geometric theory of differential equations,algebraic geometry, nonlinear functional analysis, and

nonlinear global analysis, just to name a few. This is

partly due to computers being readily available. Nowadays,

any second-year college student can use a personal computer

to program the evolution of a nonlinear difference equation

and find all kinds of chaotic behavior. With the new level

of sophistication of graphics, it can evolve into

"mathematical entertainment" or "arts". Each popular science

or engineering magazine has at least one article on

nonlinear dynamical systems a year. It not only caught the

fascination of the scientists, but also attracted the

attention of the enlightened public. Ten to fifteen years

ago, there were but a dozen papers on dynamical systems in

each major mathematics or physics journal per year. At

present, the majority of these journals contain sections on

nonlinear sciences, and indeed there are several new

journals solely devoted to this subject. Nonetheless, the

communication and infusion of knowledge between the

mathematicians working on the analytic approach and the

scientists and engineers working mostly on the applications

and numerical simulations have been less than ideal.

Part of the reason is cultural. Mathematicians tend to

approach the problem in a more generic sense, that is, they

tend to ask questions and look for answers for more general

properties and the underlying structures of the systems.

Books written by mathematicians usually treat the subject

vii

with mathematical rigor, but lack of some motivation forwanting to study the underlying mathematical structures ofthe system. The treatment by scientists and engineersusually encompasses too many details and misses theunderlying structures which may transcend the usualboundaries of various disciplines. Thus, scientists andengineers may not be aware of the advances of the same typeof problems in other disciplines. We shall give examples ofsuch underlying mathematical structures for diversedisciplines. This volume intends to bridge the gap betweenthese two categories of books treating nonlinear dynamicalsystems. Thus, we would like to bridge the gap and fostercommunication between scientists and mathematicians. In thefollowing, proofs of theorems are usually not given.Instead, examples are provided so that the readers can getthe meaning of the theorems and definitions. In other words,we would like the readers to get some sense of the conceptsand techniques of the mathematics as well as its "culture".This volume is based on the lecture notes of a graduatecourse I gave in 1983-4 while I was with TRW.

Chapter 1 introduces the concept of dynamical systemsand stability with examples from physics, biology, andeconomics. We want to point out that even though thedifferential or difference equations governing thosephenomena are different, nonetheless, the mathematicalprocedures in analyzing them are the same. We also try tomotivate the reader about the concept and the need to studythe structural stability.

In Chapter 2, we assemble most of the definitions andtheorems about basic properties of algebra, points set (alsocalled general) topology, algebraic and differentialtopology, differentiable manifolds and differentialgeometry, which are needed in the course of the lectures.Our main purpose is to establish notations, terminology,concepts and structures. We provide definitions, examples,and essential results without giving the arguments of proof.The material presented is slightly more than absolutely

viii

necessary for the course of this lecture series. Forinstance, we certainly can get by without explicitly talkingabout algebraic topology, such as homotopy and homology.Nonetheless, we do utilize the concepts of orientablemanifolds or spaces so that an everywhere non-zero volumeelement can be found. We also discuss the connectedcomponents and connected sums of a state space. Nor do wehave to discuss tubular neighborhoods, even though whendealing with return maps, the stability of orbits andperiodic orbits, etc. we implicitly use the concept oftubular neighborhoods. Nonetheless, these and many otherconcepts and terminology are frequently used in researchliterature. It is also our intent to introduce the reader toa more sophisticated mathematical framework so that when thereader ventures to research literature or further reading,he will not feel totally lost due to different "culture" orterminology. Furthermore, in global theory, the state spacesmay be differentiable manifolds with nontrivial topology. Insuch cases, concepts from algebraic topology are at timesessential to the understanding.

In light of the above remarks, for the first reading thereader may want to skip Section 2.3, part of Section 2.4, deRham cohomology part of Section 2.8 and Section 2.9.

The subjects discussed in Chapter 3 are not extensivelyutilized in the subsequent chapters, at least notexplicitly. Nonetheless, some of the concepts and eventerminology do find their way to our later discussions. Thischapter is included, and indeed is lectured, to prepare thereaders with some concepts and understanding about globalanalysis in general, and some techniques important to theglobal theory of dynamical systems. In particular, thereader may find it useful when they venture to theoreticallyoriented research literature.

In the next chapter, we shall discuss the general theoryof dynamical systems. Most of the machinary developed in thelast chapter is not used immediately. Only in the lastsection of Chapter 4, the idea of linearization of nonlinear

ix

differential operators will be utilized for the discussionof linearization of dynamical systems. Nor does Chapter 5depend on the material of Chapter 3. As a consequence, onemay want to proceed from Chapter 2 to Chapters 4 and 5,except Section 6 of Chapter 4. Then come back to Chapter 3and continue to Chapter 6.

In Section 4.7 we briefly discuss the linearizationprocess based on some results from Chapter 3. For mostscientists and engineers, the linearization process is"trivial". Everyone has done this since their freshman yearmany times. Unfortunately, the linearization process is themost misunderstood and frequently mistaken procedure forscientists and engineers in dealing with nonlinear phenomenaand nonlinear dynamical systems at large. This isparticularly true for highly nonlinear systems. Twoillustrations from our earlier training in mathematics canmake the point clear. First, in our freshman calculuscourse, we learned about the condition of continuity of afunction at a point, and its derivative at that point. Forthe derivative to have a meaning, not only the change of theindependent variable, say, ax, has to be very small (i.e.,local), but also the limits lim x»0+ ay/ ax and lim

x - o-

ay/ex agree at x = x0. In other words, the usuallinearization makes sense, only when it is done locally, andany "displacement" from the point has to be consistent. Theconsistency can be illustrated by the following example. Aswe shall discuss in Chapter 2, a differentiable manifold Mis a topological manifold (a topological space with certainnice properties) endowed with a differentiable structure.For any two points p and q in M, A, B are sufficiently smallneighborhoods of p and q in M respectively. At p and q in Aand B, a rectangular coordinate system can be attached toeach of p and q, and every variable looks linear in A and B.What makes the coordinate system or the differentiablestructure go beyond the confine of A or B is that if theintersection of A and B is non-empty, then the coordinatevalues in A and B have to agree at all points in the

x

intersection of A and B. That is, the differentiablestructure has to be consistently and continuously agreedbetween A to B. Consequently, for a highly nonlinear systemwhen one linearizes such a system, one has to be certainthat the linearization procedure is "consistent" in theabove sense. Otherwise, except at or very near theequilibrium point(s) in the phase space, one has noassurance as to the correctness of the results oflinearization.

Related to the above discussion of the linearizationprocedure, we would also like to caution the reader aboutnumerical simulation of nonlinear dynamical systems, evenwhen only dealing with a local situation. First and foremost,do not just code the differential equations and let thecomputer do the rest. One has to analyze the characteristicsof the system, namely, how many and what kind of fixedpoints, periodic orbits, attractors, etc. Then, let thecomputer do the dirty work, and compare with the analysis tosee whether or not the numerical simulation agrees with theanalysis on the number and type of characteristics.Otherwise, there is no way one can be sure the simulation iscorrect and meaningful.

Another important point sometimes scientists andengineers have overlooked is whether or not the system has aCauchy data set. If the system does not admit a Cauchy dataset, it is tantamount to say that the system does not admita unique time. Thus, the time evolution of the system losesits meaning. One may argue that who cares about t - oo. But

even for a finite t, the system may become unpredictable,with or without chaos, etc.

It is also appropriate to point out a practical pointwhich relates to the numerical schemes for studying thenonlinear dynamical system. The implicit method is a verypopular and efficient scheme to study the linear or weaklynonlinear differential equations. But for highly nonlinearsystems, this scheme may leads to erroneous results. This isbecause the interpolation definitely introduces errors,

xi

which initially are small, but their growth rate usually isalso the largest. After propagating for a short time, theamplitudes of the errors can be as large as the mainvariables. Small scale self-focusing in nonlinear optics isa very good example (Bespalov and Talanov 1966; Fleck et al1976; Lee 1977; Brown 1981]. The two volume set by R.Bellman (Methods of Nonlinear Analysis, Academic Press,1973) provides eloquent motivation and many techniquesparticularly useful for numerical simulation of nonlinearsystems. This set is highly recommended for anyone seriouslyinterested in the numerical simulation of nonlineardynamical systems.

In Chapter 5 we introduce the Liapunov's direct (orsecond) method for analyzing stability properties ofnonlinear dynamnical systems. In this chapter, we are stilldealing with the local theory of stability for nonlineardynamical systems.

The first half of Chapter 6 discusses the global theoryof stability and the very important concept of structuralstability. Sections 6.6 on bifurcations and 6.7 on chaoscould be grouped together with the local theory ofstability. Nonetheless, because some of the concepts andtechniques are also useful for the global theory, we putthese two sections in this chapter and deal with globaltheory concerned with bifurcations and chaos. A recent newdefinition of stability proposed by Zeeman, which isintended to replace the original structural stability, isvery interesting and has a great potential in analyzing theglobal stability of many practical physics problems. We didnot include any discussion on quantum chaos, because thisauthor does not understand it. We only mention fractals inpassing, not that it is unimportant, (on the contrary it isa very important topic), but mainly due to lack of space.

Chapter 7 discusses applications of stability analysesto various disciplines, and we also indicate the commonalityand the similarities of the mathematical structures forvarious problems in diverse disciplines. For instance, in

xii

Section 7.5, we not only discuss the dynamical processes ofcompetitive interacting population processes in biology andpopulation ecology and biochemical autocatalysis processes,but we have also pointed out that these equations alsodescribe some processes in laser physics and semiconductorphysics, to name a few. As another example, the discussionof permanence in Section 7.5 with minor modifications, canalso be applied to mode structures and phased arrays inlasers. The point is that scientists and engineers canlearn, or even directly apply results, from differentdisciplines to solve their problems. Putting it differently,physical scientists can learn from, and should communicatewith, biological scientists, and vice versa. One way for thephysical scientists to establish such communication is toread some biological journals. The conventional wisdom hasit that papers in biological journals are not verysophisticated mathematically. The fact is, in the pastdecade or so, there has been an influx of mathematiciansworking on mathematical biology. Consequently, the wholelandscape of the theoretical or mathematical biologyjournals has changed dramatically. Nowadays, there are manydeep and far reaching mathematical papers in a much broadercontext being published in those journals.

The second goal of this volume is to draw the attentionfor such "lateral" interactions between physical andbiological scientists.

The third goal is to provide the reader a very personalguide to study the global nonlinear dynamical systems. Weprovide the concepts and methods of analyzing problems, butwe do not provide "recipes". After all, it is not intendedto be a "cookbook". Should a cookbook be contemplated, the"menu" would be very limited.

We have tried to include some of the references theauthor has benefited from. Surely they are far fromcomprehensive, and the list is very subjective to say theleast. The omission of some of the important works indicatesthe ignorance of the author.

xiii

The following books are highly recommended forundergraduate or beginners on dynamical systems: Hirsch andSmale [1974], Iooss and Joseph [1980], Irwin [1980], Ruelle[1989], Thompson and Stewart [1986].

Without inspiring teachers like Peter Bergmann, HeinzHelfenstein, John Klauder, and Douglas Anderson, I would nothave the opportunity to learn as much. My colleagues andfriends, David Brown, Ying-Chih Chen, Da-Wen Chen, GerritSmith, and many others, are also my teachers. They not onlyhave taught me various subjects, but they also provided mewith enjoyable learning experiences throughout myprofessional career. I would also like to acknowledge theunderstanding and support of my two childern, Jennifer andPeter, without their encouragement I would not be able tocomplete this project. It is my pleasure to acknowledge theconstant encouragement, support, suggestions forimprovements, and patience, of the editorial staff of thepublisher, in particular, Mr. K. L. Choy.

Kotik K. LeeColorado Springs

xiv

CONTENTS

Preface vii

Chapter 1 Introduction 1

1.1 What is a dynamical system? 1

1.2 What is stability, and why should we care about it? 18

Chapter 2 Topics in Topology and Differential Geometry 24

2.1 Getting to the basics - algebra 25

2.2 Bird's eye view of general topology 27

2.3 Algebraic topology 40

2.4 Elementary differential topology and differentialgeometry 45

2.5 Critical points, Morse theory, and transversality 57

2.6 Group and group actions on manifolds, Lie groups 73

2.7 Fiber bundles 82

2.8 Differential forms and exterior algebra 92

2.9 Vector bundles and tubular neighborhoods 102

Chapter 3 Introduction to Global Analysis and InfiniteDimensional Manifolds 111

3.1 What is global analysis? 111

3.2 Jet bundles 112

3.3 Whitney C "topology 118

3.4 Infinite dimensional manifolds 122

3.5 Differential operators 128

Chaper 4 General Theory of Dynamical Systems 145

4.1 Introduction 145

4.2 Equivalence relations 152

4.3 Limit sets and non-wandering sets 156

xv

4.4 Velocity fields, integrals, and ordinarydifferential equations 168

4.5 Dispersive systems

4.6 Linear systems

4.7 Linearization

Chapter 5 Stability Theory and Liapunov's Direct Method

5.1 Introduction

174

178

187

198

198

5.2 Asymptotic stability and Liapunov's theorem 206

5.3 Converse theorems 221

5.4 Comparison methods 222

5.5 Total stability 225

5.6 Popov's frequency method to construct a Liapunovfunction 229

5.7 Some topological properties of regions of attraction 231

5.8 Almost periodic motions 238

Chapter 6 Introduction to the General Theory of 241Structural Stability

6.1 Introduction

6.2 Stable manifolds of diffeomorphisms and flows

6.3 Low dimensional stable systems

6.4 Anosov systems

6.5 Characterizing structural stability

6.6 Bifurcation

6.7 Chaos

6.8 A new definition of stability

Chapter Applications

241

246

252

260

260

267

280

314

324

7.1 Introduction 324

xvi

7.2 Damped oscillators and simple laser theory 327

7.3 Optical instabilities 341

7.4 Chemical reaction-diffusion equations 351

7.5 Competitive interacting populations, autocatalysis,and permanence 359

7.6 Examples in semiconductor physics and semiconductorlasers 375

7.7 Control systems with delayed feedback 379

7.8 Semiconductor laser linewidth reduction by feedbackcontrol and phased arrays 385

References 399

Index 443

xvii

Chapter 1 Introduction

1.1 what is a dynamical system?A dynamical system can be thought of as any set of

equations giving the time evolution of the state of thesystem from the knowledge of its previous history. Nearlyall observed phenomena in scientific investigation or in ourdaily lives have important dynamical aspects. Examples are:(a) in physical sciences: Newton's equations of motion for aparticle with suitably specified forces, Maxwell's equationsfor electrodynamics, Navier-Stokes equations for fluidmotions, time-dependent Schrodinger's equation in quantummechanics, and chemical kinetics; (b) in life systems:genetic transference, embryology, ecological decay, andpopulation growth; (c) and in social systems: economicalstructure, the arms race, or promotion within anorganizational hierarchy. Although these examples illustratethe pervasiveness of dynamic situations and the potentialvalue of developing the facility for modeling (representing)and analyzing the dynamic behavior, it should be emphasizedthat the general concept of dynamics and the treatment ofdynamical systems transcends the particular origin or thesetting of the processes.

In our daily lives we often quite effectively deal withmany simple dynamic situations which can be understood andanalyzed intuitively (i.e., by experience) without resortingto mathematics and the general theory of dynamical systems.Nonetheless, in order to approach complex and unfamiliarsituations efficiently, it is necessary to proceedsystematically. Mathematics can provide the requiredconceptual framework and proper language to analyze suchcomplex and unfamiliar dynamic situations.

In view of its mathematical structure, the term dynamicstakes on a dual meaning. First, as stated earlier, it is aterm for the time-evolutionary phenomena around us and aboutus; and second, it is a term for the pact of mathematics

1

which is used to represent and analyze such phenomena, andthe interplay between both aspects.

Although there are numerous examples of interestingdynamic situations arising in various areas, the number ofcorresponding general forms for mathematical representationis limited. Most commonly, dynamical systems arerepresented mathematically in terms of either differentialor difference equations. In fact, in terms of themathematical content, the elementary study of dynamics isalmost synonymous with the theory of differential anddifference equations.

Before proceeding to the quantitative description ofdynamical systems, one should note that there arequalitative structures of dynamical systems which are offundamental importance, as will be discussed later. At themoment, it is suffice to note that even though there aremany different disciplines in the natural sciences, letalone many more subfields in each discipline, Nature seemsto follow the economical principle that a tremendous numberof results can be condensed into a few simple laws whichsummarize our knowledge. These laws are qualitative innature. It should be emphasized that here qualitative doesnot mean poorly quantitative, rather topologicallyinvariant, i.e., independent of local and detaildescriptions. Furthermore, common to many natural phenomena,besides their qualitative similarity, is their universalitywhere the details of the interactions of systems undergoingspontaneous transitions are often irrelevent. This calls fortopological descriptions of the phenomena underconsideration. Hence, the concept of structural stabilityand the theories of singularity and bifurcation undoubtedlylead the way.

Simply stated, the use of either differential ordifference equations to represent dynamic behaviorcorresponds to whether the behavior is viewed as occurringin continuous or discrete time respectively. Continuous timecorresponds to our usual perception that time is often

2

viewed as flowing smoothly past us. In mathematical terms,continuous time is quantified by the continuum of realnumbers and usually denoted by the parameter t. Dynamicbehavior viewed in continuous time is usually described bydifferential equations.

Discrete time consists of an ordered set rather than acontinuous parameter represented by real numbers. Usually itis convenient to introduce discrete time when events occuror are accounted for only at discrete time periods. Forinstance, when developing a population model, it may beconvenient to work with annual population changes, and thedata is normally available annually, rather thancontinually. Discrete time is usually labeled by simpleindexing of variables in order and starting at a convenientreference point. Thus, dynamic behavior in discrete time isusually described by equations relating the value of avariable at one time to the values of variables at adjacenttimes. Such equations are called difference equations.Furthermore, in order to calculate the dynamics of a system,which are normally represented by differential equations,with an infinite degree of freedom (we shall come to thisshortly) such as fluids, it is more convenient tonumerically break down the system into small but finitecells in space and discrete periods of time. Thus one alsouses the difference method to solve differential equations.

In what follows, we shall concentrate on the aspect ofcontinuous time, i.e., the differential equations aspect ofdynamical systems.

In the late 1800's, Henri Poincare initiated thequalitative theory of ordinary differential equations in hisfamous menoir (1881, 1882]. Ever since then, differentialtopology, a modern development of calculus, has provided theproper setting for this qualitative theory. As we know,ordinary differential equations appear in many differentdisciplines, and the qualitative theory often gives someimportant insight into the physical, biological, or socialrealities of the situations studied. And the qualitative

3

theory also has a strong appeal, for it is one of the mainareas of inter-disciplinary studies between pure mathematicsand applied science.

If we are studying processes that evolve with time andwe wish to model them mathematically, then the possiblestates of the systems in which the processes are takingplace may often be represented by points of differentiablemanifolds known as state spaces of the models. For instance,if the system is a single particle constrained to move on aplane, then the state space is the Euclidean R4 and thepoint (x,,x2,v,,v2) represents the position of the particleat x = (xl,x2) with the velocity v = (vt,v2) . Note that thestate space of a model can be finite dimensional, as in theabove example, or it may be infinite dimensional, such as influid dynamics. Furthermore, it may occur that all past andfuture states of the system are completely determined by theequations governing the system and its state at any oneparticular instance. In such a case, the system is said tobe deterministic, such as in Newtonian mechanics. Thesystems modeled in quantum mechanics are not.

In the context of deterministic processes, usually theyare governed by a smooth vector field on the state space. Inclassical mechanics, the vector field is just another way ofdescribing the system. When we say a vector field governinga process we mean that as the process develops with time thepoint representing the state of the system moves along acurve (integral curve) in the state space. The velocity atany position x on the curve is a tangent vector to the statespace based at x. We say the process is governed by thevector field, if this tangent vector is the value of thevector field at x, for all x on the curve.

In the qualitative theory (or geometric theory) ofdifferential equations, we study the smooth vector fields ondifferentiable manifolds, focusing on the collection ofparameterized curves on the manifold which have the vectorfields as the tangents of the curves (integral curves).Hopefully, a geometric feature of the curves and vector

4

fields will correspond to a significant physical phenomenonand also is part of a good mathematical model for such aphysical situation.

In the following we shall provide some simple examplesof qualitative theory of differential equations andillustrate the approaches we will be taking later. Let usfirst examine some familiar examples in classical mechanicsfrom a geometrical viewpoint.

First, let us consider a pendulum which is the simplestand the most well-known dynamical system. For simplicity,let us scale the mass m and the length 1 of the pendulum tobe unity, i.e., m = 1 = 1. We shall not dwell here on thephysical basis of the equation of motion for the pendulum,which is d20/dt2 _ - gsinO except that there is no airresistance and no friction at the pivot. By using thedefinition of the angular velocity b = dA/dt, we can replacethe equation of motion by a pair of first order differentialequations:

dO/dt = b,d&)/dt = - g sing (1.1-1)

The solution of Eq.(1.1-1) is an integral curve in the (O,b)plane parameterized by t, and the parametrized coordinatesof the curve are (O(t),b(t)). The tangent vector to thecurve at t is (b(t), -g sinO(t)). From various initialvalues of 0 and b at t = 0, one obtains correspondingintegral curves and these curves form the phase-portrait ofthe system. It can be shown that the phase-portrait of thependulum looks like Fig.l-la,b. One can easily distinguishfive distinct types of integral curves. They can beinterpreted as follows:(a) the pendulum hangs vertically downward and is at rest,(b) the pendulum swings between two positions ofinstantaneous rest which are equally inclined to thevertical,(c) the pendulum continuously rotates in the same directionand never at rest,(d) the pendulum stands vertically upward and is at rest,

5

(e) the limiting case between (b) and (c) when the pendulumtakes an infinitely long time to swing from one uprightposition to another.

(a)

(b)

W

fR

(c)

s'

Fig. 1.1.1

6

There are certain features in the phase-portrait whichare unsatisfactory. First of all, the pendulum has only twoequilibrium positions, the one which hangs downward isstable and the other which stands upward is unstable.Secondly, solutions of type (c) are periodic motions of thependulum but it appears as nonperiodic curves in thisparticular form of phase-portrait. In fact, we ought toregard 0 = 9o and 9 = 2nr + 00 for any given integer n asgiving the same position of the pendulum. In other words,the configuration space, which is the differentiablemanifold representing the spatial positions, of a pendulumis really a circle rather than a straight line. Thus, weshould replace the first factor R of R' by the circle S1,which is the real module 2r. By keeping 0 and as twoparameters, we obtain the phase-portrait on the cylinderS'xR as shown in Fig.l.lc.

Consider the kinetic energy T and the potential energy Vof the pendulum, and T = 1/2?' and V = g(1- cosO). Let thetotal energy of the pendulum be E = T + V. Then it is clearfrom Eq.(1.1-1) that dE/dt = 0, i.e., E is a constant on anyintegral curve. Any system with constant total energy iscalled a conservative or Hamiltonian system.

In fact, for a pendulum, one can easily construct thephase- portrait by determining the energy levels (i.e.,energy contours). One can represent the state space cylinderS' x R as a bent tube and interpret the height as energy.This is illustrated in Fig.1.1.2a. The two arms of the tuberepresent solutions of the same energy E > 2g, where 2g isthe potential energy of the unstable equilibrium, with thependulum rotating in the opposite direction.

The stability properties of individual solutions areparticularly apparent from the above picture. Any integralcurve through a point close to the stable equilibriumposition A remains close to A at all times, since the energyfunction E attains its absolute minimum at A and isstationary at B. In fact, B is a saddle point. Thus, thereare points arbitrarily close to the unstable equilibrium

7

point B such that integral curves through them depart from agiven small neighborhood of B.

R

2g

(a) (b)

Fig. 1.1.2

The above example does not include the effects of airresistance and friction at the pivot of the pendulum. Let usnow take these dissipative forces into consideration, andfor simplicity let us assume they are directly proportionalto the angular velocity. Thus, Egs.(1.1-1) become

dO/dt = & and g sinO - a (1.1-2)

where a is a positive constant. Now we find that the energyno longer remains constant along any integral curve and thesystem is called dissipative. This is because for b + 0,dE/dt = -a?2 which is negative and the energy is dissipatedaway along integral curves. If we represent E as a heightfunction as before, the inequality E < 0 implies that theintegral curves cross the horizontal contours of E"downward" as in Fig.1.1.2b.

Now the stable equilibrium becomes asymptotically stablein the sense that nearby solutions tend toward equilibriumsolution A as time goes by. Yet, we still have the unstableequilibrium solution B and other solutions that tend eithertoward or away from B. Nonetheless, we would not expect torealize any such solutions, since we could not hope to

8

satisfy the precise initial conditions needed.By comparing the systems of Egs.(1.1-1) and (1.1-2), one

obtains some hint of what is involved in the importantnotion of structural stability. Roughly speaking, a systemis structurally stable if the phase-portrait remainsqualitatively (or topologically) the same when the system ismodified by any sufficiently small perturbation. Byqualitatively (or topologically) the same, we mean that somehomeomorphism of the state space map integral curves of theone onto integral curves of the other. The existence ofsystems (1.1-2) shows that the original system (1.1-1) isnot structurally stable since the constant a can be as smallas we want. Yet, the systems (1.1-2) are themselvesstructurally stable. To distinguish between the systems(1.1-1) and (1.1-2), we observe that most solutions of theformer are periodic whereas the only periodic solutions ofthe latter are the equilibria (We shall discuss this inChapter 6]. In fact, this last property holds true for anydissipative system, since E is decreasing along integralcurves.

In the above example, it is more convenient anddesirable to use a state space other than Euclidean space,but it is not essential. In studying more complicatedsystems, the need for non-Euclidean state spaces becomesmore apparent. Indeed, it is often impossible to studycomplicated systems globally using only Euclidean statespaces. We need non-Euclidean spaces on which systems ofdifferential equations are defined globally, and this is oneof the reasons for studying differentiable manifolds. Weshall give a brief outline of it in the next chapter.

To illustrate the necessity of a non-Euclidean statespace globally, let us consider the spherical pendulum. Weget the spherical pendulum from the pendulum by removing therestriction that the rod moves in a plane through the pivot.Thus the pendulum is constrained to move on a unit 2-sphereof radius one in Euclidean 3-space. Here once again weassume that the length of the rod is unity. The 2-sphere can

9

be represented parametrically by Euler angles 0 and 0. Theequations of motion for the spherical pendulum are:

d'9/dt' = sing cos9(do/dt)' + g sing (1.1-3)

d' p/dt' = -2 (cot()) dO/dt dO/dt.We can replace this system of second order equations by theequivalent system of four first order equations.

dO/dt = b,dO/dt = µ, (1.1-4)

dt)/dt = µ2 sin9 cosO + g sing

dµ/dt = - 2bg cotO.The state of the system is determined by the position of thependulum on the sphere, together with the velocity which isspecified by a point in the 2-dim tangent plane of S2 at theposition of the pendulum. In fact, the state space is nothomeomorphic to R4, nor to S2xR2 , but is the tangent bundleof S', TS'. This is the set of all planes tangent to S2 andit is an example of a non-trivial vector bundle. We shalldiscuss these concepts in the next chapter. Locally, TS' is

topologically indistinguishable from R4 and one can use thefour variables 0, 0, ta, µ as local coordinates in TS' exceptat the north and south poles of S'.

The total energy of the system in terms of localcoordinates is

E = (C)' + µ2 sin9)/2 + g(1+ cosO)

and it is straightforward to show that E = 0 alongintegral curves, i.e., the system is conservative. Thus,every solution is contained in a contour of E = constant. E= 0 is again a single point at which E is the absoluteminimum, corresponding to the pendulum hanging verticallydownwards in stable equilibrium. Again E = 2g contains theother equilibrium point, where the pendulum standsvertically upward in unstable equilibrium, and at this pointE is stationary but not minimal. From Morse theory (seeChapter 2 and Hirsch [1976], Milnor [1963)) one knows thatfor 0 < c < 2g the contour E-1(c) is homeomorphic to S3.

The spherical pendulum is symmetrical about the verticalaxis through the pivot point. This symmetry manifests itself

10

in Eq.(1.1-4), for they are unaltered if we replace 0 by 0 +k (k = constant) or if we replace 0 and µ by -0 and -µ. Thusthe orthogonal group 0(2) acts on the system as a group ofsymmetries about the vertical axis. Such symmetries canreveal important features of the phase-portrait. Here forany c with 0 < c < 2g, the 3-sphere E-1(c) is partitionedinto a family of tori, together with two exceptional circlesas in Fig.1.1.3. This decomposes R3 into a family of tori,with a circle through p and q and the line 1. Combining witha "point at - " turns R3 into a topological 3-sphere and theline 1 into a topological circle. The submanifolds of thispartition are each generated by a single integral curveunder the action of SO(2) and they are the intersections ofE-1(c) with the contours of the angular momentum function onTS2 . The two exceptional circles correspond to the pendulumrevolving in a horizontal circle in two opposite directions.Between them comes a form corresponding to that of apendulum in the various planes through 1.

Fig. 1.1.3

11

In the examples given above, the dynamical state of thesystem is represented by a point of the state space which isthe tangent bundle (such as S' x R1 or T(S2)) of theconfiguration space (S' or S2 respectively). And theequation of motion represented by a vector field on thestate space and its integral curves give the possiblemotions of the system.

A useful way of visualizing a vector field v on anarbitrary manifold X is to imagine a fluid flowing on X. Letus assume that the velocity of the fluid at each point x e Xis independent of time and equals to the value v(x). Thenthe integral curves of v are precisely the paths followed byparticles of the fluid. Let 0(t,x) be the point of X reachedat time t by a particle of the fluid that leaves X at time0. Obviously, 0(0,x) is always x. Since the velocity isassumed to be independent of time, O(s,y) is the pointreached at time s+t by a particle starting at y at time t.if we set y = p(t,x), as the particle started from x at time0, then 0(s,0(t,x)) = O(s+t,x). We also expect smoothness of0. We shall make these observations more precise when wediscuss one-parameter groups of motion in Ch. 2.

The map 0 may not be defined on the whole space X x R,because particles may flow off X in a finite time. But, if 0is a well-defined smooth map from X x R to X with the aboveproperties, we shall call it, an analogy to the fluid, asmooth flow on X; othherwise, it is a partial flow on X. 0is the integral flow of v or the dynamical system given byV.

If 0 : X x R - X is a smooth flow on X, then for any t eR, we may define a map of : X -+ X by ot(x) = 0(t,x) and itis a diffeomorphism with inverse O"t. If we put f = Oa forsome a e R, then by induction, O(na,x) = f'(x) for allintegers n. Thus, if a is small and non-zero, we often get agood idea of the properties of 0 by studying the iterates f"of f (just as real events can be represented well by thesuccessive still frames of a motion picture). The theory ofdiscrete dynamical systems or discrete flow resembles the

12

theory of flow in many ways; and we shall cover both of

them.Now let us turn our attention to an example of a dynamic

system in the context of population growth. Let us look atthe simplest example of the rich theory of interactingpopulations, the predator-prey model. Let us imagine anisland populated by goats and wolves only. The goats surviveby eating the island vegetation and the wolves survive byeating the goats. The modeling of this kind of populationsystem goes back to Volterra in response to the observationthat populations of species often oscillate. Let N, (t) andN2(t) represent the populations of the prey (goats) andpredators (wolves), respectively. Volterra described thesituation in the following way:

dN1(t)/dt = aN1(t) - bN1(t)N2(t)

dN2(t)/dt = - cN2(t) + dN1(t)N2(t) (1.1-5)

where the constants a, b, c and d are positive. The model isbased on the assumption that in the absence of predators(wolves), the prey (goats) population will increaseexponentially with a growth rate a. Likewise, in the absenceof prey, the predator population will diminish at a rate c.When both populations are present, the frequency of"encounters" is assumed to be proportional to the product ofthe two populations. Each encounter decreases the prey(goats) population and increases the predator (wolves)population. The effects of these encounters are accountedfor by the second terms in the differential equations.

Of course, these equations are highly simplified and donot take into account a number of external factors such asgeneral environment conditions, supply of other food forboth predator and prey, migration of the populations,disease, and crowding. An important application of a modelof this type is the study and control of pests and feed onagricultural crops. The pest population is often controlledby introducing predators, and such a predator-prey modeloften forms the foundation of ecological intervention.

The nonlinear dynamic Eq.(1.1-5) cannot be solved

13

analytically in terms of elementary functions. Nonetheless,it is easy to see that there are equilibrium points. For thesteady state situation, by setting dN,/dt = dN2/dt = 0, wehave one equilibrium point at N, = N2 = 0 and another at N, _c/d, N2 = a/b. It is convenient to normalize variables byletting

x, = dN1/c, xz = bN2/a.Then the dynamic Equations (1.1-5) become

dxl/dt = axe (1 - x2)dx2/dt = -cxz (1 - x1) . (1. 1-6 )

Clearly the nontrivial equilibrium point is at x1 1, x2 =

1.

Let us study the stability of the two equilibrium points(0,0) and (1,1). It is clear that (0,0) is unstable, for ifxi is increased slightly it will grow exponentially. Thepoint (1,1) requires more elaborate analysis. Alinearization of the system in terms of displacements x1, x

2

from the equilibrium point (1,1) can be obtained byevaluating the first partial derivatives of Eq.(1.1-6) at(1,1), and we have

(edx,/dt) = - a (exz)(Adx2/dt) = c(exl) .

The linearized system has eigenvalues ±iac representing amarginally stable system. From linear analysis (such as thefirst method of Liapunov) it is not possible to inferwhether the equilibrium point is stable or unstable.Therefore we have to study the nonlinearity more explicitly.

We can find a constant of motion by writing Eq.(1.1-6)dxz/dt/dxl/dt = [ -cxz (1-x1) ] / [ ax, (1-xz) ]

and rearranging terms leads tocdx,/dt - cdxl/dt/x, + adxz/dt - adxz/dt/xz = 0.

Integrating, we havecx1 - c log xi + axz - a log xz = log k

where k is a constant. For xi > 0, x2> 0, we can define the

functionV (x1, xz) = cx1 - c logxl + ax2 - a logxz.

Clearly V is a constant of motion. Thus, the trajectory of

14

population distribution lies on a curve defined by V = k.

The predator-prey cycles look like Fig.1.1.4. Since thetrajectories circle around the equilibrium point, it isstable but not asymptotically so. The function V, similar tothe energy function E in the case of a pendulum, attains aminimum at the equilibrium point (1,1). V also serves as aLiapunov function for the predator-prey system andestablishes stability. We shall discuss Liapunov functionsin Ch. 5.

X2

1

1

X,

Fig. 1.1.4

We shall discuss crowding, multispecies cases, and theirinterrelationships with biochemical reactions, semiconductorphysics and laser physics in Chapter 7. For an introduction,an interested reader should consult another text [e.g.,Luenberger].

Next we shall discuss a classical dynamic model ofsupply and demand interaction, which also serves as anexample of a difference equation. The model is concernedwith a single commodity, say corn. The demand d for thecommodity depends on the price p through a function d(p)If the price increases, consumers will buy less, thus d(p)decreases as p increases. For simplicity, in this example weassume that the demand function is linear, i.e., d(p) = do -ap, where do and a are positive constants. Likewise, thesupply of the commodity, s, also depends on the price pthrough a function S(p). Usually, the supply increases whenthe price increases. For instance, a higher price willinduce farmers to plant more corn. Note, there is a time lag

15

involved (we shall come to this point shortly). Let usassume that the supply function is also linear, i.e., s(p) _so + by where b is positive and so can have any value, butusually negative.

In equilibrium, the demand must equal the supply, thiscorresponds to the point where these two lines intersect.But the equilibrium price is attained only after a series ofadjustments made by both consumers and producers. It is thedynamics of this adjustment process, movement along theappropriate demand and supply curves, that we wish todescribe.

Assume at period k there is a prevailing price p(k) forthe commodity. The farmers base their production (orplanting) in period k on this price. Due to the time lag inthe production process (growing corn), the resulting supplyis not available until next period, when that supply isavailable, its price will be determined by the demandfunction. That is, the price will adjust so that all of theavailable supply will be sold. This new price at period k+lwill determine the production for the next period. Thus anew cycle begins.

Let us set up the supply and demand equations accordingto the cycles described above. The supply equation can bewritten as

s(k+l) = so + bp(k)and the demand equation

d(k+1) = do - ap(k+l).The condition of equilibrium leads to the dynamic equation

so + bp(k) = do - ap(k+l) .

This equation can be restated in the standard form fordifference equation p(k+l) = - bp(k)/a + (do - so)/a.By setting p(k) = p(k+l), one obtains the equilibriumprice,

p = (do - so)/(a + b),which would persist indefinitely. It is natural to askwhether this price will ever be established or even if oversuccessive periods the price will tend toward this

16

equilibrium price and not diverge away from it. From thegeneral solution of the first- order equation we have

p(k) = (-b/a)kp(0) + [1-(-b/a)k](do - so)/(a + b).If b < a, it follows that as k -+ oo the solution will tendtoward the equilibrium value since all (- b/a)k terms go tozero and the equilibrium value is independent of the initialprice. Clearly, b < a is both necessary and sufficient forthis convergence property to hold.

Let us trace the path of supply and demand oversuccessive periods on graphs and interpret the results. Thegraphs are shown in Fig.1.1.5b and 1.1.5c, which represent aconverging and a diverging situation, respectively. Theinitial price p(O) determines the supply s that will beavailable in the next period. This supply determines thedemand d and thus the price p(1), and so on. Thus we are ledto trace out a rectangular spiral. If b < a, the spiralwill converge inward, but if b > a, it will diverge outward.

From this stability analysis, we can deduce an importantconclusion for the economic model we have been considering.In order for the equilibrium to be attained, the slope b ofthe supply curve must be less than the slope a of the demandcurve. In other words, the producers must be less sensitiveto price changes than the consumers.

Strangely enough, some dynamical behavior of Booleannetworks have the same structure of the supply and demandproblem [Martland 1989].

17

P

(b)

(c

Fig. 1.1.5

1.2 What is stability, and why should we care about it?Very early in scientific history, the stability concept

was specialized in mathematics to describe some types ofequilibrium of a material particle or system. For instance,a particle subject to some forces and possessing anequilibrium point po. The equilibrium is called stable if,after any sufficiently small perturbations of its positionand velocity, the particle remains forever arbitrarily nearpo, with arbitrarily small velocity. In Sec. 1 we havediscussed the dynamics of a pendulum and its stability inthis light.

When formulated in precise mathematical terms, this

18

mechanical definition of stability was found useful in manysituations, but inadequate in many others. This is why, overthe years, a host of other concepts have been introduced,each of them related to the first mechanical definition andto the common sense of stability.

In contrast to the mechanical definition of stability,the concept known as Liapunov's stability has the followingcharacteristics: (i) it pertains not to a material particleor the particular equation, but to a general differentialequation; (ii) it applies to a solution, thus not only to anequilibrium or critical point. More precisely, let

dx/dt = f(t,x)(1.2-1) where x and f are real n-vectors, t e R is the time,f is defined on R x R. We also assume f is smooth enough toensure the exsistence, uniqueness and continuous dependenceof the solutions of the initial value problem associatedwith Eq.(1.2-1) over R x R". Let 11.11 denote any norm on R.A solution x(t) of Eq.(1.2-1) is stable at to, or at t = toin the sense of Liapunov if, for every e > 0, there is a 6 >0 such that if x1(t) is any other solution with IIx'(to) -z(to) II < 6, then IIx' (t) - x(t) lI < e for all t > to.Otherwise, x(t) is unstable at to. Thus, stability at to isnothing but continuous dependence of the solutions on x'o =z'(to), uniform with respect to t e [t,oo ).

Notice that in the case of the pendulum, the equilibriumpoint (0,0) in the phase space is such that no neighboringsolution approaches it when t -' - , except if some frictionwere present. In many practical situations, it is useful torequire, in addition to Liapunov stability of a solutionx(t), that all neighboring solutions x'(t) tend to x(t) whent oo . This leads to the notion of asymptotic stability.

Many other examples can illustrate the necessity ofcreating new specific concepts. Indeed, the stability ofrelative equilibria in celestial mechanics is subtle, to saythe least, depending on deep properties of Hamiltoniansystems (as has been shown by Kolmogorov [1954], Moser[1962], Arnold [1963a,b] and Russman [1970]), it is known as

19

the Kolmogorov-Arnold-Moser (KAM) theorem. We shall brieflydiscuss this theorem in Section 2.8. For further reading ofKAM theorem, consult, for instance, Abraham and Marsden(1978], or Arnold (1978]. From common sense, the solarsystem is considered stable because it is durable, i.e.,none of its planets escapes to infinity, nor do any two suchplanets collide. But the velocities are unbounded iff twobodies approach each other. Therefore, the Lagrangestability simply means that the coordinates and velocitiesof the bodies are bounded. Thus boundedness of the solutionappears as a legitimate and natural type of stability. Formany other definitions of stability and attractivity, pleasesee Rouche, Habets & Laloy [1977].

The most comprehensive of many different notions ofstability is the problem of structural stability. Thisproblem asks: If a dynamical system X has a known phaseportrait P, and X is then perturbed to a slightly differentsystem X' (such as, changing the coefficients in thedifferential equations slightly), then is the new phaseportrait P' close to P in some topological sense? Thisproblem has obvious importance, because in practice thequalitative information obtained for P is not applied to Xbut to some nearby system X'. This is because thecoefficients of the equation are determined experimentallythus approximately. .

An important role physics plays in various disciplinesof science is that most systems and structures in natureenjoy an inherent "physical stability", i.e., they preservetheir quality under slight perturbations - i.e., they arestructurally stable. Otherwise we could hardly think aboutor describe them, and reproducibility and confirmation ofexperiments would not be possible. Thus we have to acceptstructural stability as a fundamental principle, which notonly complements the known physical laws, but also serves asa foundation upon which these physical laws are built. Thusuniversal phenomena have a common topological origin, andthey are describable and classifiable by unfoldings of

20

singularities, which organize the bifurcation processesexhibited by dynamical systems. Here bifurcation refers tothe changes in the qualitative structure of solutions ofdifferential equations describing the governing dynamicalsystems. A phenomenon is said to be structurally stable ifit persists under all allowed perturbations in the system.

Another important approach or "utility" of structuralstability analysis is the following. Since most of thenonlinear equations of nature are not amenable to aquantitative analysis, only a few are known. Consequently itis often unclear which particular quantitative effects oneought to study. Nonetheless, since the nonlinear equationsare derived from geometrical or topological invarianceprinciples, they must process structurally stable solutions.In determining these stable solutions qualitatively, it willprovide us with conceptual guidance to single out the mostsignificant phenomena in complex systems to answer thequestions of structure formation and recognition.

Traditionally, the usefulness of a theory is judged bythe criterion of adequacy, i.e., the verifiability of thepredictions, or the quality of the agreement between theinterpreted conclusions of the model and the data of theexperiments. Duhem adds the criterion of stability. Thiscriterion refers to the stability or continuity of thepredictions, or their adequacy when the model is slightlyperturbed. And the general applicability of this type ofcriterion has been suggested by Thom [1973]. This stabilityconcerns variation of the model only, the interpretation andexperimental domain being fixed. Therefore it mainlyconcerns the model, and is primarily a mathematical orlogical question. It is safe to say that a clear enunciationof this criterion in the correct generality has not yet beenmade, although some progress has been made recently.

A tacit assumption (or criterion) which has beenimplicitly adapted by physicists may be called the doctrineof stability. For instance, in a model of a system withdifferential equations where the model depends on some

21

parameters or some coefficients of the differentialequations, each set of values corresponds to a differentmodel. As these parameters can be determined approximately,the theory is useful only if the equations are structurallystable, which cannot be proved at present in many importantcases. Thus physicists must rely on faith at this moment.Thom [1973] offered an alternative to the doctrine ofstability. He suggested that stability, when preciselyformulated in a specific theory, could be added to the modelas an additional hypothesis. This formalization reduces thecriterion of stability to an aspect of the criterion ofadequacy, and may admit additional theorems or predictionsin the model. Although no implications of this axiom isknown for celestial mechanics as yet, Thom has describedsome conclusions in his model for biological systems. Acareful statement of this notion of stability in the generalcontext of physical sciences and epistemology, just to namea few disciplines, could be quite useful in technicalapplications of mechanics as well as in the formation of newqualitative theories in physical, biological, and socialsciences.

In all fields of physics, waves are used to investigate(probe) some unknown structures. Such a structure or objectimpresses geometrical singularity upon smooth incidentwavefields. The question is then, what information about thestructure under study can be inferred from these geometricalsingularities; this is the so-called inverse scatteringproblem. In order that the reconstruction of structures fromthe backscattered or transmitted curves will be physicallyrepeatable, the scattering process has to be structurallystable, that is, qualitatively insensitive, to slightperturbations of the wavefields.

Imposing this structural stability principle on theinverse scattering process allows us to classify thegeometrical singularities, impresses on the sensingwavefields by the unknown structure, into a few universaltopological normal forms described by catastrophe

22

polynomials. Moreover, the topological singularities providean explanation for the similarity and universality of thepatterns encountered in geophysics and seismology [Dangelmayand Guttinger 1982, Hilterman 1975], ocean acoustics [Kellerand Papadakis 1977], optics [Baltes 1980, Berry 1977, Nye1978, Berry 1980], and various topography, etc.

On a more fundamental level, nonlinear dynamical systemsand their stability analyses have clearly demonstrated theunexpected fact that systems governed by the Newtoniandynamics do not necessarily exhibit the usual"predictability" property as expected. Indeed, a wideclasses of even very simple systems, which satisfy thoseNewtonian equations, predictability is impossible beyond acertain definite time horizon. This failure ofpredictability in Newtonian dynamics has been very wellelucidated in a review paper by Lighthill [1986]. Moreover,recently, there have been several papers attempting torelate such unpredictability to statistical mechanics.Nonetheless, a much more fundamental framework together withappropriate mathematical structure have to be establishedbefore any such correlation can be made.

In the next chapter, we shall assemble some of thedefinitions and theorems, without proof, about basicproperties of algebra, topology, and differential geometry,which are essential to our discussion of the geometrictheory of nonlinear dynamical systems.

23

Chapter 2 Topics in Topology and Differential Geometry

In this chapter, we assemble most of the definitions andtheorems about basic properties of algebra, points set (alsocalled general) topology, algebraic and differentialtopology, differentiable manifolds, and differentialgeometry, which are needed in the course of the lectures.Our main purpose is to establish notation, terminology,concepts, and structures. We provide definitions, examples,and essential results without giving the arguments of proof.The material presented is slightly more than the essentialknowledge for this lecture series. For instance, wecertainly can get by without explicitly talking aboutalgebraic topology, such as homotopy and homology. However,we do utilize the concepts of orientable manifolds orspaces, so that everywhere the non-zero volume element canbe found. We also discuss the connected components andconnected sums of a state space. Although we do discusstubular neighborhoods at the end of this chapter, we do notexplicitly use some of the results in discussing dynamicalsystems; nonetheless when dealing with return maps, thestability of orbits and periodic orbits, etc., we implicitlyuse the concept of tubular neighborhoods. Nonetheless, theseand many other concepts and terminology are frequently usedin research literature. It is also our intent to introducethe reader to a more sophisticated mathematical framework sothat when the reader ventures to research literature orfurther reading, he will not feel totally lost due todifferent "culture" or terminology. Furthermore, in globaltheory, the state spaces may be differentiable manifoldswith nontrivial topology. In such cases, concepts fromalgebraic topology at times are essential to theunderstanding.

In light of the above remarks, for the first reading,the reader may want to skip Section 2.3, part of Section2.4, the de Rham cohomology part of Section 2.8, and Section2.9.

24

There are several well written introductory books ongeneral topology, algebraic topology, differential topology,and differential geometry, which can provide more examples,further details and more general results, e.g., Hicks[1971], Lefschetz [1949], Munkres [1966], Nach and Sen[1983], Singer and Thorpe [1967], Wallace [1957]. For moreadvanced readers, the following books are highlyrecommended: Bishop and Crittenden [1964], Eilenberg andSteenrod [1952], Greenberg [1967], Hirsch [1976], Kelley[1955], and Steenrod [1951].

2.1 Getting to the basics - algebraSome results and notations from group theory will be

needed later, here we sketch some of them.Recall that a group is a set of elements, denoted by G,

closed under an operation (or +) usually calledmultiplication (or addition), satisfying the followingaxioms:(1) for all x,y,z e G;(2) There exists a unique element e e G called the identitysuch that

x e

x a Gx such that x-Lx = e.

A subset H of G, H c G, is called a subgroup if H is agroup (with the same operation as in G). A subset H c G is asubgroup of G iff ab-1e H for all a,b a H. Let H, G be twogroups (with the same operation ), a mapping f : G into- H

is a homomorphism if for all x, y e G. Ife is the identity of H, then f1(e) c G and f-1(e) is calledthe kernel of f, and f(G) is the image of f. It can be shownthat f(G) c H. Graphically, it looks like this:

f (into) /11

25

Two groups G and H are called isomorphic if there is aone- to-one, onto map f:G °"LO- H such that f f (x) - f (y)for all x, y e G, and f is called an isomorphism and denotedby G ~ H. A homomorphism f: G - H is an isomorphism iff f isonto and its kernel contains only the identity element of G.Graphically,

f (1 to 1 onto ) H

If G is a group and S a set (finite or infinite) ofelements of G such that every element of G can be expressedas a product (here we use the multiplication operation) ofelements of S and their inverses, then G is said to begenerated by S and elements of S are called the generatorsof G.

If a group G satisfies another axiom, in addition to(1), (2) and (3) stated above, specifically, the commutativelaw, i.e., y e G, then G is abelian. Ifthe group operation is + instead of , such an abelian groupis called an additive abelian group and its identity elementis denoted by 0, and the inverse of x is -x, for all x e G.If G is any additive abelian group and H c G, G can be splitup into a family of subsets called cosets of H, where anytwo elements x,y e G belong to the same coset if x-y a H.The coset of H with the additive operation form a groupcalled the quotient group of G with respect to H, denoted byG/H.

By considering a sequence of homomorphisms, one cancalculate a group from other related groups. This is thenotion of exact sequence.

Let f: A B be a homomorphism, then from the definitionof the images and kernel of f, we have Im f a A/Ker f. Asequence A f-+ B g-+ C is exact at B iff Im f = Ker g. Anexact sequence has the following properties:(i) a sequence id by A f- B is exact at A iff f is 1-to-1;

26

(ii) a sequence A f-+ B 9-+ id is exact at B if f f is onto ;(iii) a sequence id A f-+ B - id is exact (everywhere) iff

f is an isomorphism;(iv) a short exact sequence id - A f- B 9- C id has the

property that C ~ B/Im f.We shall utilize these properties of a sequence in

Chapter 3 and thereafter. In the next section, we shalldiscuss some fundamentals of general topology.

2.2 Bird's eye view of general topologyA topological space consists of a set X with an

assignment of a non-empty family of subsets of X to eachelement of X. The subsets assigned to each point p e X willbe called neighborhoods of p. The assignment ofneighborhoods to each point p e X must satisfy:(1) If U is a neighborhood of p, then p e U;(2) Any subset of X containing a neighborhood of p is itselfa neighborhood of p;(3) If U and V are neighborhoods of p, so is U n V;(4) If U is a neighborhood of p, there is a neighborhood Vof p such that U is a neighborhood of every point of V.

Example: X = R" , U = (a sphere of center p) then theabove conditions are satisfied, so X is a topological space.

Exercise: Let X be a set and suppose that each pair ofelements x and y of X is assigned a real number d(x,y)called the distance function between x and y and satisfyingthe following conditions:

(1) d(x,y) ? 0 and d(x,y) = 0 iff x = y;(2) d(x,y) = d(y,x) for all x,y a X;(3) d(x,z) <_ d(x,y) + d(y,z) for any x, y, z e X.

Now define an e-neighborhood U of x e X as U =(y a Xj d(x,y)< e). Then define a neighborhood of x e X to be any set in Xcontaining an e-neighborhood of x. Prove that with thesedefinitions, X becomes a topological space.

Remark: A topological space defined in such a way iscalled a metric space. Also, prove that Euclidean space is ametric space.

27

If A and B are any sets, then the set of all pairs (a,b)with a e A and b e B is denoted by A x B, and is called theproduct set of A and B. Let X and Y be two topologicalspaces. Let a set W c X x Y a neighborhood of (x,y) e X x Ywhere x c X, y e Y if there is a neighborhood U of x e X anda neighborhood V of y e Y such that U x V c W. The productspace X x Y becomes a topological space and X x Y is calledthe topological product of X and Y. Clearly, X = Euclidean2-plane is the product of R x R where R is the space of realnumbers.

Let A be a set of points in a topological space X. Apoint p e A is an interior point of A if there is aneighborhood U of p such that U c A. The interior of A = A =(all interior points of A).

Example: X = R'. A = { (x y) J x2 + y' S 1) .A = jx' + y' < 1).

A set A in a topological space is an open set if, foreach p e A, there is a neighborhood U of p such that U c A(i.e., there is no "boundary").

Example: A = (xl a < x < b a,b a R). Also, any solidopen n-sphere (i.e., E x;' < a' ) in R' is an open set.

The following theorems can also be taken as a set ofaxioms to give an alternative definition of a topologicalspace.

Theorem 2.2.1 Let X be a topological space. Then (1)the union of an arbitrary collection of open sets in X is anopen set; (2) the intersection of a finite collection ofopen sets in X is an open set; (3) the whole space X is anopen set; (4) the empty set is an open set.

As a consequence, we have the following theorem:Theorem 2.2.2 If A is any set in a topological space,

then A is an open set and it is the largest open setcontained in A.

We have mentioned prior to Theorem 2.2.1 that theproperties of open sets could be used to provide analternative definition of a topological space. Such aprocedure is based on:

28

Theorem 2.2.3 Let X be an abstract set and let 0 be afamily of subsets of X such that(1) The union of any collection of sets belonging to 0 is a

set belonging to 0;(2) The intersection of a finite collection of sets

belonging to 0 is a set belonging to 0;(3) The whole set x belongs to 0;(4) The empty set 0 belongs to 0.Then there is one and only one way of making X into atopological space (i.e., by assigning neighborhoods to theelements of X) so that 0 is the family of open sets of X: byrequiring that a set U is a neighborhood of p e X iff thereis a set W beloning to 0 such that p e W c U.

Intuitively, it is easy to see with the help of Theorem2.2.1, nonetheless the proof is somewhat tedious. Theessence of this theorem is that a topological space can bedefined by giving the open sets instead of assigningneighborhoods to each point. The original definition of atopological space in terms of neighborhoods is probably themost convenient and the most intuitive one, which depends ona common sense notion of nearness. But there are advantagesfrom the viewpoint of open sets. First, it is more simplelogically to name one single family of open sets in X thanto name a family of sets attached to every point of X.Moreover, many definitions can be made and theorems provenmore easily and elegantly in terms of open sets than interms of neighborhoods.

Let X be a topological space and S a subset. If p e S, asubset U of S is a neighborhood of p in S iff U = S n V forsome neighborhood V of p in X. Then the neighborhoods in Sof points of S define a topology on S. This is the topologyinduced on S by X, and S with this topology is a subspace ofX.

Remark: One must be careful, in speaking of a space anda subspace, to distinguish between neighborhoods in thespace and neighborhoods in the subspace; likewise, todistinguish between open sets in the space and open sets in

29

the subspace.Example: Let X = R2, i.e., the (x,y)-plane, and let S =

RI be the x-axis. As usual, a neighborhood of a point p e Xis any set containing a circular disk with center p, while aneighborhood of a point p e S is any set in S containing aninterval with midpoint p. If U is an open interval in S(i.e., interval without endpoints), U is open in S but notin X because U contains no circular disk. The usualtopologies on X and S make S a subspace of X. This can beseen by noting that if p e S, the intersection of S with adisk of center p is an interval with midpoint at p.

Theorem 2.2.4 Let X be a topological space and S asubspace. Then a subset U of S is open in S iff there is anopen set V inX such that U = V n S.

Exercise: Let X be a topological space and S a subset ofX. Define U c S to be open in S iff U = V n S for some set Vopen in X. Show that the sets so defined as being open in Ssatisfy the conditions of Theorem 2.2.3 thus defining atopology on S.

A sequence of points p,, p2 .... in a topological spaceX is said to have a limit p, or to converge to p, if for anypreassigned neighborhood U of p, there is an integer N suchthat: pn e U for all n >_ N.

Remark: There are two warnings to be made concerning thelimits of an arbitrary topological space. First, the Cauchycriterion for the convergence of a sequence of real numbershas no analogue in a topological space in general. This isbecause in general topological space there is no uniformstandard of nearness which can be applied to a variable pairof points. (But Cauchy criteria do have a uniform standard,i.e., the metric). Second, there is no guarantee that thelimit of a sequence of points, if it exists, is unique.

It is desirable to restrict our attention to topologicalspaces satisfying a condition wich will enable theuniqueness of the limit of a sequence of points to beproven. The following definition will provide the required

30

condition.A topological space X is a Hausdorff space, sometimes

denoted by T2 space, if for every pair p, q e X with p + q,there is a neighborhood U of p and a neighborhood V of qsuch that U n V = 0.

Theorem 2.2.5 Let X be a Hausdorff space, and supposethat a sequence (pn) has a limit p. Then this limit isunique.

Example: R1 is T2. In general, R" in its usual topologyis a T2 space.

Let A be a set of points in a topological space X. Thena point p e X is a limit point (or accumulation point) of Aif every neighborhood of p contains a point of A differentfrom p.

Example: Take X = R1. Then the above definition becomesthe one usually given in analysis. One can take a decreasingsequence of intervals with p as midpoint of length, say 1,1/2, 1/3, 1/4 ..... If p is a limit point of the set ofnumbers A, each of these intervals will contain a number x1,x2, ... respectively, different from p and belonging to A.If U is any neighborhood of the number p, U will containevery interval of length 1/n and midpoint p for sufficientlylarge n. Thus the sequence x1, x2, ... has p as its limit.Incidentally, every neighborhood of p contains infinitelymany points of A.

There are pathological cases one should be aware of (inparticular for non-T2 spaces).

Corollary 2.2.6 In a T2 space, only infinite sets ofpoints are capable of having limit points at all.

Note also that a limit point of a set A in a topologicalspace need not be the limit of any sequence of points of A.The construction of an example is a bit complicated. Let X

.be the set of all real valued functions of the real variablex, defined for all values of x. If f is any point of X(i.e., a function of x) the (e, x1, x2,....)-neighborhood off will be defined to be the set of all points 0 of X suchthat 10(x,) - f(x,)I < e, i = 1,2,... n. Such a neighborhood

31

of f can be defined for every positive number e, and everyfinite collection of number x1, x2,...xn. The topology of Xcan be defined by saying that U is a neighborhood of f iff Ucontains a (e, x1, x21...,xn)- neighborhood of f for some e,x1, x21...,xn. It can be verified that this assignment ofneighborhood satisfies the conditions of a topologicalspace. Furthermore, one can show that this space is a T2space.

Now let fo be the function of x which is identicallyzero, and define a subset A of X as follows. Let x1,x21...,x, be any finite collection of real numbers and letfxixy..xn be the function of x defined by setting fx,x:...xw(x;)

0 (i = 1,2,..,n) and fx,x.x

"(x)= 1 for all other values of

X. Let A be the set of all fx,xl..xti for all possible dinitesets x1, x2..,xn of real numbers. fo certainly does notbelong to A. But fo is a limit point of A in the topologicalspace X. Let U be a neighborhood of fo. Then U contains the(e,x1,x2,..,xn)-neighborhood of fo for some a and x1, 2,..,x,

and the function fx x belongs to this (e,xl,..,xn)-neighborhood, since If, 'xM(x1) - fo(x;)I < e because bothterms of the difference are zero for i =1,2,..,n. Clearly fx

x is a member of A and belongs to the given neighborhood Uof 0. Thus fo is a limit point of A.

Now we want to show that fo is not the limit of anysequence selected from A. Suppose f,, f2,.. is a sequencebelonging to A and having fo as the limit. By the definitionof A, each fn is a function of x equal to zero at a finiteset Sn of values of x and equal to 1 for all other values.The union of all the Sn for n = 1,2,... is, at most, adenumerable set of points, and sothere is a real number xonot belonging to any of the S.. Let V be theneighborhood of fo for some e < 1. By the choice of x0,fi(xo)= 1 for all i, and so none of the members of thesequence f1, f2,... belongs to V, it follows that fo cannotbe the limit of this sequence.

It might be added that the topological space constructed

32

in the above example appears quite naturally and is usedfrequently in analysis. To say that a sequence f1, f2,... ofpoints of this space converges to the limit f means exactlythat the functions f1, f2,.. converge oointwise to f, i.e.,for every fixed value of x, the sequence f1(x), f2(x),...ofreal number converges to f(x).

Just in passing, suppose a topological space X satisfiesthe following condition: For each point p e X there is asequence N,(p), N2(p),... of neighborhoods of p such thatN,1(p) c Nn(p) for each n, and given any neighborhood U of pthere is an n such that N,(p) c U. One can prove that inthis case if p is a limit point of a set A in X, p is thelimit of some sequence of points of A. The above conditionimposed on X is called the first axiom of countability. (Thesecond one is a condition on the family of open sets. Weshall come to that later).

Exercise: Prove that a Euclidean space satisfies thefirst countability axiom. In fact, one can show that theaxiom holds for any metric space.

Let A be a set in a topological space X. Then theclosure of A, A, is defined to be the set in X consisting ofall the points of A along with all the limit points of A.

Examples: (1) X = R in the usual topology, A = (xla < x< b), then A = {xla _< x <_ b). (2) X = R" and A is the set ofpoints at distance less than r from some fixed point p, thenA is the set of points whose distance from p are less thanor equal to r.

A set A in a topological space X with the property thatA = A is called a closed set of X.

Theorem 2.2.7 A set A in a topological space X isclosed iff the complement of A in X is an open set.

This theorem is sometimes used to define a closed set.One can also translate Theorem 2.2.1 into a theorem aboutclosed sets by taking the complements of all the open setsmentioned there.

Once again, it should be pointed out that if one isconsidering a subspace S of a space X as well as S itself,

33

the notion of closure and closed sets, like those ofneighborhoods and open sets, are relative.

Up to now, we have been considering the topologicalspace, its subsets and their properties. In the following,we shall briefly discuss the relationships between spacesand the topological properties of spaces.

Let X and Y be two topological spaces. A mapping f of Xinto Y is a rule which assigns to each point p e X a welldefined point f(p) in Y. The mappping f is continuous at oif, for each neighborhood U of f(p) in Y, there is aneighborhood V of p in X such that f(V) c U. The mapping fof X into Y is continuous if it is continuous at all pointsof X.

Let X and Y be two topological spaces, and let f: X - Y

(not necessarily continuous), and let X' be a subspace of X.Then f induces a map f' of X' into Y, defined by settingf'(p)= f(p) for all p e X'. f' is called the restriction off to X'.

Note that f and f' are different mappings, because ifonly f' is known, f is by no means uniquely determined. Forexample, f:R - R defined by f(x)= x2 for all x e R, and X'=[0,1] and f':X' - R by f'(x)= x2 for x e [0,1]. Clearly f'is a restriction of f to X'. Let g:R - R be defined bysetting g(x) = 0 for x < 0, g(x) = x2 for 0 <_ x <- 1, andg(x) = 1 for x > 1. Clearly f' is also a restriction of g toX'.

Theorem 2.2.8 Let X and Y be two topological spaces. Amap f:X -, Y is continuous iff the inverse image of everyopen set in Y is open in X.

Note that the inverse image is not in general a mapping.If it is to be a mapping, f must be one-to-one. Furthermore,f is onto. But in topology, special interest is attached tothose mappings which are not only one-to-one and onto, butalso have the property that both the mapping and its inverseare continuous.

Let X and Y be two topological spaces and let f be aone-to- one mapping of X onto Y. Then if both f and f1 are

34

continuous, f is a homeomorphism of X onto Y, and X and Yare homeomorphic under f.

Clearly, if f is a homeomorphism of X onto Y, then f-1is a homeomorphism of Y onto X since if g = f"1, then g"'= f.

From Theorem 2.2.8 and the definition just given, itfollows that a homeomorphism of X onto Y sets up aone-to-one correspondence between the open sets of X andthose of Y. The idea that these two spaces are homeomorphicto each other under f means that not only that the pointsare in one-to-one correspondence, but also the neighborhoodsof corresponding points are very similar to one another.These are properties of a topological space X which dependonly on the definition of X as a topological space (i.e.,depend only on the knowledge of which sets in X are open),but depend in no way on any other properties the elements ofX may have. Such a property is characterized by thefollowing definition:

A property of a topological space X is a topologicalproperty of X if it also belongs to every topological spacehomeomorphic to X.

Since two homeomorphic spaces are to be regarded ashaving the same topological structure, homeomorphism (ofspaces) plays the part in topology analogous to that playedby an isomorphism (of groups) in algebra.

We shall discuss some elementary topological propertieswhich are of considerable importance in topology. First, wewould like to generalize the notion of a closed bounded setin a Euclidean space to other topological spaces. Animportant property of Euclidean spaces (normally mentionedin analysis) is given by the Heine-Borel theorem: A set A ina Euclidean space R" is closed and bounded iff whenever A iscontained in the union of an arbitrary collection of opensets in R", then it is also contained in the union of afinite number of open sets chosen from the given collection.One can avoid the explicit mentioning of the open sets ofthe Euclidean space and the wording of the theorem can alsobe tidied up with the following definition. (This definition

35

is not just for tidying up the theorem, its importance willbe manifested later).

Let X be any topological space, and let F be a family ofsets in X such that X is their union. Then F is a coveringof X. If all sets in F are open, then F is an open coveringof X. If F and F' are two coverings of X such that every setbelonging to F' also belongs to F, then F' is a subcoveringof X. Then the Heine- Borel theorem can be restated asfollows: A set A in a Euclidean space is closed and boundediff every open covering of A contains a finite subcovering.

We note that the property of being closed and bounded ina Euclidean space is equivalent to a topological property.This topological property can be formulated for anytopological space.

A topological space X is a compact space if it is aHausdorff space, and if every open covering of X contains afinite subcovering. If A is a set in X, then A is a compactset if A, as a subspace of X, is a compact space.

So the Heine-Borel theorem says that a set A in R" iscompact iff it is closed and bounded. We have just said thatbecause compactness (or a set is closed and bounded) isdefined entirely in terms of open sets, thus we refer it tobe a topological property. But we want to check whether if aspace is homeomorphic to a given compact space is alsocompact. Indeed, we have the following stronger theorem:

Theorem 2.2.9 Let f be a continuous mapping of atopological space X onto a topological space Y. Then if X iscompact, and Y is Hausdorff, Y is compact.

Proof: Let F be a given open covering of Y. Then f-1 ofeach set of F is open in X (Theorem 2.2.8), and f-'(F) forma covering F'of X. F' is thus an open covering of thecompact space X, so it contains a finite subcovering of X.But the images under f of the sets in this subcovering of Xare known to be sets of the covering of F (from thedefinition of F') and form a covering of Y. Thus the givencovering of Y contains a finite subcovering, and Y isHausdorff, so Y is compact.

36

There are some "trivial" corollaries:Corollary 2.2.10 If X and Y are homeomorphic spaces and

X is compact, so is Y.Corollary 2.2.11 If f: X - Y is a continuous mapping of

a compact X into a Hausdorff space Y, then f(X) is a compactset.

One of the most important properties of product spacesis given by the Tychonoff Theorem: The product of compactspaces is compact.

Next we shall dicuss and make precise the idea based onour observation that certain sets of points, say in a plane,have the property that any two of their points can be joinedby a curve lying entirely in the set, while certain othersets fail to do so. E.g., if A is a disk, either open orclosed, it is clear that every pair of points of A can bejoined by a curve (in fact, by a line segment) lyingentirely in A. But if At be a set consisting of two disjointcircular disks, then any path joining a point of one ofthese disks to a point of the other disk must cross the gapbetween the disks, thus not lying entirely in A'. First, wehave to define what is a path or a curve.

Let X be a given topological space, and I be the unitinterval 0 <_ t <_ 1, regarded as a subspace of the space ofreal numbers in the usual topology. Then a path in X joiningtwo points p and a of X is defined to be a continuousmapping f of I into X such that f(0) = p and f(l) = q. Thepath is said to lie in a subset A of X if f(I) c A. It isimportant to note that the path is the mapping.

A topological space X is arcwise connected if, for everypair of points p, q e X there is a path in X joining p andq. If A is a set in X, then A is arcwise connected if everypair of points of A can be joined by a path in A.

Now we shall show that arcwise connectedness is atopological property.

Theorem 2.2.12 Let X and Y be two spaces and f is acontinuous mapping of X onto Y. Then if X is arcwiseconnected, so is Y.

37

To give the proof is as easy as giving an example, so weopt for the proof. Let p and q be points in Y. Since f isonto, there are points p' and q' in X such that f(p')= p andf(q')= q. Since X is arcwise connected, there is a path g (acontinuous map) in X joining p' and q'. Then the compositemap is a continuous map of I into Y such thatp and q. [As an exercise, one can prove that if X,Y, Z are three spaces and f:X - Y, g:Y - Z are twocontinuous maps, then the composite map is a continuousmap of X into Z.] That is, is a path in Y joining p andq. Since p and q are arbitrary points of Y, the theorem isproved.

Corollary 2.2.13 If X and Y are homeomorphic spaces,then X is arcwise connected iff Y is. That is, arcwiseconnectedness is a topological property.

As we shall discuss shortly, arcwise connectedness is auseful condition for a physical space. But it is toorestrictive for an abstract space. A much weaker and simplercondition in terms of open sets is about to be discussed.

A topological space X is connected if it cannot beexpressed as the union of two disjoint non-empty open sets.A set A in X is connected if, when regarded as a subspace ofX in the induced topology, A is a connected space.

Example: Let A consist of two circular disks in theplane such that the distance between their centers isstrictly greater than the sum of their radii. Then A is nota connected set.

Theorem 2.2.14 Let X and Y be two topological spacesand f is a continuous map of X onto Y. Then if X isconnected so is Y.

Corollary 2.2.15 If X and Y are homeomorphic, then X isconnected iff Y is.

Thus, connectedness is also a topological property. Aswe have pointed out earlier, connectedness is weaker thanarcwise connectedness, and it is intuitively easy to realizethe following theorem.

Theorem 2.2.16 An arcwise connected space is connected.

38

Note that the converse is not true. For example: Let Abe the set in the (x,y)-plane such that y = sin(1/x) for 0 -<

x <- 1 along with the line segment (0,y) for -1 <_ y < 1.Clearly A is connected, nonetheless A is not arcwiseconnected. This is because for any map f of the interval 0 <_

t 5 1 into A such that f(0) is a point of A with x + 0 andf(l) is a point with x = 0 is necessarily discontinuous at t= 1. But arcwise connectedness requires the mapping f to becontinuous for 0 -< t <_ 1.

A simpler example is the following:A = ((x,y)I(0,y): -1 < y < 1; (x,0): 0 <_ x 5 1).

We have introduced some basic concepts in point set (orgeneral) topology. Before we go on introducing some basicnotions of differential geometry, let us reflect on thoseconcepts we have just discussed and try to relate them tothe world models (be they physical, biological, or social)we want to construct and understand.

As we have discussed in the introduction, any worldmodel must be a topological space. Before put more structureon it, intuition as well as some reality dictates theelimination of a few classes of spaces.(1) One would not wish to construct a world model based on anon- Hausdorff space. This is because non-Hausdorff spacewill not allow us to describe "distinct events", which areof fundamental importance in physical sciences. Nor willnon-T2 spaces allow any statistical inference because thereis no distinct sampling, (let alone discrete or continuoussampling spaces), nor can one construct such concepts asdistribution. So the Hausdorff property of the space isequally important for biological and social sciences.(2) Nor can one build a model of nature based on anon-connected space. This is because no "communication" (orinfluence) can be carried out between separated componentsor distinct events. It is clear that such a model isunacceptable.

As we go along, we shall discuss a few more restrictionsas to the acceptable spaces upon which world models can be

39

built. Next, we shall introduce some basic concepts inalbegraic topology and familiarize ourselves with some ofthe terminology and fundamental results.

2.3 Algebraic topologyRoughly speaking, algebraic topology is a branch of

mathematics which deals with some "equivalence" in algebraicfashion. A typical process in algebraic topology is toassociate certain groups with a given space. It studieshomotopy, homology, cohomology, exact sequences, spectralsequences, excision, obstruction, characteristic classses,duality etc. It has a close relationship with differentialtopology, which differs from differential geometry bystudying the differentiable structure of the space, insteadof being particularly interested in the geometric constructof the space.

In the following pages we shall briefly describe some ofthe concepts in algebraic topology and useful results forfuture use.

2.3.1. Homology theoryIf a simple closed curve, such as an ellipse or a

polygon, is drawn on the plane, then it has an "inside" and"outside". That is to say the closed curve forms the commonboundary of these two portions of the plane. Similarly, if aclosed curve is drawn on the surface of a sphere, the curveis the boundary of two portions of that surface. In contrastto this situation, by drawing the closed curve a on thesurface torus, a does not necessarily divide the surfaceinto two portions, or a is not the boundary of any portionof the surface of the torus. The possibility of drawing aclosed curve on a surface or the maximum number of closedcurves along which the surface may be cut without dividingthe surface into disjoint portions is clearly a topologicalproperty.

A 1-chain may be a line segment, a curve, or a closed

40

loop. The boundary of a line segment is its endpoints. A1-chain without endpoints, thus has no boundary, is called a1-cycle. A 1-chain which is a boundary of some 2-dim surfacelying in the domain D is called a 1-boundary. Two cycles aresaid to be homologous (equivalent) if their difference is aboundary. Thus c2 is homologous to c3.

The constituent elements in the simplicial homologytheory are n-simplexes in R. A p-dimensional simplex,p-simplex, is denoted by aP. a° is a point (vertex), a' is aline interval whose two end points are excluded, a2 is theinterior of a triangle, a3 is the interior of a solidtetrahedron, and so on. Since a p-simplex can be uniquelydetermined by p+l distinct vertices, one may write ap-simplex aP as aP = <v° vi ... VP> where vi is the i-thvertex of the simplex. Clearly, any subset of k+1 verticesof aP forms a k-simplex. Each such sub-simplex is called aface and will be denoted by <vo ... v'; ... vi ... vP>, wherev'; means that the v; vertex is absent in the subset. It iseasy to see that the faces of a2 are its three sides(1-faces) and three vertices (0-faces).

A simplicial p-complex KP is a collection of simplexes

41

(Cl Oialb, 0 k, . , aPm) satisfying the following conditions:(a) The simplexes of KP are disjoint and no two have all thesame vertices. (b) If a simplex is in KP, all its faces arealso in KP. From this definition, it is clear that a simplexcannot cross or end in the interior of another simplex.

A chain cp on complex K is a finite collection ofp-simplexes aiP written as a formal sum with constantcoefficients g;:

cP E g.aiP.The collection of all p-chains cP on K forms an Abeliangroup denoted by CP(K).

Assume a boundary is properly defined, (this involvesthe incidence number, we shall omit here, seeWallace[1957]), a p- chain is a cycle zP if azP = 0. Ap-chain is a boundary bP if there exists a (p+l)-chain calsuch that acP.1 = bP.

Theorem 2.3.1: a(acP) = 0 for all p.

The collection of all p-cycles zP on K forms a group ZP(K) cCP(K). The collection of all p-boundaries bP on K forms thegroup BP(K). Since as = 0, clearly BP c ZP

A p-dim. homology group HP(K) of a complex K is definedto be the factor group HP(K)= ZP(K)/BP(K). Thus each elementof HP(K) is an equivalence class of p-cycles; two cycles aresaid to be homologous if they differ by a boundary. Thegeneral form of HP(K) is Z e ... e Z s GTP. The number ofgenerators of HP(K) is called the u-th Betti number = PP.G

TP is the torsion subgroup of HP(K) - an Abelian group with

only finite elements.

2.3.2. Homotopy:What topological property of a space can be used to

distinguish between, say a closed disc and an annulus (adisc with a hole in it)? The natural answer to this questionis by considering the possibility of shrinking closed loopsdrawn on the two spaces to a point. (For an arcwiseconnected space, it is independent of the base point. Weshall come to this point later.)

42

Let a, r be paths in arcwise connected topological spaceX (i.e., maps of I into X) with the same endpoints (i.e.,a(0) = r(0) = x0, a(1) = r(1) = x1). We say a and r arehomotopic with endpoints held fixed a ~ r rel (0,1), ifthere is a map F: I x I - X such that

F(s,0) = a(s) all sF(s,l) = r(s) all sF(0,t) = xo all tF(l,t) = xt all t

F is called a homotopy from a to r. For each t,s - F(s,t) isa path FL from xo to x1, and FO= a, F,= r. We write Ft: a = rrel (0,1). Pictorially,

1

xo

0 a

x1

4

In particular if a is a loop at xo (i.e., xt = x0) and ris the constant loop r(s) = xo for all s, and if a = rrel(0,1) we say that "a can be shrunk to a point", or ishomotopically trivial.

For instance, the correct statement of Cauchy's theoremin complex analysis is that: fcf(z)dz = 0 for all loops C inthe domain D of analyticity of f which are homotopicallytrivial.

Let a path a start from xa and end at x1, and anotherpath r starts from xi and ends at x2, then the compositepath ar (multiplication operation) starts from xo and endsat x2.

Theorem 2.3.2 Let ,rl(X,xo) be the set of homotopyclasses of loops in X at xa. If multiplication in ,rl(X,xo) isdefined as above, ,r1(X,xo) becomes a group, the neutralelement is the class of the constant loop at xo and theinverse of a class [a] is the class of the loop a"1 definedby a"1(t) = all - t), 0 <- t 5 1 (i.e., travel backward along

43

a).

Theorem 2.3.3 If X is pathwise connected, the group7r1(X,xo) is independent of x0, up to isomorphism. It is

denoted by 7r1(X) - the fundamental group of X.A space X is simply-connected if it is pathwise

connected and 7rl(X) = 0 (i.e., any closed loop in X can beshrunk to a point).

Theorem 2.3.4 7r, (S) = Z, 7r7 (S") = 0, n ? 2.Proposition 2.3.5 7r1 (X x Y) = 7r, (X) x 7r1 (Y) .In Section 2.2 we have defined covering and subcovering

to define the concept of compact space. Here we shallbriefly discuss the concept of covering space and itsrelation to homotopy.

E P-+ X is a covering space of X if every x e X has anopen neighborhood U such that p"'(U) is a disjoint union ofopen sets S i in E, each of which is mapped homeomorphicallyonto U by p. Si are called sheets over U. If X has acovering space X - X such that X is simply-connected, thenX is unique up to equivalence, and it is called theuniversal covering space of X.

Theorem 2.3.6 Every connected space (manifold) has auniversal covering space (manifold).

Remark: We shall discuss the covering manifold later.Examples: (i) SO(3), group of rotations of R3, 7r1(SO(3))

Z/2. The universal covering space of SO(3) is S3. (ii)

Proper Lorantz group LL z P3 x R3 its universal coveringgroup is S L(2,C) tt S3 x R3.

Higher homotopy groups are commutative (7r1(X) may not).In fact:

Theorem 2.3.7 (Hurewicz) If n ? 2 and 7rq(X)= 0 for allq < n, then 7rq(X) z Hq(X) for all q _< n.

Theorem 2.3.8 If X is pathwise connected, then 07r1 (X) - H, (X) is an isomorphism iff 7r, (X) is commutative.

In algebraic topology, in order to calculate aparticular group G, one often proceeds by first finding anexact sequence with G in it, then evaluating all the easiergroups in the sequence neighboring G, then determining G by

44

using the properties of an exact sequence. In the nextsection, we shall briefly discuss some of the properties ofa differentiable manifold and some geometric structures.

2.4 Elementary differential topology and differentialgeometry

A topological manifold of dimension m is a topologicalspace M, (we are going to define manifolds, so instead ofdenoting the topological space by X, we shall use M, N,...to denote manifolds), which satisfies: (i) M is a Hausdorffspace; (ii) if p e M, then there is an open set U c M, p e Usuch that U is homeomorphic to Rm; (iii) M has a countablebasis for its topology, i.e., there is a countable family ofopen sets (U.) such that every open set is a union of someof the UQ's.

A topological manifold M of dimension m has some set-theoretical and topological properties, such as:

(i) M is locally compact. That is, if p e M, any openneighborhood U of p which is homeomorphic to Rm, thencentered at p in U we may choose an open ball of finiteradius. The closure of such ancompact.

(ii)

(iii)

(iv)

(v)

open ball is of course

separable.regular.a normal space [Kelley 1955].the help of the Urysohn's metrization

M isM isM isWith

theorem, one can show that M is a metric space [Kelley1955].

A T2 space X is called paracompact if for each covering(UQ)QEA of X, there exists a locally finite covering (Vp)$,B,

which is a refinement of (U,,)QEA (i.e., each VB is containedin some UQ). The following theorems are the results fromgeneral topology:

Theorem 2.4.1 A paracompact T2 space is normal [Kelley

1955].Theorem 2.4.2 A topological space is paracompact iff

45

every open cover has a subordinate partition of unity[Kelley 1955].

Theorem 2.4.3 Every metric space is paracompact [Kelley1955].

Since "metric" notion is fundamental to any world model,one does not want to construct a world model which is non-paracompact.

Recall Theorem 2.2.16 and the subsequent example, anarcwise connected, T2 space is necessarily to be connected.But the strcuture of a topological manifold is rich enoughsuch that:

Theorem 2.4.4 A topological manifold is connected iffit is arcwise connected.

Nonetheless, a topological manifold need not beconnected. For example, the group space of O(n) is atopological manifold, which has two connected componentscorresponding to the positive and negative determinants.

Let U and U' be open subsets of R10 and R" respectively,and let f be a mapping of U into U'. The map f isdifferentiable if the coordinates y,(f(p)) of f(p) aredifferentiable (sometimes call smooth, i.e., infinitelydifferentiable, denoted by C) functions of the coordinatesx1(p), p e U. A differentiable map f: U U' is adiffeomorphism of U onto U' if f(U) = U', f is one-to-one,and the inverse map f'1 is differentiable.

Let M be a Hausdorff space. An open chart on M is a pair(U,t) where U c M and open, and 0 is a homeomorphism of Uonto an open subset of Rm. So, the dimension of M is m. Adifferentiable structure on M of dimension m is a collectionof open charts (Us, oc)a,A on M where OQ(U,,) is an open subsetof R'" such that (i) M = U U,; (ii) for each pair a,(3 e A thecomposite map is a differentiable map of 0a(UQ n UO)onto Op(UQ n U.) (see the following figure); (iii) thecollection (U(,, 0Q) is a maximal family of open charts forwhich (i) and (ii) hold. Note: (iii) means that the set ofopen charts {U,,) covers M.

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R' rrnr R'

Fig.2.4.1Theorem 2.4.5 [Whitney 1936] Each CP-structure is CP

equivalent to a C '-structure (p ? 1).In other words, if a manifold has a CP-structure, then a

compatible C°-structure can be found. Consequently, oneusually considers the C°-structure except where otherwisestated. When one considers a coarser or finer topology of aset of additional structures, one may want to specify thedefinite differentiability as we shall see later. Forconvenience, we assume the differentiable structure to be C°unless stated otherwise.

A related notion to paracompactness is: a family ofdifferentiable functions (f.)QEA on a C°-manifold M is calleda partition of unity subordinate to the covering (UQ)a(A if:

(i) 0 5 fQ <- 1 on M for every a e A;(ii) the support of each fQ, i.e., the closure of the

set (p a MIf,(p) + 0) is contained in thecorresponding UQ;

(iii) EQ f'(p) = 1.

A differentiable manifold (or e manifold, or smoothmanifold) of dimension m is a Hausdorff space with a

47

differentiable structure of dimension m. In other words, adifferentiable manifold of dimension m is a topologicalmanifold endowed with a differentiable structure. Similarly,an analytic manifold is defined by replacing"differentiable" by "analytic".

The essential difference between a topological manifoldof dimension m and a differentiable manifold of dimension mis that the differentiable manifold requires, in addition,that the composite map of overlapping region U. n U0is differentiable. Clearly, a C°-manifold is a topologicalmanifold. Not surprising from a logical viewpoint, butremarkable from intuition, is that the reverse assertion isfalse [Kervaire 1961]. It is also interesting to ask whetherthe C°-structure on a C- manifold is unique or not. Milnor[1956] has shown that S7 has more than one distinctC°-structure .

Recall from earlier, we have defined a differentiablemapping in relation to open sets in R' and Rn. Here we usethe same concept to define a differentiable map betweensmooth manifolds.

Let M and N are smooth manifolds of dimensions m and nrespectively and f : M - N be continuous. Then f is adifferentiable (or smooth, or C°) map if for every p e M andevery coordinate charts 0a: Ua - R1° with p e Ua and tj,: Up

R" with f(p) e U,6, the composite map Oa 1 isdifferentiable.

The intuitive outcome of this definition is that we canuse the coordinate charts to transfer various notions frommanifolds to the easily understood framework in Euclideanspaces.

Let M1 and M2 be two C°-manifolds. They are diffeomorphicif there are C°-maps f: M, M2 and g: M2 M1 such thatidI and id2 where id1 and id2 are identity maps on M1and M2 respectively. The maps f and g are diffeomorphisms.If the manifolds and mappings are C°, then thediffeomorphism is just a homeomorphism. Furthermore, we donot need to explicitly assume that the manifolds have the

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same dimension. This follows from the fact that if twomanifolds are homeomorphic, then they have the samedimension.

The trivial example of diffeomorphism is a coordinatetransformation. Moreover, the composition of twodiffeomorphisms is a diffeomorphism, thus diffeomorphismsform a group called the group of diffeomorphisms. We shallmeet it again later.

It is easy to convince ourselves that equivalently onedefines a diffeomorphism of Mi onto M2 as a one-to-one mapf: M, - M2 such that f and f"1 are C°. Without requiring f"1

be also C, f is just a C° homeomorphism. ButC°-homeomorphism needs not be a diffeomorphism. This can beeasily seen by the following example. For n > 1, let u21: R- R. The inverse is not differentiable at the origin. Thisalso shows that if f is C,

f"1 needs not be C.Since our goal is to discuss the dynamical systems,

differential equations, their structural stability and theirvarious applications, we shall deal with either theconfiguration spaces or their phase spaces, which are atleast Ck(k >_ 1), thus without any confusion, a manifoldshall mean a differentiable manifold hereafter exceptotherwise stated.

Let M be a manifold, p e M and denote F(M,p) be the setof all e real functions with domain in a neighborhood of p.A Ck curve in M is a map of a closed interval [a,b] into M.A tangent vector at p is a real function X on F(M,p) (i.e.,

X : F(M,p) - R) having the following properties:(i) X(f + g) = Xf + Xg,(ii) X(af) = a(Xf),(iii) X(fg) = (Xf) g(p) + f(p)(Xg)

where f, g e F(M,p) and a e R. The above rules are thedefinition of a derivation, thus a tangent vector is oftencalled a derivation of F(M,p). All the tangents at p form alinear space denoted by MP.

If 0 = (xi,..,xm) is a coordinate syatem, the partial

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derivative at p with respect to x,, Dx (p), is the tangentdefined by (Dx (p) ) f = /au,) (O(p)) , which is alsodenoted by Dx f (p) , where x1= ui O. In the more conventionalnotation, it will be denoted by

(a/axi)Pf = [a(f.O-,)/au;lm(p)It is easy to see that Dx Xj(p) = 6,,, the Kronecker delta,and hence (Dx (p)) is linearly independent.

RR,

Theorem 2.4.6 Let M be a e n-manifold and let(xl,..,xn) be a coordinate system about p e M. Then if X in

MP, where XP = (Xx1) Dx (p) (or XP = (Xx,) (a/(3x,) P) where repeatedindices are summed, and the coordinate vectors Dx (p) (or

(a/ax,)P) form a basis for MP which has dimension n.A vector field X on a subset A of M is a mapping that

assigns to each point p e A a vector XP in MP. If X is avector field on an open set A, and f e F(M,p) on B then thereal function (Xf)(p) = X

Pf is C' on A n B. If X is a vector

field with its domain contained in the coordinate system, wemay write in terms of its basis X = f; Dx (or X = fi (3/ax;),

where the f. are real functions. It is easy to show that Xis e iff fi is C°. If f is a e map of M into Rk, so f =(f1,..,fk) with f. real functions, and X a vector field on M,then we write Xf for (Xfl,..,Xfk). It is clear that if X isCO, then Xf is Co.

Let V(M) denotes the set of all C° vector fields on M.If X, Y, Z e V(M), we define a C° vector field [X,Y], the

50

bracket of X and Y, on the intersection of their domains by[X,Y] = XY - YX. Let us also denote F(M) be the set of C°real valued functions on M. If f,g e F(M), it is trivialthat [X,Y](f+g) = [X,Y]f + [X,Y]g and [X,Y](af) = a[X,Y]ffor a e R. One only has to check that the product property[X,Y](fg) = f[X,Y]g + g[X,Y]f to establish [X,Y] a V(M). Itis clear that [X,Y]= -[Y,X] and [X,X]= 0. Moreover, [fX,gY]= fg[X,Y] + f(Xg)Y - g(Yf)X. The bracket operation alsosatisfies the Jacobi identity,

[X,[Y,Z]] + [Z,[X,Y]] + [Y,[Z,X]]= 0.Let M and N be e manifolds of dimension m and n

respectively. We have defined the concept of a Co map ffrom M into N. Such a map induces a linear map from eachtangent space MP into the tangent space Nf(P). This linear mapis called the differential of f and it is defined by: if t eMP, 0 e F(N,f(p)), then df(t)(O)= By selecting acoordinate system (x,, .. , xm) about p and (y1, .. , y.) aboutf(p), we can determine a matrix representation for df, whichis called the Jacobian matrix of df, with respect to thechosen coordinate systems and the bases (Dx (p)) and (DY(f (p)) ), by the Jacobian (Dx (p)) for 1 -< j <- m and 1 <_

i <- n. We call a C° map f:M - N non- singular at p if df atp is non-singular i.e., the Jacobian at p is non-singular.

Let f:M - N be Co into map.(a) f is an immersion if dfP is non-singular for each p

e M.

(b) The pair (M,f) is a submanifold of N if f is anone-to-one immersion.

(c) f is an imbedding if f is an one-to-one immersion,which is also a homeomorphism into, i.e., f is open as a mapinto f(M) with the relative topology.

For example, one can immerse the real line R into theEuclidean plane as illustrated in the following figure. Notethat the first case is an immersion but not a submanifold,the second is a submanifold but not an imbedding, and thethird is an imbedding.

51

R1

I a=

D

immersion but submanifold but imbeddingnot submanifold not imbeddingThe ideas in imbedding or immersion arose from the

desire to view manifolds as submanifolds of a simpler one,namely the Euclidean space. The notion of an imbedding islike this: a manifold Mm may be imbedded in a "larger"manifold N' if If is identical to a submanifold of N. Onthe other hand, an immersion is a map that locally appearsto be an imbedding failing to be one- to-one. That is, afterone moves a distance away from a given point, the mappingbegins to fold back on itself. The importance of putting amanifold into a simpler one (either by an imbedding orimmersion) cannot be overemphasized. Once a manifold is asubmanifold of a Euclidean space, one can introduce variousanalytic and differential geometric concepts which may notbe clear or accessible otherwise. Furthermore, sometimes animbedding or immersion gives the best intuitive hold on aconstruction. Here we should mention the remarkable result:

Theorem 2.4.7 A C° manifold of dimension m can alwaysbe C°-immersed in R2m, and C-imbedded in R2" 1 .

Whitney [1944] showed that this theorem can besharpened: for m > 0, every paracompact T2 m-manifold imbedsin R2m and immerses in R2m-1 if m > 1. Of course, if themanifold has further properties, such as orientable,

52

parallelizable, (these will be discussed later), then theimbedding (or immersion) theory can be improved evenfurther. E.g., a theorem due to Hirsch [1959] states thatany parallelizable manifold can be immersed in an Euclideanspace of one higher dimension. For a nice survey of work onimbeddings, see Lashof [1965].

Inverse Function Theorem 2.4.8 Let (xi,..,xm) be acoordinate system at p e M, f1,..,fm a F(M,p). Then 0 _(fl,..,fm) restricts to a coordinate system at p iffdet(Dx.f1(p)) + 0, i.e., do is non-singular on MP.

We can restate the Inverse Function Theorem in a morefamilier form:

Theorem 2.4.9. Let f : Rm - Rm be a e map. For p e Rm ifDx.f,(p) is non-singular, i.e., as a linear transformation Dx.f,1(p) is onto. Then there are open subsets U and V of Rm,and p e U and q = f(p) a V, such that the restriction flu isa diffeomorphism from U onto V; i.e., there is a C° map g:V- U which is an inverse to fIU, and Dx.g1(q) is the inverseto the matrix Dx fi(p).

For example, in functions of several variables, say whenwe deal with curvilinear coordinates, a pair of functions x= f(u,v), y = g(u,v) can be regarded as mappings from theuv-plane to the xy-plane, and the inverse function u =O(x,y), v = 4' (x,y) with proper domains.

V

U X

The well-known result 1,

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i.e., the Jacobian of the inverse mapping is the reciprocalof the Jacobian of the mapping.

Corollary 2.4.10 (Inverse Function Theorem forManifolds). If p e M, 0 is a e map from M N, then 0 is adiffeomorphism of an open neighborhood of p onto an openneighborhood of O(p) iff do is an isomorphism onto at p.

In defining arcwise connectedness, we have defined apath as a mapping. Likewise in these notes, curves will alsobe viewed as a special case of mappings. In particular, wewill deal almost exclusively with parameterized curves, inparticular, we shall discuss the various trajectories andorbits in the phase space of a dynamical system.

A C° mapping a: [a,b] - M is a C° curve in M. Let t e[a,b]. Then the tangent vector to the curve a at t is thevector da(t)/dt where da(t)/dt = da(d/dt)t a N. Note thatda(t)/dt is the usual "velocity" vector associated with aparameterized curve in R3.

t

a b

The "reverse" process of finding the tangent vector to acurve is to "filling in a vector". Let X e V(M). A CO curvea in M is an integral curve of X iff da(t)/dt = X(a(t)) foreach t in the domain of a. It is clear that the curve a"fits" X and the physical idea is the following. For a givenvelocity vector field X of a steady fluid flow, thestreamlines of the flow give the integral curves. The localexistence of integral curves is guaranteed by the theory ofordinary differential equations via the following theorem.

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Theorem 2.4.11 Let X e V(M) and let p be a point in thedomain of X. Then for any real number a there exists a realnumber r > 0 and a unique curve a:(a-r, a+r) -+ M such thata(a) = p, and a is an integral curve of X.

As we shall see, dynamical systems are often governed bythe type of equations for integral curves, i.e., da(t)/dt =X(a(T)). An integral curve is called a trajectory or orbitof the system. We shall come to these again later.

It is also convenient to define a broken e curve a onan interval (a,b] to be a continuous map a from [a,b] into Mwhich is C° on each of a finite number of subintervals

[a,b1], [b1,bz].... [bk-1,b] .Let X e V(M), we associate with X a local one-parameter

group of transformations T, which for every p e M and t e Rsufficiently close to 0 assigns the points T(p,t)= a(t)where a is the integral curve of X starting at p. Theorem2.4.11 tells us that for every p there is a positive numberr and a neighborhood U of p such that T is defined and e onUx(-r,r). From our notation, since the real numbers used asthe second variable of T, are parameter value along a curve,they must satisfy additive property, that is: if q e U, t,s, s+t a (-r,r) then T(T(q,t),s) = T(q,s+t).

The set of pairs (p,t), p e M, t e I, is an open subsetof MxR containing p, hence a smooth manifold Ex of dimensionm+l. The mapping a: Ex - M by (p,t) - a(t) is the flow of X.Since M and X are CO, so the flow is also C°.

Let us look at this description in terms of fluid flowagain. As before, let us suppose that the fluid is steadystate, i.e., the velocity of the fluid at each point p e Mis independent of time and equal to the value X(p) of thevector field. In this case, the integral curves of X(p) arethe paths followed by the particles of the fluid. Now letO(p,t) be the point of M reached at time t by a particle ofthe fluid which leaves p at time 0. We notice that ¢(p,0) isalways p. Since velocity is independent of time, O(q,s) isthe point reached at time s+t by particle starting at q attime t. If we put q = O(p,t), so the particle started from

55

the point p at time 0, we can conclude thatO(¢(p,t),s) = O(p,s+t). Also, the smoothness of 0, asfunctions of p and t, will be influenced by the smoothnessof X.

A p-dimensional distribution on a manifold M (p <- dim M)is a function D defined on M which assigns to each m e M ap- dimensional linear subspace D(m) of Mm. A p-dimensionaldistribution D on M is of class C' at m e M if there are COvector fields X1, ..., XP defined in a neighborhood U of mand such that for every n e U, X,(n), ..., XP(n) span D(n).An integral manifold N of D is a submanifold of M such thatdi(Nd = D(i(n)) for every n e N. We say that a vector fieldX belongs to the distribution D and write X c D, if forevery m in the domain of X, X(m) a D(m). A distribution D isinvolutive if for all C' vector fields X, Y which belong toD, we have [X,Y] a D. A distribution D is integrable if forevery m e M there is an integral manifold of D contaning in.It is easy to see that an integrable e distribution isinvolutive. Clearly, every one- dimensional C' distributionis both involutive and integrable, by the existence ofintegral curves. We would like to mention the classicaltheorem of Frobenius:

Theorem 2.4.11 A C' involutive distribution D on M isintegrable. Furthermore, through every m e M there passes aunique maximal connected integral manifold of D and everyother connected integral manifold containing m is an opensubmanifold of this maximal one.

The following local theorem gives more information as tohow the integral manifolds are situated with respect to eachother:

Theorem 2.4.12 If D is a C' involutive distribution onM, and m e M, then there is a coordinate system (x,, ...,

xd) on a neighborhood of in, such that xi(m) = 0 and forevery m' in the coordinate neighborhood the slice (p c Mix1(p) = x,(m') for every i > dim D) is an integral manifoldof D, when given the obvious manifold structure induced bythe coordinate map.

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Before we take off from the concept of flow to the basicidea of dynamical systems and structural stability, weshould prepare ourselves with more conceptual notions andtools in differential geometry so that when we are facingthe geometric theory of differential equations, which is anintegral part of dynamical systems as we have pointed outearlier, we will be ready for it. There are several topicswe would like to briefly discuss, namely critical values,Morse Lemma, groups and group action on spaces, fiberbundles and jets, and differential operators on manifolds.These last two subjects will be discussed in Chapter 3.

2.5 Critical points, Morse theory, and transversalityThe idea of critical points to be introduced here is an

extension of the concept of maxima and minima of a function.As we know in calculus, if a differentiable function f ofone variable x has a maximum of minimum for x = x0, thendf/dx = 0 at x0. Similarly, if a function of two variablesx, y has a maximum or minimum at (x0, yo), then of/ax =of/ay = 0 at this point. Geometrically, what we are sayingis that the tangent plane to the surface z = f(x,y) ishorizontal at (x0, yo). Of course the same condition is alsosatisfied at a saddle point, a point that behaves like amaximum when approached in one way and like a minimum whenapproached in another. Moreover, this situation can bethought of as corresponding to an embedding of M in 3-spacesuch that the function f is identified with one of thecoordinates z, and the horizontal plane z = f(xo, yo) is atangent plane to M at (xo, yo, f (xo,yo)) . More precisely, wehave the following definitions.

Let M10, N" are C°-manifolds, and f is a C° map. A pointa e re is a critical point of f if df, = 0, (i.e., df is notonto at a, or the Jacobian matrix representing df has rankless than the maximum (n)). b e N" is a critical value, if b= f(a) for a e M'". A value b is a regular value if f"'(b)contains no critical points. Thus f maps the set of criticalpoints onto the set of critical values.

57

For example, if M = N = R1, then a critical point is apoint where the derivative vanishes. In calculus or advancedcalculus, the main interest in critical point and criticalvalue is centered on the search for extrema. Although theyare important in their applications, they are of equalimportance in answering geometric questions such as theimmersions, submanifolds, and hypersurfaces as the followingtheorem illustrates.

Theorem 2.5.1 Let f : M '" - N" be a C° map and b e N" be aregular value. Then f-'(b) is a submanifold of M'" whosedimension is (m-n).

Next we use a special but remarkably simple notion ofLebesque measure in real analysis. This particular notion ofmeasure zero gives us a very simple yet intuitive definitionfor our purpose without resorting to a host of machinery.

Let W; be a cube in R" and denote its volume by µ(W1). Aset S c R" is said to have measure zero, µ(S) = 0, if forany given e > 0, there is a countable family of W, such that(i) S a U1Wi; (ii) Ejµ(W1) < E.

It should be noted that it is possible for thecontinuous (C°) image of a set of measure zero to havepositive measure [Royden 1963]. Nonetheless, such apossibility is excluded when the maps are e as thefollowing theeorem shows.

Theorem 2.5.2 Let S c U c R", where µ(S)= 0 and U isopen, and let f: U -+ Rm be Cr (r ? 1). Then µ(f(S))= 0.

Theorem 2.5.3 (Sard) Let f: M - N be C. Then the setof critical values of f has measure zero in N.

Let C be the set of critical points of f, then f(C) isthe set of critical values of f, and the complement N - f(C)is the set of regular values of f. Since M can be covered bycountable neighborhoods each diffeomorphic to an open subsetof Rm, we have

Corollary 2.5.4 (Brown) The set of regular values of aC° map f: M - N is everywhere dense in N.

Corollary 2.5.5 Let f: M'" - N" (n ? 1) be onto and C°.Then except for a subset of N" of measure zero, for all y e

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N", fl(y) is a submanifold of M. Moreover, there is alwayssome y e N" such that f-1(y) is a proper submanifold of M.

Corollary 2.5.6 Let the n-disk be D" = (x 6 R"j 1lxII <-

1) and its boundary 3D" = S""1, an (n-1) -sphere. Let is S"-1 -+D" be the inclusion map. Then there is no continuous map r:D" - S""1 such that r i = id on S"-1, i.e., no continuous rsuch that for each x e S"-1, r(i(x)) = x.

If such an r exists, it is called a retraction of D"onto S""1. For n = 2, this corollary can be worded asfollows: The circle is not a retraction of the closed unitdisk (normally a theorem in elementary homotopic theory). Asa corollary: Any continuous map f of the closed disk intoitself has a fixed point, i. e. , f (xo) = x0 for some x0 a D' .This is the n = 2 case of the Brouwer Fixed Point Theorem.

Corollary 2.5.7 (Brouwer Fixed Point Theorem) Let D" _

(x E R"I lixil -< 1) be the n-disk, let f: D" -+ D" becontinuous. Then f has a fixed point, i.e., there is some x0E D" such that f (xo) = x0.

It has been realized for some time that a topologicalspace can often be characterized by the properties ofcontinuous functions on it. But it was Morse [1934) whofirst called attention to the importance of nondegeneratecritical points and invariant index, which completelycharacterizes local behavior near that point. Moreover, thenumber of critical points of different indices relates tothe topology of the manifold by means of the Morseinequalities. In addition, a sufficiently isolated criticalpoint indicates the addition of a cell to the celldecomposition of the manifold. Consequently, this shows howa manifold is put together, as a cell complex, in terms ofthe critical points of a sufficiently well behaved function.On the other hand, Morse theory also treats geodesics on aRiemannian manifold. Although Morse did a great deal more,here we shall only touch on a few items directly concerningour main emphasis. There is some material from algebraictopology, such as homology, Betti numbers, Eulercharacteristics, which will be needed when we get to the

59

Morse inequalities. At the appropriate places, we shallstate all the basic facts without proof.

Let us recall the concept of a critical point. Let M bea m-dimension C° manifold and f: M - R be a C° function.Then a e M is a critical point of f if f is not onto at a.Since the range of df is a 1-dimension vector space at a, ais a critical point when df is the zero map at a. From amore conventional viewpoint, a e M is a critical point ifthere is a coordinate chart 0Q : UQ -+ R'", x e U. such thatall first partial derivatives of f 0Q"1 vanish at 0.(a). Anda real number b = f(a), where a is a critical point, iscalled a critical value.

Clearly, the first partial derivatives at a criticalpoint have degenerate behavior. Nonetheless, when the secondpartial derivatives are better behaved, it is called anondegenerate critical point. More precisely, if a e M is acritical point for f : M R, f e F(M), and (31(f-O.-1) /axiaxi ;the Hessian at a, is non-singular, then a is a nondegeneratecritical point of f. It can be shown that this definition isindependent of the choice of the coordinate chart.

For example, let S2 be the unit sphere centered at theorigin in R3, and let f assign to any point its z = constantplanes, i.e., f(x,y,z) = z. It is easy to see that there areonly two critical points al(0,0,1) and a2(0,0,-l) and theircritical values z = ±1. Moreover, both of the criticalpoints are nondegenerate.

z

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As another example, let T2 be a 2-dimension torusimbedded as a submanifold of R3. This T2 can be thought ofas the surface traced by the circle of center (2,0) andradius 1 in the (x,y)- plane as this plane is rotated aboutthe y-axis.

The surface has the equation(x2 + y' + z2 + 3)2 = 16(x' + z').

It is easy to show (and easy to see from the figure) thatthere are just four z = constant horizontal planes H,, H21H3, and H4 that are tangent planes of T2 at p1, p2, p3 and p4respectively, coresponding to four critical points for thefunction z on T2 and H, (i=1,2,3,4) are critical levels.Furthermore, one can show that these four critical points pi(i=1,2,3,4) are nondegenerate. Notice that, in this example,if N, is a non-critical level of z, it is surrounded byneighboring noncritical levels, all of which arehomeomorphic to each other. For example, see the abovefigure, between Hi and H2 all the noncritical levels arecircles. But as soon as we cross a critical level, a changetakes place. The noncritical levels immediately below H2 arequite different from those immediately above. In fact, thisobservation is valid in general.

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Let the differentiable manifold M be a 2-sphere and takea zero-dimension sphere S° in M. S° has a neighborhoodconsisting of two disjoint disks. This is of the form S°xE2.(E2 is a 2- dim disk). Let us call this neighborhood B. ThenM -Int B is a sphere with two holes in it. E1xS1 is acylinder (here El is a line segment), and when its ends areattached to the circumferences of the two holes, theresulting surface is a sphere with one handle, i.e., atorus. Thus the torus can be obtained from the 2-sphere by aspherical modification of type 0. See Fig.2.5.1. Let usdefine this term as in the following:

Let N be an n-dim C' manifold and S' is a directlyembedded submanifold of M. Sr has a neighborhood in M whichis diffeomorphic to SrxEn-r and we call it B, where En-r is a(n-r)-cell. The boundary of B is the manifold S'XS"-`'-1. Thus

M - Int B is a manifold with boundary and the boundary isSrXSn-r-1. But SrXSn-r-t is also a boundary of the C' manifold,

Er+lxSn-r-l. So the two manifolds M -Int B and Er'lxSn-r-l can bejoined together by identifying their boundaries. Such ajoined space is a CO manifold M'. M' is said to be obtainedfrom M by a spherical modification of type r.

T2 obtained by spherical modification of type 0 from Sz.

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0 0 E'xS°

S2 obtained by spherical modification of type 1 from T2.Fig.2.5.1

From the examples above, it is clear that if M' isobtained from M by a spherical modification, then M can beobtained from M' by another spherical modification.Furthermore, we have noticed from the examples that if M isa C° manifold and f a e function on it and if Ma and Mbare noncritical level manifolds of f separated by onecritical level, then Mb can be obtained from M. by aspherical modification. This fascinating subfield ofalgebraic and differential topology is called surgery and ithas been an active field since 1960. There are some quitefar reaching results. Related to this subject is the conceptof cobounding of a manifold. If two compact differentiablemanifolds MO and M1 are said to cobound if there is acompact differentiable manifold M such that the boundary ofM is the disjoint union MO U M,. In general, to testingwhether a pair of manifolds are cobounding is verycomplicated, nonetheless, the following very interestingresult can be stated: If MO and M1 are compactdifferentiable manifolds, then they are cobounding iff each

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can be obtained from the other by a finite number ofspherical modifications. A well written introductory book isWallace [1968]. Hirsch [1976] has a couple of chaptersdiscussing surgery (also called cobordism in literature). Avery advanced classic is the one by Wall [1970]. There maybe revised edition. There were some potential applicationsin physics the author thought of while he was a graduatestudent. However, the author has not been following thedevelopments lately. It seems that it may also be useful forthe description of super-strings in particle physics.

Before we are too far off the field from the subject ofnondegenerate critical points, let us look at some of itsproperties from the Morse Lemma.

It has been known for some time that a topological spacemay be characterized by the algebra of continuous functionson it. From the examples given earlier, it is not surprisingthat one can learn a great deal about a smooth manifold fromthe smooth real functions defined on it. Morse [1934] firstrealized the importance of nondegenerate critical points andthe numerical invariant called the index, which completelycharacterizes the local behavior near that point. Moreover,the number of critical points of various indexes relates tothe topology of a manifold by means of the Morseinequalities. A sufficiently isolated critical point alsosignals the addition of a cell to the decomposition of themanifold. Thus, this shows how a manifold is put together asa cell complex, in terms of the critical points of asufficiently well-behaved function. Furthermore, from Sard'stheorem (Theorem 2.5.3), these well-behaved functions areactually very common. In the rest of this section, we willdiscuss the Morse lemma and inequalities, and transversalityproperties.

A symmetric bilinear form represented by a matrix B hasindex i if B has i negative eigenvalues. We asy that B hasnullity k if k of the eigenvalues are zero. Thus, p is anondegenerate critical point for f when p is a criticalpoint and the nullity of the Hessian of f at p is zero.

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Lemma 2.5.8 (Morse) Let M be n-dim e manifold, f: M -+R be smooth and xo e M a nondegenerate critical point. Thenthere is a coordinate chart U. containing xo with 0Q(xo) = 0and such that f (0Q"'(u)) = f (OQ-'(0)) - u12 u12 + ui+12+..+ un' .where u = (u,,..,ud a R" and i is the Morse index of f atxo.

Corollary 2.5.9 The non-degenerate critical points of asmooth function are isolated.

Corollary 2.5.10 If f is smooth on a compact smoothmanifold with all critical points non-degenerate, then f hasonly finitely many critical points.

As an example, consider the family of surfaces in R3,x' - y' - Z' = c

with -1 <_ c <_ 1. Note that the two surfaces obtained byputting c = -1 and 1 are hyperboloids of one and two sheetsrespectively. Then the critical points are easily obtained.

SI

So

These two S° are nondegenerate critical points which areisolated. These infinitely many critical points of S' aredegenerate.

If all critical points of f are non-degenerate and allcritical values are distinct, then f is called a Morsefunction. A fundamentally important result is the followingset of Morse inequalities:

Theorem 2.5.11 (Morse inequalities) Let M be a C°compact m- manifold (without boundary) and f: M R be aMorse function. Let ck, k = 0,1,..,m, denote the number ofcritical points of f of index k and Qk be the kth Betti

65

number of M (i.e., the number of independent generators ofthe kth homology group of M). Then

Co > PbC1 - CoC2 - C1 + Co >_ P2 - P1 + P.........................Cm - Cm_1 +...+(-1)mCo > PM - QM-1 +..+(-l)mf30.Before we continue, we would like to briefly discuss

orientation, duality, and Euler characteristics, and some oftheir geometric interpretations. Orientation of a manifoldis of fundamental importance. Duality relates (or moreappropriately pairing) the Betti numbers of an orientable,compact manifold. Euler characteristic is a topologicalinvariant quantity built upon from Betti numbers, and itsexistence on a manifold has profound geometric implications.For lower dimensional manifolds, in particular,two-dimensional manifolds, it is directly related to thecurvature of the 2-manifold.

Proposition 2.5.12 Every manifold has a unique Z/2 -orientation (the number of orientation = no. of elements inH°(X,Z/2) ).

There are several ways to define orientation. The aboveis a topological one. We shall discuss a geometric onelater.

Theorem 2.5.13 Let X be a connected non-orientablemanifold. Then there is a 2-fold connected covering space EP- X such that E is orientable.

This theorem tells us that if we are interested in thedetailed local geometry of the manifold (assumed to bearcwise- connected) as a model of physical state space, thenwe may as well assume the manifold to be orientable. This isbecause one can always find an orientable covering spacewhich has same local geometry. Indeed, as the next corollaryshows, one can always goes to its universal covering space,which is simply-connected.

Corollary 2.5.14 Every simply-connected X isorientable.

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Proposition 2.5.15 Hn(X) = 0 if X is connected and non-compact, (dim X = n).

Poincard Duality Theorem 2.5.16 If X is an oriented,n-dim manifold, then the homomorphism D : Hcq(X) - Hn_q(X) is

an isomorphism for all q, where Hcq(X) is the cohomologygroup with compact support.

We shall discuss differential forms, and the de Rhamcohomology, and the cohomology groups of a manifold.

Corollary 2.5.17 If X is compact, orientable, then theBetti numbers of X satisfy bq = bn_q for all q.

The Euler characteristic of X x(X) is defined as analternating sum of all the Betti numbers of X, i.e.,

x(X)= E (-1)q bq , n = dim X.

Examples: For q 1, n >_ 1

Hq (S") _ R, q = n0, q + n.

Thus, x(S") 0, n = odd2, n = even.

Remark: x(X) is a very useful topological invariant.E.g.:

Theorem 2.5.18 A differentiable manifold (anydimension) admits a non-zero continuous vector field iff itsEuler characteristics are zero [Steenrod 1951].

As a consequence, we have: For compact manifold M, thereexists a non-vanishing vector field iff x(M)= 0. Thus onlyodd- dim spheres admit non-vanishing vector fields. Indeed,one can prove that this is equivalent to that the tangentbundle of M splits, which is also equivalent to the manifoldadmits a metric of Lorentz signature (i.e.,pseudo-Riemannianmetric). Since it is well-known that any differentiablemanifold admits a Riemannian metric, the Lorentz metric canbe constructed by

t7(Y,Z) = g(Y,Z) - 2g(X,Y)g(X,Z)/g(X,X)or nsi = g1i - 2 X;Xj/IXI:.where X, Y, and Z are non-zero vector fields on M. We shallcome back to this later, when we discuss characteristicclasses.

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Gauss-Bonnet theorem 2.5.19: Let M be a compactconnected oriented Riemannian 2-manifold with Riemannian(Gausssian) curvature function K. Then f K = 2lrx(M) [Hicks

1971].

Generalized Gauss-Bonnet Theorem 2.5.20: If M is aneven dimension (n = 2k) compact connected orientedRiemannian manifold, then f, QDD = 2",rk (k!) x (M) where

Q = E(-l)TRT(1)r(2)ARr(3)T(4)A..."r(n-1)*(n) e A"(M)where ,r(i) are permutations [Chern 1951].

Let us get back to the Morse lemma and Morseinequalities. If furthermore, M is orientable, then,applying the Poincard duality theorem, we can furthersimplify the inequalities. As an example, if M = T2, the2-dim torus, then any Morse function on T2 has at least fourdistinct critical points since Pa = Q2 = 1 and Q1 = 2. Wehave already demonstrated and discussed this earlier.

Let us get back to Sard's theorem. Although manyarguments in imbedding and immersion can be reformulated andoccasionally made more precise by using Sard's theorem, weshall turn to a very important concept of transversality.This is a theory which investigates the way submanifolds ofa manifold cross each other.

Let f be C' map of e manifolds, f: W N", and WP be asubmanifold of N. Roughly speaking, f is transverse to W atx e M, f Ax W, means that the intersection in N of f(M) andW has the lowest possible dimension in a neighborhood off(x) a W, but the sum of the dimensions of f(M) and W is atleast n. More precisely: Given a C' map f: Mm - N" betweentwo C' manifolds and a submanifold WP of N, we say f istransverse to W if for each x e M, y = f(x) e W such thatdf(Mx) + WY = NY. Here + means that we take the set of allvectors in NY that are sums of a vector in the image of dfand a tangent vector to the submanifold W. In other words,the tangent space NY is spanned by WY and the image df(Wx).If f(M) does not intersect W, i.e., f(M) n W = 0, then f isautomatically transverse to W.

As a simple example, let M = Si, N = R2, W = x-axis in

68

R. Then this position is transverse

w Q f(S')

but this position is not.

0It seems that the concept of transversality requires theintersection be in the most general position.

Some more examples of transversality:(a) Let M = R = W, N = R2, f(x)=(x,x2). Then f * W at

all nonzero x.

w

Note that f can be slightly perturbed so that it istransversal to W; e.g.,

\\,w

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(b) M, N, W as in (a) and f is defined by the graph

Then f ; W everywhere except on the segments within thebrackets.

(c) If M = W = R, N = R3, then if f is any mapping of MN, it is transversal to W only if f(M) n W = 0. Note that

here a nontransversal mapping can also be approximatedclosely by a transversal mapping because in 3-space f canavoid W even easier by just going around it and f only hasto move a little bit to accomplish this. We shall make thismore precise in the next proposition.

From these simple examples, it becomes apparent that therelative dimensions of M, N and W play an important role indetermining the conditions as well as meaning for f to betransversal to W. Moreover, for any M, N and W, the set oftransversal mappings is a very large one. Thom'stransversality theorem is a formalized observation of thisfact. Before discussing Thom's theorem, we first give someproperties of the set of maps which is transversal to W.

Proposition 2.5.21 Let M and N be smooth manifolds, W cN a submanifold. Suppose dim W + dim M < dim N. Let f : M -N be smooth and suppose that f d+ W. Then f(M) n W = 0.

This can be seen by the fact that suppose f(p) a W, thenby the definition of tangent space and the assumption,

dim(Wf(P) + (df) (MP)) -< dim Wf(P) + dim MP= dim W + dim M < dim N = dim Nf(x) ,

thus it is impossible for Wf(x) + df(MP) = Nf(x). Thus if f ,' W

at p, then f (x) I W.It is also appropriate for us to relate the notion of

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transversality with Sard's theorem. The simplest example isletting f : R - R be C1. If yo is a regular value, then thehorizontal line R x (yo) c R x R (i.e., y = yo line) istransverse to the graph of f.

Thus, the Sard's theorem or its corollary (Brown's theorem)implies that "most" horizontal lines are transverse to thegraph. For f : RI - R1, the Sard's theorem says that mosthorizontal planes R2 x (zo) c R2 x R1 are transverse to thegraph of f. If we replace e in the Sard's theorem by Cr, wemay want to know whether the theorem will change or not. Forf : R' - R1, the theorem seems plausible for f to be C2 . In

fact, intuitively it even seems plausible for f being onlyC1. But Whitney [1935] has found an ingenious and veryinteresting counter-example. He constructed a C1 map f : R2

- R1 whose critical set contains a topological arc r, yetfjr is not constant. Thus f(Cf) contains an open subset ofR, where Cf = (critical points of f). This leads to aninteresting paradox. The graph of f is a surface S c R3 onwhich there is an arc r such that at every point of r thesurface has a horizontal tangent plane, yet r is not at aconstant height! We shall not go into any more detail aboutthis example, but to say that for Cr mappings, there is adifferentiability condition for Sard's theorem. Let us stateSard's theorem for Cr (r < m) maps.

Theorem 2.5.22 (Sard) Let M and N be smooth manifoldsof dimensions m and n respectively, and f : M - N be a Crmap. If r > max (0, m-n), then f(Cf) (the set of criticalvalues of f) has measure zero in N. The set of regular

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values of f is dense..Theorem 2.5.23 Let f, M, N and W as before. If f is

transverse to W, then f"'(W) is a submanifold of M withdimension m - n + p.

Corollary 2.5.24 If M and W are both submanifolds of Nand for each x e M n W such that W,, + Mx = N. Then M n W isa submanifold of N.

Before we can get to Thom's Transversality theorem, weneed some refinement on the topology of the space ofdifferentiable maps between differentiable manifolds.

Let C°(R",Rk) be the set of C°-maps (or C'-functions)from R" to Rk. The set is topologized as follows: If e(x) isa positive, continuous function defined on R", and p > 0 isany integer, letB(O,e(x),p) = (feC°(R",Rk) I ID°fj(x) I<e(x)

for all IaIS p and j)where f

iis the j-th coordinate of f. This set forms a basis

for the neighborhood of the constant function 0. A similarbasis neighborhood for g e C(R",Rk) can be defined byB(g,e(x),p) = (f a C°(R",Rk)I (f-g) a B(O,e(x),p)). Here theinteger p is allowed to vary. To generalize the topology ofC° (M'", N") , where M'", N" are two C°-manifolds, one can proceedas above by choosing coordinate charts to cover N" anddemand that the above construction holds in all thecoordinate charts near any point. Of course one may alsoreduce this problem to the above construction in much largerEuclidean spaces by using the imbedding theorem.

We want to point out that the above constructionprovides a rather fine topology on C'(M'",N"), this is becausefunction such as e(x) may decrease to zero rapidly, eventhough e(x) > 0 for all x.

Theorem 2.5.25 Let M"', N" be C' manifolds, and WP N" asubmanifold. The set FW(M"',N"), consisting all maps inC'(M'",N") that are transverse to W, is an open subset ofC'(M'",N") .

Theorem 2.5.26 (Thom's transversality theorem) FW(Mm,N")

is dense in C' (M'", N") .

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The transversality theorem in its many variations arenot only of fundamental importance to structural stabilityand bifurcations, but are also of crucial importance to manyareas of differential topology such as Thom's constructionof cobordism theory. Two compact manifolds M1, M2 are calledcobordant, if there is a compact manifold N such that aN z(M1 x 0) U (M2 x 1). Loosely speaking, this means that thedisjoint union of M1 and M. is the boundary of N. We call Na cobordism from Mi to M2. The spherical modification wementioned earlier is a special situation of cobordismtheory.

In the next section, we shall discuss more geometricdetails of a differentiable manifold, in particular, thegroup actions on manifolds. This will provide the fundationfor the discussion of fiber bundles, which is the sectionafter next.

2.6 Group and group actions on manifolds, Lie groupsLie groups exist naturally in many areas of mathematics

and physics where natural group structures may be found oncertain manifolds. Lie groups are also very useful inphysics, in particular, as transformation groups, internalsymmetry groups and their representations for particleclassifications, gauge groups, etc. Even in classicalmechanics, the usual Lagrange and Poisson bracekets not onlyprovide the system's symmetry or conservation laws, but alsoprovide a geometric manifestation of the dynamical processesby noticing that a Poisson bracket of a pair of vectorfields is the dragging of a vector field along the integralcurve of another vector field. There are many well writtenbooks on Lie groups. As a beginning, many differentialgeometry books have a chapter or two on Lie groups, theirgeometry, and representations, e.g., Bishop and Crittenden[1964]. There are several sections in various chapters ofChoquet-Bruhat, De Witt-Morette and Dillard-Bleick [1977].For more advanced readers, Chevalley [1946] is still the

73

best. Pontgryagin [1966] is also a very good book. Helgason[1962] is a detailed treatment of the differential geometryof the group spaces.

A topological group G is a group which is also endowedwith the structure of a topological space and the map 0 :

GxG into G defined by O(x,y)= xy'1 is continuous. Clearly,the multiplication and inverse maps defined by (x,y) xy

and x - x"1 are both continuous.For example, the additive group of real numbers is a

topological group, and the group of invertible nxn realmatrices is also a topological group.

Let G be a topological group and X a topological space.G acts on X to the left if there is a continuous map 0 : G xX - X and we write O(g,x)= gx such that: (i) 0(g, 0(h,x))=O(gh,x) or (gh)x = g(hx) for all g,h a G, x e X and (ii) ife e G is the identity element in G, and x e X, 0(e,x) = ex =x. Sometimes G is called a left transformation group. Note,a right action would be defined by a map r : X x G - X withthe appropriate properties.

Given an action 0 : G x X - X and a set S c X, then GS =(g a GI O(g,y)= y for all y e S). If GX = (e), i.e., only

the identity element leaves X fixed, we say the action iseffective. If x, y e X, there is some g c G - O(g,x) = y,then we say the action is transitive. A trivial example of atransitive action which is not effective, is when G isnontrivial but x is a single point. Of course, if G = (e),then G is effective on any X.

Given an action 0 : G x X X and a subgroup H c G, thenthere is an induced action 0H : H x X - X defined by O(h,x)where h e H. If 0 is effective, so is OH, and if 0H istransitive, so is 0.

Let X be a Hausdorff space, 0 G x X - X a transitiveaction. Fix some xo a X, and let H = (g c GI O(g,xo) = x0),the isotropy group of x0, then H is a closed subgroup of G.The coset space, G/H, is defined to be the quotient space ofG by the equivalence relation g _ h if f g'1h a H. G/H isHausdorff because the mapping w : G G/H is continuous and

74

open. If G/H is compact, then the map G/H - X is ahomeomorphism. If H is a closed subgroup of G, then G actstransitively on G/H. A space with a transitive group ofoperators is called homogeneous, such as G/H for H being aclosed subgroup of G.

For example, the action of O(n), the group of nxnorthogonal matrices, on S""%. The subgroup of O(n) whichleaves a unit vector v = (1,0,..,0) fixed is isomorphic to0(n-1). The coset space is 0(n)/O(n-1) = S'-1 which is ahomogeneous space. More generally, a Stiefel manifold Vmk isdefined to be O(m)/O(k) . Thus, VV«t,m = Sm (Chevalley 1946,Steenrod 1951].

A Lie group G is a differentiable (or analytic) manifoldand also endowed with a group structure such that 0 : G X G- G defined by (g,h) - gh"1, where g,h a G, is e (oranalytic).

For example:(i) R" is a Lie group under vector addition.(ii) The manifold GL(n,R) of all nxn non-singular real

matrices is a Lie group under matrix multiplication.(iii) The product G X H of two Lie groups is itself a

Lie group with the product manifold structure and the directproduct group structure, i.e., (g1,h1) (g2,h2) = (g1g2, (hlh2) -1)where 9,.92 e G, h1,h2 a H.

(iv) The unit circle S' is a Lie group with the additionof angles.

(v) The n-torus T" (n an integer > 0) is the Lie groupwhich is the product of the Lie group S' with itself ntimes.

If f is C°, then we may be able to extend Taylor'sformula into a convergent series. If we can, then f is saidto be analytic. A standard example of a e function which isnot analytic is the following.

- I exp (-1/(1- 1X12)), lXI < 1,

0(x) = 1 0, 1 X I ? 1.

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1

Let g e G. Left translation by g and right translationby g are the diffeomorphisms 1g and r9 of G defined by l9(h)= gh, r9(h) = hg for all h e G. If U is a subset of G, wedenote 19(U) and rg(U) by gU and Ug respectively. A vectorfield X on G is called left invariant if for each g e G, Xis lg-related to itself, i.e., if 19: G - G

dl9: Gh - Ggh, the tangent space of Gdlg X = X ' 19.

Clearly, the translation map gives a prescribed way oftranslating the tangent space at one point to the tangentspace at another point on a Lie group G. We shall come backto this later after we have introduced the concept oftangent bundles and fiber bundles.

Recall that we have defined the tangent vector fields asderivatives and the bracket operation of two vector fieldsearlier, now we shall use these concepts to define the Liealgebra.

A Lie algebra g over R is a real vector space g togetherwith a bilinear bracket operator [ , ] : g x g - g such thatfor all X, Y, Z e g,

(i) the bracket is anti-commuting, i.e., [X,Y] _

-[Y,X],(ii) the Jacobian identity is satisfied, i.e.,

[[X,Y],Z] + [[Y,Z],X] + [[Z,X],Y] = 0.The importance of the concept of Lie algebra is its

intimate association with a Lie group. For instance, theconnected, simply connected Lie group are completelydetermined (up to isomorphism) by their Lie algebras. Thusthe study of these Lie groups reduces in large part to thestudy of their Lie algebras.

Examples of Lie algebra:

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(i) The vector space of all smooth vector fields on themanifold M forms a Lie algebra under the bracket operationon vector fields.

(ii) Any vector space becomes a Lie algebra if allbrackets are set equal to 0. Such a Lie algebra is calledabelian.

(iii) The vector space gl(n,R) of all nxn matrices forma Lie algebra if we set [A,B] = AB - BA where A, B egl(n,R).

(iv) R3 with the bilinear operatioon X x Y of thevector cross product is a Lie algebra.

Theorem 2.6.1 Let G be a Lie group and g its set ofleft invariant vector fields. Then,

(i) g is a real vector space, and the map a : g - Gedefined by a(X)= X(e) is an isomorphism of g with thetangent space Ge of G at the identity. Thus, dim g = dim Ge

dim G.

(ii) Left invariant vector fields are smooth.(iii) The Lie bracket of two left invariant vector

fields is itself a left invariant vector field.(iv) g forms a Lie algebra under the Lie bracket

operation on vector fields.It should be noted that the correspondence between Lie

groups and Lie algebras of the above theorem is not unique.For instance, all 1-dim Lie groups such as S' or R1 have thesame Lie algebra. Similarly, the plane R2, torus T2, andcylinder S'xR' all have the same abelian Lie algebra. It isnot difficult to see that two Lie groups that haveisomorphic Lie algebra are themselves isomorphic at least ina sufficiently small neighborhood of their identityelements. Such a notion can be formulated more precisely interms of a local Lie group in a neighborhood of e. Then thecorrespondence between such local Lie groups and Liealgebras is indeed one-to-one and onto. We have

Theorem 2.6.2: Two Lie groups are locally isomorphiciff their Lie algebras are isomorphic.

Let us now consider the subgroups and subalgebras of a

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given Lie group and its algebra.Let G be a Lie group. A subgroup H c G is a Lie subgroup

of G if the set of elements of H is a submanifold of thesmooth manifold G. Let g be a Lie algebra. h c g is a Liesubalgebra of g if (i) h is a vector subspace of g, and (ii)if X,Y a h, then [X,Y] a h.

A Lie subgroup is clearly a Lie group with the structureinherited from the bigger group. A Lie subgroup is alsoclosed. Conversely, it is known that a closed subgroup of aLie group is a Lie subgroup.

It is obvious that if g is an abelian Lie algebra, thenany vector subspace of g is a Lie subalgebra. Clearly, anyone- dimensional subspace of a Lie algebra is a Liesubalgebra.

Theorem 2.6.3 (a) Let G be a Lie group and H a Liesubgroup. Then the Lie algebra of H, h, is a subalgebra ofthe Lie algebra of G, g. (b) Let G be a Lie group with Liealgebra g. Suppose h is a Lie subalgebra of g. Then there isa Lie group H whose Lie algebra is isomorphic to h and a 1-1map of H to G.

Nonetheless, the image of H and G need not be asubmanifold of G, thus H is not necessarily a Lie subgroupof G.

Theorem 2.6.4 A closed subgroup of a Lie group is a Liesubgroup.

From the idea of the imbedding theorem, one may betempted to think that any given Lie group is isomorphic to aLie subgroup of GL(n,R) for sufficiently large n. But thisis not to be the case. Certainly, there are Lie groups withthe same Lie algebra which are mapped to some GL(n,R) by aone-to-one map (Theorem 2.5.3(b)), moreover, the image neednot be a Lie subgroup as pointed out earlier.

Don't despair, as a corollary of a famous theorem ofPeter and Weyl[1927] for representations of Lie groups, oneis assured that a compact Lie group is isomorphic to asubgroup of some GL(n,R) for sufficiently large n. By usinga corollary of the duality theorem of Tannaka [Chevalley

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1946, Hochschild 1965] and the reduction theorem of Liegroups, one can sharpen the result by stating that a compactLie group is isomorphic to some Lie subgroup of O(k) (see

also, Chevalley 1946, 1951, Hochschild 1965].Let G be a Lie group. A continuous homomorphism a : G -

GL(n,R), for some positive integer n, is called arepresentation of G. a is trivial if the image of a isconstantly the identity matrix and a is faithful if a isone-to-one.

Roughly speaking, an application of the theorem of Peterand Weyl asserts that compact Lie groups always have atleast one faithful representation.

The theory of representation is an interesting subfieldof mathematics, for those who are interested in thissubject, the theory of characters, orthogonality relations,theorem of Peter and Weyl, can be found in [Chevalley 1951,Pontryagin 1966]. There are many important applications ofrepresentation theory in physics such as in quantummechanics, atomic spectra, solid state physics, nuclear andparticle physics [Bargmann 1970, Hamermesh 1962, Hermann1966, Am. Math. Soc. Translation Ser. 1, Vol 9, 1962]. Weshall not go into this any further.

Let us take a closer look at closed subgroups of Liegroups. As a preliminary, we shall discuss the exponentialmap which is a generalization of the power series for exwhere x is a matrix.

Let G be a Lie group, and let g be its Lie algebra. Ahomomorphism 0: R - G is called a one-parameter subgroup ofG. Let X e g, then Ad/dt - AX is a homomorphism of the Liealgebra of R into g. Since the real line issimply-connected, then there exists a unique one-parametersubgroup expx : R -. G such that d expx(Ad/dt) = AX. Thatis, t - expx(t) is the unique one-parameter subgroup of Gwhose tangent vector at 0 is X(e). The exponential map isdefined as exp: g - G by setting exp(X) = expx(1).

We shall show that the exponential map for the GL(n,R)is actually given by exponentiation of matrices.

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In a more geometric setting the exponential map can bedefined by: Let G be a Lie group and g its Lie algebra. LetX e g. Let rX be the integral curve of X starting at theidentity. The exponential map exp: g - G is the map whichassigns r (1) to X, i.e., exp x = rx(1).

There is also an exponential map from the tangent spaceto the base manifold. Let M be a e manifold, p e M, theexponential may expP : MP - M is defined as: if X e MP, thereis a unique geodesic r such that the tangent of r at 0 is X.Then expPX = t(1). In a local sense, a geodesic is a curveof shortest distance between two points on a manifold. Wewill not have time and space to discuss this interesting andimportant topic in differential geometry. It is also afoundation of general relativity. Interested readers shouldconsult, for instance [Burke 1985, Hicks 1971, Willmore1959].

MP

I exPP

Theorem 2.6.5 Let X e g the Lie algebra of the Liegroup G. Then:

(i) exp(aX) = expx(a), for all a c R.(ii) exp(al + a2)X = (exp a1X) (exp a2X), for all a,,a2 e

R.(iii) exp(-aX) _ (exp aX)-1 for all a e R.(iv) exp : g y G is C° and d exp: go -+ G. is the

identity map, so exp gives a diffeomorphism of aneighborhood of 0 in g onto a neighborhood of e in G.

(v) is the unique integral curve of X whichtakes the value a at 0. As a consequence, left invariant

80

vector fields are always complete.(vi) The one-parameter group of diffeomorphism X.

associated with the left invariant vector field X is given

by X. = rezp (a)'Theorem 2.6.6 Let 0: H - G be a homomorphism, then the

following diagram commutes:

H - 0 - G

exp t t exp

h - dm - g

In the case of groups of matrices, we shall see that theexponential map exp : gl(n,R) - GL(n,R) for the generallinear group is given by exponentiation of matrices. Let I(rather than e) denote the identity matrix in GL(n,R), let

eA = I + A + A2/2! + ... + Ak/k! + .... (2.5-1)for A e gl(n,R). We want to show that the right hand side ofthe series converges. In fact, the series convergesuniformly for A in a bounded region of gl(n,R). For a givenbounded region (1 of gl(n,R), there is a µ > 0 such that forany matrix A in i1, IA;j <- µ for each element (or component)of A. It follows by induction that I (Ak) ,jI < n(k-1)µk. Then bythe Weierstrass M-test, each of the series of components

Ek=p (Ak);j/k! , (1 <- i, j <- n) (2.5-2)converges uniformly for A in 1. Thus the series in (2.5.1)converges uniformly for A in 1. Let Sk(A) be the k-thpartial sum of the series (2.5.1), i.e.,

Sk(A)= I + A + A2/2! +....+ Ak/k! (2.5-3)and let B,C e gl(n,R). Since C - BC is a continuous map ofgl(n,R) into itself, it follows that

B (limk,m Sk(A)) = limk. (BSk(A)) .In particular, if B is non-singular, then

B (limk Sk(A) ) B-1 = limk,o BSk(A) B'1.Because BAkB-1 = (BAB")k, it then follows that BeAB_1= eBAB.

From this, it is easy to show that if the eigenvalues ofA are 11, ... An, then the eigenvalues of eA are el, , .. , ex^ ea'is an eigenvalue of eA follows easily if the upper left

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corner of A is 11 and the rest of the first column are zero.Likewise for the others. The determinant of eA is e raisedto the trace of A, by observing that the determinant is theproduct of the eigenvalues and the trace is the sum of theeigenvalues. eA is always non-singular. If A and B commute,then eA+B = eAeB. Consider the map OA : t to of R intoGL(n,R). It is smooth because etA can be represented by apower series in t with infinite radii of convergence. Itstangent vector at 0 is A (simply differentiate the powerseries term by term), and this map is a homomorphism,because OA(t) = etA,

OA(tl + t2) = e(t,+tl)A =et'A+t2A = OA(t1)OA(t2) , etc.

Thus OA : t -+ etA is the unique one-parameter subgroup ofGL(n,R) whose tangent vector at 0 is A. So the exponentialmap exp: gl(n,R) y Gl(n,R) is given by exponentiation ofmatrices: exp(A) _ OA(1) = eA where A e gl(n,R). Thus thisis the historical justification for the terminology.

It should be pointed out that exp : g -+ G need not beonto even when G is connected. An example for gl(2,R) hasbeen constructed to demonstrate this [Kahn 1980, p.272].

2.7 Fiber bundlesFiber bundles provide a convenient framework for

discussing the concepts of relativity, invariance, gaugetransformations and group representations. Fiber bundleswere originally introduced in order to formulate and solveglobal topological problems. Nonetheless, the notion of afiber bundle is also very appropriate for local problems ofdifferential geometry. The concept of induced representationof Lie groups, which is important for particle physics andfield theory, can be very easily represented and explainedby using fiber bundle language. The canonical formalism ofclassical mechanics assumes the cotangent bundle of theconfiguration space as the underlying manifold. Classicalelectrodynamics may be interpreted as the theory of aninfinitesimal connection in a principal fiber bundle with

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structure group U(1). A similar interpretation can be givento the Yang-Mills fields, and in general, to all fieldsresulting from "gauge transformations".

Bundles are a generalization of the concept of Cartesianproduct. An example from the evolution of the framework ofphysics can clarify as well as provide the need for such ageneralization.

In Aristotelian physics, space and time are consideredto be absolute, i.e., every physical event being defined byan instant of time and a location in space. This isequivalent to say that space-time M is a Cartesian productof T (time) and S (space). In Galilean (or Newtonian)physics, the time remains absolute, but space is not. Thiscan be described by saying that there is a projection map it: M - T which to any event p e M associates thecorresponding instant of time t = ir(p). T is called the basespace. The inverse image of t, 1r-'(t), is called a fiber.Each fiber is isomorphic to R3, is therefore called atypical fiber. This is the usual spatial part of thespace-time, where the Galilean transformations (translationsand rotations) map a point on a typical fiber to anotherpoint on the same fiber. And the Galilean invariantquantities make sense. In special relativity, neither thespace nor the time is absolute. Nonetheless, the bundle oflinear frames of the space-time is a product bundle. But ingeneral relativity, it is so complicated that only theprincipal bundle of the bundle of frames of the space-timeis a product bundle.

The word "global" came as a specific mathematical notionwhen differential geometry broke away from its historicalconfinement as a "local" discipline. The contrast betweenthe local and global aspects of differential geometry isperhaps best illustrated by the analytic tools that arebeing used for studying local and global aspects of surfacesor manifolds in general.

The analytic machinery of local differential geometrymanifests itself as a collection of differential expressions

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relating to local geometric properties in the neighborhoodof points, lines, surfaces or manifolds in general. Inglobal differential geometry, by contrast, one is interestedin the integrals of those local differential relations andquestions arise whether those integrals exist and whetherthey are unique.

For lower dimensional spaces (n <_ 3) there is a wellbalanced relationship between local and global results indifferential geometry (e.g., Gauss-Bonnet theorem). However,for higher dimensional space, it is much more difficult togeneralize the local results to global ones.

In Newtonian or Poincare-invariant theories thespace-time is considered to be given a priori, and thephysical dynamics are defined on this background. In generalrelativity, the topological and geometric structure ofspace-time is to be established as part of the dynamics.Global structures place restrictions on the class ofdifferentiable manifolds suitable for space-times. Globalstructures are presented in mathematical form but theirimposition is usually based upon physical intuition and onobservations. Once they are imposed, however, the resultantclass of admissible spaces further clarifies thesignificance of any given global structure and could evenlead to its rejection.

Before getting into formal definitions and majorresults, let us describe the concepts we shall encounter inan intuitive way. We have defined the tangent space of amanifold M at a point p e M, i.e., MP. The tangent bundle ofM, denoted by TM, is defined as the union of all tangentspaces of M, i.e., TM = U MP for all p c M. A vector bundleof a manifold M is a family of vector spaces V each attachedto a point of M such that locally the vector bundle ishomeomorphic to U x V where U is a neighborhood of p e M.The principal bundle of M with structural group G is locallyhomeomorphic to the attachment to each point in M adifferent copy of G, i.e., the bundle is locallyhomeomorphic to U x G where U is a neighborhood of p e M. We

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say a bundle is trivial, we mean that instead of the bundlespace "is locally homeomorphic to" by "is homeomorphic to".

In the following, we shall discuss fiber bundles on asmooth manifold with smooth mappings. These notions are veryuseful for our discussion as well as for differentialgeometry. But it should be pointed out that fiber bundlescan be defined on topological manifolds only with all mapscontinuous [Steenrod 1951].

For example, let B, X and F are topological manifoldsand let a continuous map it : B -+ X of B onto X called theprojection and B be the bundle space and X the base space.The B is a fiber bundle over X with fiber F, and projectionit if for every p e X, r-1(p) is homeomorphic to F, and thereexists a neighborhood U of p and a homeomorphism Ou : Bu

UxF where Bu = 7r-1(U) such that the following diagram

commutesOu

U x F1 rru

7 * Uwhere ru is an obvious projection.

A C° principal fiber bundle is a set (B, M, G) where B,M are C° manifolds, G a Lie group

(i) G acts freely (and C) to the right on B, i.e., Bx G -, B defined by (b,g) - bg = R9b e B where b e B, g e G.

(ii) M is the quotient space of B by equivalence underG and the projection r : B - M is C, so for p e M, G issimply transitive on r-1(p) where 7r-'(p) is a fiber over p eM

(iii) B is locally trivial, i.e., for any p e M, thereis a neighborhood U of p and e map Fu : T" (U) - G such thatFu commutes with R. for every g e G and the map of r"' (U) -+

UxG given by b -+ (7r(b),Fu(b)) is a diffeomorphism. Here B iscalled the bundle space, M the base space, and G thestructural group. Note, the fibers r'1(p) are diffeomorphicto G in a special way, i.e., via the map b : G - 7r-'(7r(p)) c

B defined by b(g)= R9b. This definition can be graphically

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illustrated by the following figure (Fig.2.7.1).

B

M

G

Fig. 2.7.1

As before, G a Lie group, M a manifold, then B = M X Gprovided with the right action of G on itself in the secondfactor, i.e., (p,g)h = (p,gh), is the bundle space of aprincipal bundle which is called the trivial bundle. Across-section of a bundle is a continuous map s : M B suchthat 7r- s = id on M (i.e., w(s(p)) = p for all

Let us discuss some examples of bundles:p(i)

e M).From the

above pictorial representation of a fiber bundle, we can letthe base space be S1, the fiber being a line segment [0,1],and the bundle space being S' x [0,1]. Clearly, this bundleis a trivial bundle. (ii) For a flexible rectangular sheet,we hold one side of the sheet fixed and twist the other sidethrough 180° and then identify (glue together) these twoopposing sides. Then the resulting two-dimensional space isa bundle space, and is the well-known Mobius band. The basespace is the circle (S), the fiber is a line segment [0,1],and the group action is the "twist" (Z2). Aside from the"seam", it is clear that the inverse of the projection mapof a neighborhood of a point on the circle is a smallrectangle (i.e., locally a product bundle). But clearly,

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this fiber bundle is not a trivial bundle.Theorem 2.7.1 A bundle is isomorphic (or homeomorphic)

to a trivial bundle if f there is a e (or continuous)cross-section.

The above theorem means that a trivial bundle has globalcross-sections. The two simple examples illustrate exactlythis point. Both bundles have local cross-sections, but onlythe first one has a global cross-section. By the way, theglobal cross- section, if it exists, is homeomorphic to thebase space.

Let Mm be a C' manifold, B be the set of (m+l)-tuples(p, el , .. , em) , where p e M, and (el , .. , em) is a basis of MP,and let it : B - M be defined by lr (p, ei , .. , em) = p. Let g eGL(m,R) = G, then GL(m,R) acts to the right on B byRg(p,e1,..,em) = (p, E g,1e,,.., E g1me1) where g is viewed asa matrix g =(g). Let (xi, .. , xm) be a coordinate systemdefined in a neighborhood U of p, Then FU is defined byletting Fu(p',f1,..,fm) = (dx1(fi)) = (g1j) a GL(m,R), wherep' a U. Using the e structure given to B by the localproduct representation (7r,Fu), we see that B is the bundlespace of a principal bundle, called the bundle of bases ofM, B(M).

For a classical notion of fiber bundles or tangentbundles one can find in Anslander and MacKenzie [1963]. Itis also convenient at times to view B as the set ofnonsingular linear transformations of Rm into the tangentspaces of M, i.e., we identify b = (p,ei,..,em) with the mapb : (ri , .. , rm) -+ Er,e,. If this is done, it is natural toconsider GL(m,R) as the nonsingular linear transformationsof Rm, where bg(rl,..,rm) = E. jr1gj1ej = Ei(Eirig1j)e1 _b(g(rl,..,rm)). In other words, bg (as a map) = b (as amap) g.

If G is a Lie group, H a closed subgroup, then there isa principal bundle with base space G/H (left cosets), bundlespace G, and the structure group H such that ir: G - G/H isthe projection and right action is given by (g,h) - gh. Thusa homogeneous space is an example of a principal fiber

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bundle. [Helgason 1962, Steenrod 1951].Let (B,M,G) be a principal bundle and let F be a

manifold on which G acts to the left. The fiber bundleassociated to (B.M.G) with fiber F is defined as follows:Let P'= B x F, andconsider the right action of G on P' by(b,f)g = (bg,g"1f) where b e B, f e F, g e G. Let P = P'/G,the quotient space under equivalence by G, then P is thebundle space of the associated bundle. The projection a': P- M is defined by 7r'((b,f)G) = ,r(b). For p e M, we take aneighborhood U of p as in (iii) of the definition of aprincipal bundle, with FU: ,r-'(U) - G. Likewise we have Fe':

,r' (U) - F by Fu' ((b, f) G) = Fu (b) f so that (7r')-'(U) ishomeomorphic to U x F, and define P as a manifold byrequiring these homeomorphisms to be diffeomorphisms. Thusir' and the projection 0: P'- P are C.

First of all, (B,M,G) as above, let G act on itself byleft translation, then (B,M,G) is the bundle associated toitself with fiber G.

Let us look at a tangent bundle as a bundle associatedto the bundle of basis B(M) with fiber Rm. Since GL(m,R) isthe group of nonsingular linear transformations of RIO, andhence act on Rm to the left. The bundle space of theassociated bundle with fiber Rm is denoted by TM and it iscalled the tangent bundle of M. TM can be identified withthe space of all pairs (p,t) where p e M, t e MP as follows:

((pre,,..,em),(r1,..,r,))GL(m,R) - (p, E r;e;).Hence the fiber of TM above p e M may be viewed as thelinear space of tangents at p, i.e., MP, and TM as the unionof all the tangent spaces together with a manifoldstructure. Moreover, the coordinates of TM can easily beadopted by letting U be a coordinate neighborhood in M withcoordinates xl, .. , xm. Define coordinates Y11-1Y2, on(,r') "' (U) in such a way that if (p, t) a (7r') "'(U) , thenyi(p,t) = x,(p), y,1(p,t) = dx,(t) where i = 1,..,m. Clearly,a C' vector field may be regarded as a cross- section of Yr'.For more on tangent bundles see [Bishop and Crittenden 1964,Yano and Ishihara 1973, and other modern differential

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geometry books]. It is interesting to note the following

theorem.Theorem 2.7.2 TM is orientable even if M is not.When R'", in the tangent bundle of M, is replaced by a

vector space constructed from Rm via multilinear algebra,i.e., the tensor product of Rm and its dual with variousmultiplicities, we get a tensor bundle. A cross-sectionwhich is C° on an open set is called a CO tensor field, andthe type is given according to the number of times Rm andits dual occur. The structural group of a tensor bundle is,of course,. GL(m,R) and it acts on each factor of the tensorproduct independently. GL(m,R) acts on R'" as with thetangent bundle, and it acts on the dual via the transpose ofthe inverse, i.e., if v e Rm* = the dual of R10, x e Rm, g eGL(m,R), then gv(x) = v(g"1x). Clearly, TM is a special caseof a tensor bundle; this is similar to that a tangent vectoris a special tensor, a contravariant tensor of rank one.

Vector bundles, which we shall encounter later, in whichthe fiber is a vector space and they are frequently definedwith no explicit mention made of the structural group(although often it is a subgroup of the general linear groupof the vector space). It is usually defined as the union ofvector spaces, all of the same dimension, each associated toan element of the base space and defining the manifoldstructure via smooth, linearly independent cross-sectionsover a covering system of coordinate neighborhoods. In fact,we did this for TM, a special case of a vector bundle.

The quotient space bundle of an imbedding, sometimesconsidered as a normal bundle for Riemannian manifold, maybe defined as follows: Let is N - M be the imbedding of thesubmanifold N in M. The fiber over q c N is the quotientspace M,(q)/di (Nq) , and the bundle space is the union of thesefibers, so the bundle space can be considered as thecollection of pairs (q,t+di(Nq)), where t e Mj(q).

The Whitney (or direct) sum of two bundles E =(B,M,G,,r,F) and = (B',M,G',Tr',F') over the same base spaceM is the bundle a r whose fiber over x e M is F a F'. If

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0, are charts for E, C over U respectively, a chart p fort e Z over U is px = 0x a *x : F + F' -+ Rm a R. Thus dim(EC) = dim B + dim B'.

Let two bundles and C over the same base space M withB c B', then E is a sub-bundle of C if each fiber F is asub-vector-space of the corresponding F'.

Lemma 2.7.3 Let E and C be sub-bundle of n such thateach vector space F(n), fiber of n, is equal to the directsum of the subspaces F amd F'. Then n is isomorphic to theWhitney sum E e C.

Then the question arises, given a sub-bundle E c n doesthere exist a complementary sub-bundle so that n splits as aWhitney sum? If n is provided with a Euclidean metric(provided the base space is paracompact) then such acomplementary summand can be constructed by letting F(E1) bethe subspace of F(n) consisting of all vectors v such that

0 for all o e F(E). Let B(E1) c B(n) be the union ofall F(E1). One can show that B(t1) is the total space of asub-bundle t1 c n. Moreover, n is isomorphic to the Whitneysum E ® r1 [Milnor and Stasheff 1974]. Here E1 is called theorthogonal complement of E in n.

Suppose N c M are smooth manifolds and M is providedwith a Riemannian matric. Then the tangent bundle TN is asub-bundle of the restriction TMIN. Then the orthogonalcomplement TN1 c TMIN is the normal bundle of N in M, i.e.,

1(N)= ( ( q , t ) a TMIt a Mq for some q e N and t i Nq). Wewould like to mention that the notion of a normal bundle isnot only useful in differential geometry (such as indiscussing geodesics and completeness of the Riemannianmanifold) but also very useful in algebraic and differentialtopology (such as using the Whitney duality theorem torelate the immersibility of a n-dim manifold in R"+k).Interested readers may want to consult the following books:Steenrod [1951], Milnor and Stasheff [1974]. Now let us getback to some properties of tangent bundles.

Theorem 2.7.4 [Steenrod 1951] The tangent bundle to adifferentiable manifold admits a nonzero cross-section and

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is equivalent to the existence of a nowhere zero vectorfield on M iff the Euler characteristic of M is zero.

As we have defined earlier, the Euler characteristic ofa manifold M is defined in Section 2.5. For example, we havepointed out that (i) for a noncompact manifold, its Eulercharacteristic vanishes; (ii) for compact manifolds, onlyodd dimensional ones have a vanishing Euler characteristic.Thus, only odd dimensional spheres have a nowhere zerovector field. One can convince oneself that there does notexist a nowhere zero tangent vector on 2-dim sphere. Milnor(1965] gives an interesting and illuminating view on this.

When TN" is homeomorphic to Mm X Rm, it is a trivialbundle. In such a case, one says that the manifold has atrivial tangent bundle, or the manifold is parallelizable.

Although all odd-dimensional spheres have a nowhere zerovector field, but it is a deep result (Bott and Milnor 1958,Adams 1962] that only S1, S3, and S7 have trivial tangentbundle (i.e., they are the only parallelizable n-spheres).

Normally, some heavy machinery in algebraic topologysuch as characteristic classes [e.g., Milnor and Stasheff1974, Steenrod 1951, Husemoller 1975] and obstructions[e.g., Milnor and Stasheff 1974, Steenrod 1951] are neededto prove the following theorem. We shall relate thehistorical origin of the term parallelizable manifold with anon-zero vector field.

Let us assume M is parallelizable, i.e., TM = M x Rm.Thus M has m linearly independent tangent vector fieldst1(p) = (p,(0,..,l,..,0)) with 1 in the ith place, for anyp e M. In other words, at any point the m vectors t,(p) area basis for n-'(p) c TM. Thus, a nonzero vector v e r"1(p)can be expressed by v = E ait1(p), then one can transport itparallel to itself over the entire manifold to obtain anowhere zero vector field by setting v(p) = E a,t,(p) forall p e M. This is the global notion of parallel transportin such a manifold.

From the above demonstration, it is almost trivial thatif M is parallelizable, it has a global C° base field.

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Theorem 2.7.5 A manifold W' is said to beparallelizable, (i.e., has a trivial tangent bundle TM = M xRm) iff M admits a global C° base vector field [e.g., Milnorand Stasheff 1974].

Theorem 2.7.6 (generalization of a theorem due toCartan) Any connected Lie group G is topologically a productspace H X E where H is a compact subgroup of G and E is aEuclidean space.

E.g., In physics, the proper Lorentz group Lo, orSO(3,1), SO(3) x R3, and its universal covering groupSL(2,C) Z S3 X R3.

If (B,M,G) is a principal bundle, H a subgroup of G,then G is reducible to H iff there exists a principal bundle(B',M,H) which admits a bundle map f: (B',M,H) - (B,M,G)such that fm is the identity map on M, fB is one-to-one, andfG is the inclusion map H `- G.

Theorem 2.7.7 [Steenrod 1951] If (B,M,G) is a principalbundle, H a maximal compact subgroup of G, then G can bereduced to a bundle with structure group H.

Corollary 2.7.8 Every principal bundle with GL(m,R) asthe structure group, e.g., bundle of bases B(M), can bereduced to a bundle with the structural group being theorthogonal group O(m).

The reduced bundle with O(m) as structural group iscalled the bundle of orthonormal bases and denoted by O(M).

Many modern differential geometry books have at leastone or two chapters covering fiber bundles. For instance,Bishop and Crittenden [1964], Helgason [1962], Kobayashi andNomizu [1963]. For specific details in tangent and cotangentbundles, see Yano and Ishihara [1973]. For more advancedreaders, the classic by Steenrod [1951] is highlyrecommended. For a more modern and broader treatment,Husemoller [1975] is also recommended.

2.8 Differential forms and exterior algebraTensor analysis is part of the usual mathematical

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repertoire of a physicist or engineer. Differential formsare special types of tensors. Yet, its utility andconceptual implications are far beyond the capabilities oftensors. Not only does it provide more compact formulationsof electrodynamics, Hamiltonian mechanics, etc. and simplermathematical manipulations, but it also provides topologicalimplications. In this section, we shall briefly define anddiscuss some properties of differential forms and exterioralgebra, and illustrate its power. It is very tempting tobriefly discuss de Rham cohomology theorem. Once again, thereader is urged to consult those differential geometry bookswe have just mentioned earlier for further details.

For p e M, the dual vector space MPG of MP is called thecotangent space (or the space of covectors at p). Anassignment of a covector at each p is called a one-form. If(ul,..,u") is a local coordinate system in a neighborhood ofp, then du',..,du' form a basis for MP*, and they are thedual basis of the basis a/au1, ..., a/au" of MP. So in acoordinate neighborhood, a 1- form can be written as a = E1f,du'. Clearly, a is C°, if f,'s are. Note, one-form can alsobe defined as an F(M) linear mapping of the F(M)-module X(M)into F(M). That is,

(a(x))P <ar, XP> , where X e X(M), p e M.

The exterior product is defined by A A B = (A x B)B,here a denotes that it is antisymmetrized, where A and B areskew-symmetric, covariant tensors. It has the followingproperties:

(a) associativity: (A A B) A C A A (B A C),(b) distributivity: (A + B) A C A A C + B A C,(c) anticommutativity: If A is of degree p, and B is of

degree q, then A A B = (-1)a'B A A.Of course, together with addition and scalar multiplicationoperations they form the algebra.

An r-form can be defined as a skew-symmetric r-linearmapping over F(M) of X(M)x...xX(M) (r-times) into F(M). Ifal,..,ar are 1-forms, Xi,..,Xr a X(M), then

(at A a2 A ... A ar) (X,,. . , Xr)

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= E(i ... i) sign(ij, , ir) (ail (Xi5) ) ... (aip(Xiq) ) .

For instance, if r = 2, a, p E AB(M), X, Y E X(M) then

(aA(3) (X, Y) =a(X) (X)a (Y) a (Y)

In general, if a is a p-form, Q is a q-form on M, andX1,..,Xp+q E X(M), then

(a A Q) (X1,..,Xp+q)Eci ...i )

sign (i,..i q) a(Xi Q(X1 ,..Xj ).We shall denote AP(M*) to be the collection of all p-formson M. When there is no danger of ambiguity, we usually willdenote it by AP. We shall now introduce a derivative calledan exterior derivative.

Let M be C. The exterior derivative is a map: d: A - Awith the following properties:

(i) For q = 0, f is a e function, then df is a1-form;

(ii) d is a linear map such that d(Aq) c Aq`1;(iii) For a E Aq, Q E AS

d(a A Q) = da A /3 + (-1)qa A dQ;(iv) d(df) = 0 and it follows that d(da) = 0 for any a.

Any form a is said to be closed if da = 0; if a is a p-formand there is a (p-1)-form Q such that a = dQ, then we call aan exact p-form. From (iv) above, clearly every exact p-formis also closed. Let us denote AeP be the space of exactp-forms, A,P be the space of closed p-forms. Clearly, AeP c

AcP c AP.DP(M) = ASP/AeP is called the p-dimensional de Rham

cohomology group of M obtained by using differential forms.Clearly DP is an abelian group because AeP and AcP arethemselves abelian.

Theorem 2.8.1 (de Rham) For a paracompact and T2differentiable manifold M, DP(M) = HP(M,R).

Remark: Thus DP is a topological invariant. It alsofollows immediately that from properties (iii) and (iv)

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closed form A closed form = closed form, closed form A exactform = exact.

Another way to define or to determine whether a manifoldis orientalble or not is the following useful theorem.

Theorem 2.8.2 A paracompact, T2, C° manifold M of dim mis orientable iff M admits a continuous, nonvashing globallydefined n-form (for instance, a volume element).

In physics, a space-time M is a 4-dim, connected,paracompact, C°-manifold with Lorentz signatures. Theorientability of a physical "space-time" is concerning theconsistent assignment of an "arrow of time" and of"handedness" throughout the "space-time".

The strong principle of equivalence [Dicke 1959, 1962]asserts that local experiments should be the same throughoutthe space-time. From the strong principle of equivalence,some local experiments in particle physics, C, P ((3-decay),and CP(KL, KS decays) noninvariance and T noninvariance, andCPT invariance. We can conclude that the space-time must beorientable.

Remark: Assuming space-time being orientable is alsoconvenient; otherwise we can always find its universalcovering space which is orientable.

Now we shall introduce a linear mapping which is also anantiderivation.

Let X eX(M)of a differentiable manifold M. A linearmapping (inner product) i(X) : A - A such that

(i) i(X) : AP - AP', p 1; and i(X) (f) = 0;(ii) i(X) is an antiderivation, i.e., if 0 e AP, 0 e

Aq, then i(X)(0 A 0) = (i(X)6) AO + (-l)P6 A (i(X)O);(iii) if 0 e A', i(X)0 = 0(x).Lie derivative is a geometric procedure which is very

useful in finding symmetries, such as Killing vector field£Xg = 0. We shall define and state the properties of Liederivatives.

£xY ° [X,Y], also called the "dragging" of Y along X(more appropriately, along the integral curve of X), asindicated in the following diagram.

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(O .Lsm)

( {lsm )(M)

here (µs), (At) are one parameter groups of X & Y.If £xY = [X,Y] = 0 then (0-t N'-s et µ's m) = m.

Proposition 2.8.3 Let X, Y, Yo, ..., YP a X(M), f eF(M), a E AP(M), then:

(i) £xf = X(f)(ii) £x = i(X) d + d i(X)(iii) £x commutes with d(iv) (Ca) (Y1,..,YP)

,+ EP _1 a(Yj,..,Y;-tI(v) d a(Y,,..,YP) = EP p (-1)1Y1 a(Y.,..,Yi,...,YP)

+ Ej,j (-1)'i, a([Y;.Yj],Y0,..,Yi .... Yj,..YP).

Symplectic manifoldsSymplectic manifold is a very natural framework in which

to discuss classical mechanics, in particular, theHamiltonian systems. Here we shall very briefly define anddiscuss symplectic manifolds and Hamiltonian systems.

Definition 2.8.4 A volume on n-dim M is an n-form fl e

An(M) such that fl(p) + 0 for all p e M. M is calledorientable iff there is a volume on M.

Definition 2.8.5 Let a e A2(M), a is non-degenerate iffa(el,e2) = 0 for all e2 a X(M), thus e, = 0.

Proposition 2.8.6 a is nondegenerate on M iff M iseven-dim (say 2n) and a" = a A a A ... A a is a volume on

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M. Thus if a is nondegenerate, then M is orientable.Definition 2.8.7 A svmplectic form (structure) on an

even- dim M is a non-degenerate, closed 2-form a on M. Asvmplectic manifold (M, a) is an even-dim M together with asymplectic form a.

Theorem 2.8.8: (Darboux) Suppose a is a non-degenerate2-form on 2n-dim M. Then da = 0 iff there is a chart (U, 0)at each p e M such that 0(p) = 0 and with 0(u) _ (x1(u),

x"(u), y1(u), ..., y"(u)), we have alU = E"dx' A dy'.Remark: The charts guaranteed by the theorem are called

symplectic charts and x', y' are canonical coordinates.Thus, in a symplectic chart a = E dx' A dy' and fl., = dxl A

A dx" A dy' A ... A dy" is the volume element.Remark: In most mechanics problems, the phase space is

T*M and the coordinates are (q1,..., q", p1,..., p") then thecanonical form on T*M is

6 = En i=1 p;dq' and a = E" ;=i dq' A dp'.Proposition 2.8.9 T*M of any manifold is orientable.Theorem 2.8.10 Lie derivative of X, restricted to A(M*)

is a derivation, i.e., (£,,: AP AP), £x i(X) d + d- i(X) .Example: In electrodynamics, f e A2 (M) and in the

symplectic chart, f = 1/2 F." dx" A dx". The Maxwellequations can be written as: df = 0, in terms of symplecticcoordinate chart, it can be written as F,,,, + F + F =µv,a va,µ aµ,v

0. In the usual vector notation, it is equivalent to thefollowing set: div B = 0 and curl E = - dB/dt. Now definethe star operator * : AP - A"-P, where dim M = n and theHodge's operator 6 = d *, then the other set of Maxwellequations can be represented by: 6f = d*f = 47r *J. It isequivalent to: div E = 47rq and curl B = dE/dt + 47rJ.For source-free, then Fµ"," = 0, i.e., f is a harmonic 2-form(f is closed and coclosed). The existence of vectorpotential is equivalent to the non-existence of a magneticmonopole.

Definition 2.8.11 (M, a) and (N, /3) be symplectic. A C°map F : M - N is symplectic (or canonical transformation)iff F, /3 = a where F is the pullback of forms, i.e., F,:

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Ak (N) - Ak (M) .

Theorem 2.8.12 If F: M - N is symplectic, then F isvolume preserving, and F is a local diffeomorphism.

Definition 2.8.13 (M, a) is symplectic. f, g e F(M).Xf, X9 a X(M). The Poisson bracket of f and g is thefunction (f,g) = -i f ixI a.

Proposition 2.8.14 As above:

(i) (f,g) = - £xf g = £x3 f,(ii) d(f,g) =

(

df,dg)

(iii) X(f,9) -[Xf,X9](iv) The real vector space F(M) and the composition

form a Lie algebra.(v) (U, 0) be a symplectic chart, then the Poisson

bracket has the usual form,(f,g) = E" _, [ (af/ax') (ag/ay') - ((3f/aY') ((3g/ax') ]

See for instance, Goldstein [1950].Hamiltonian systemsDefinition 2.8.15 M be a manifold, X e X(M). Let a c

Ak(M). We call a an invariant k-form of X iff £x a = 0.Definition 2.8.16 Let (M, a) be symplectic and X e

X(M). We say that X is locally Hamiltonian iff a is aninvariant 2-form of X, i.e., £x a = 0. The set of locallyHamiltonian vector fields on M is denoted by XLH (M).

Proposition 2.8.17 (i) Let X E XLH (M) on 2n-dim (M,a). Then a, a,, ..., a" are invariant forms of X, (ii)

XLH (M) is a Lie subalgebra of X(M).Proposition 2.8.18 The following are equivalent:(i) X E XL H (M) ,(ii) ix a is closed,(iii) U a neighborhood of p e M and H e F(U) such that

XlU = XH.Definition 2.8.19 Let X e XLH (M). A function H as

above is called a local Hamiltonian of X.Proposition 2.8.20 H a local Hamiltonian of X. Then H

is constant along the integral curves of X in U (i.e., H isa constant of motion, or "energy" is conserved). This isbecause £xH = £x H = {H, H) = 0.

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Proposition 2.8.21 Let X c XLH (M), H e F(U) be a localHamiltonian on a symplectic manifold (M, a). Let V c U, (V,

0) be a symplectic chart with 0(V) c Rzn, 0(v) =(q'(v),..,gn(v), p,(v)..pn(v)). Then a curve c(t) on V is anintegral curve of X iff

(dq'/dt)c(t) = (aH/aP1)c(t), i = 1,...,nand (dp,/dt)c(t) = -(aH/aq')c(t), i = 1,...,nwhere q'(t)= q'(c(t)), pi(t)= p1(c(t)).

Proposition 2.8.22 X, H, (M, a) as above. Then for f eF(U), we have £Xf = {f,H) on U.

Before ending this subsection, for our future use aswell as for general interest, we are going to discussbriefly the Kolmogorov-Arnold-Moser theorem. This theoremwas originally proved for Hamiltonian of two degrees offreedom, and it was later extended to n degrees of freedomsystems with (2n - 2)- dimensional Poincare maps. Forsimplicity, here we shall only state the theorem for twodegrees of freedom. Let us consider the Hamiltonian H` is asmall perturbation of an integrable Hamiltonian H°. Forsimplicity, let us assume that the perturbed Hamiltonian hasthe following form:

H'(q,P,0,I) = F(q,P) + G(I) + eH1(q,p,0,1),where H' is 2w periodic in 0 and the unperturbed systemH°(q,p,9) = F(q,p) + G(I) decouples directly into twoindependent systems with integrals F and G. Furthermore, weassume nondegeneracy, i.e., dG/dI + 0. This implies that fore small, H' is invertible and can be solved for I. Bytransforming (q,p) to a second set of action angle variablesJ and 0, and the unperturbed system H° becomes

dJ/dt = 0, dO/dt = 27rT,and it associates with the Poincare map P0. And theperturbed system modifies the Poincare map to P,. The KAMtheorem asserts that for sufficiently small e, most of theclosed curves J = constant of P° are preseved for P,. Inother words, the Poincare map under consideration is an areapreserving diffeomorphism.

Theorem (KAM) If an unperturbed Hamiltonian system is

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nondegenerate and a is sufficiently small, then theperturbed map P, has a set of invariant closed curvespositive Lebesgue measure µ(e) close to the original

of

set J =

JO; moreover, µ(e)/µ(J) - 1 as e - 0. The survivinginvariant closed curves are filled with dense irrationalorbits.

Instead of these technical versions, we can put the KAMtheorem in the following words:

Theorem (KAM) If an uperturbed system is nondegenerate,then for sufficiently small conservative Hamiltonianperturbations, most non-resonant invariant tori do notvanish, but are only slightly deformed, so that in the phasespace of the perturbed system, there are invariant toridensely filled with phase curves winding around themconditionally periodically too, with the number ofindependent frequencies equal to the number of degrees offreedom. These invariant tori form a majority in the sensethat the measure of the complement of their union is smallwhen the perturbation is small.

As we shall see, the major part of these lectures willbe concerned with dissipative dynamical systems and with thestructure of the nonwandering and attracting sets occuringin such systems. Nonetheless, it is often very useful toconsider such systems as perturbations of Hamiltoniansystems, this is because the existence of energy integralsor other constants of motion enables us to obtain globalinformation on the structure of solutions. Thus, the KAMtheorem will be utilized many times in later discussionseither explicitly or implicitly. Recently, transport inthree-dimensional nonintegrable Hamiltonian flows is studiedand the destruction of KAM barriers in the presence ofstochastic perturbations are described [Gyorgyi and Tishby1989]. They further extended the action principle toHamiltonians with small noise, which provided a framework todetermine universal scaling of characteristic times as afunction of the noise.

For the example of a Hamiltonian system with two degrees

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of freedom, e.g., two coupled nonlinear oscillators, Greene[1979] developed a method for deciding the existence of anygiven KAM surface computationally. One finds, when giventhat KAM orbits exist, that the guiding hypothesis is thatthe disapearance of a KAM surface is associated with asudden change from stability to instability of nearbyperiodic orbits. The relation between KAM surfaces andperiodic orbits has been explored extensively in this paperby the numerical computation of a particular mapping.

Lagrangian systemWe will be concerned with an alternative description of

classical mechanics on another phase space, the tangentbundle of the configuration space M of the system, TM. Weconsider a function L on TM and solutions to a certainsecond-order equation. From L we can derive an energyfunction E on TM which, when translated to T*M by means ofthe "fiber derivative" FL: TM - T*M ( usually called theLegendre transformation), the derivative of L in each fiberof TM yeilds a suitable Hamiltonian. Then the solutioncurves in T*M (i.e., solutions of Hamiltonian equations) andin TM (i.e., solutions of Lagrangian equations) willcoincide when projected to M. The following diagram will behelpful.

R

z lT*M ( TM L % R

WV

M

Definition 2.8.23 Let M be a manifold and L e F(TM).Then the map FL: TM - T*M : om - DLm(am) a L(TmM1R) = Tm*M iscalled the fiber derivative of L. Lm denotes the restrictionof L to the fiber over m e M.

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Proposition 2.8.24 FL: TM - T*M is a fiber preservingmap

It should be noted that FL is not necessarily a vectorbundle mapping.

For further detail of "geometric theory of classicalmechanics", Abraham and Marsden [1978] and Arnold [1978] arehighly recommended.

2.9 Vector bundles and tubular neighborhoodsAs we have mentioned earlier, a vector bundle can be

thought of as a family of disjoint vector spaces {Vx}XEMparameterized by the base space M. The union of these vectorspaces is a space B, and the map it : B - M, v(Vx) = x iscontinuous. Furthermore, it is locally trivial in the sensethat locally B looks like a product of B with R"; and thereare open sets U covering M and homeomorphisms v-'(U) = U xR", mapping each fiber Vx linearly onto (x)xR". A morohismfrom one vector bundle to another is a map taking fiberslinearly into fibers.

A vector bundle is similar to a manifold in that bothare building up from elementary objects glued together byspecial maps. For manifolds, the elementary objects are opensubsets of R", the gluing maps are diffeomorphisms; forvector bundles the elementary objects are local "trivialbundles" UxR", and the gluing maps are morphisms UxR" - UXR"of the form (x,y) - (x,g(x)y) where g: U - GL(n,R).

In both manifolds and vector bundles, linear maps play acentral role. Linear maps enter into manifolds in a subtleway as derivatives, the linearity in vector bundle is moreexplicit. This makes the category of vector bundles moreflexible and easier to analyze than that of manifolds. Infact, many natural constructions with vector bundles, suchas direct sum, quotients, and pullbacks, are impossible tomake for manifolds.

By introducing tubular neighborhoods, a new connectionbetween vector bundles and the topology of manifolds is

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established. If M c N is a submanifold and M has a certainneighborhood in N which looks like a normal vector bundle ofM in N. Furthermore, such neighborhoods are essentiallyunique. Consequently, the study of the kinds ofneighborhoods that M can have as a submanifold of a largermanifold is reduced to the classification of vector bundlesover M. For example, the question whether the inclusion mapM `- N can be approximated by imbedding is tantamount towhether the normal bundle of M in N has a nonvanshingsection.

Let F be a field (can be either real R, complex C orquaternion Q). A k-dimension vector bundle ^ over F is abundle (B,M,ir) together with the structure of ak-dimensional vector space over F on each fiber Tr-'(p) suchthat the local triviality condition is satisfied: For eachpoint of M ther is an open neighborhood U and an isomorphismis UxFk - it-, (U) such that the restriction pxFk 7-1(p) is avector space isomorphism for each p e U.

If F = R it is a real vector bundle, F = C a complexvector bundle, F = Q a quaternionic vector bundle. And theisomorphism is UxFk _7r-l(U) is a local coordinate chart of

n.

One can also define vector bundles as a special case ofa fiber bundle, or a special principal bundle. For instance,a real vector bundle is a fiber bundle where the fiber is areal vector space Vk and the structure group is GL(k,R).

It is helpful to introduce the notion of an exactsequence of vector bundles morphisms in just the way as anexact sequence of groups introduced earlier, i.e., a finiteor infinite sequence

....ten;-, f yn; f , n;,,y...of morphisms such that for each p e M we have Im(f;_,)P =Ker(f,)P for all i. In particular, we are interested in theshort exact sequence 0 - a f Q 9 µ - 0 where 0 denotesa 0-dim bundle over M. Such an exact sequence means that fis one-to-one, onto, and Im f = Ker g. One can show that theshort sequence is unique up to isomorphism.

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In the short exact sequence we call a the quotientbundle of the one-to-one map (monomorphism) f. Then everymonomorphism has a quotient bundle and quotient bundle isunique up to isomorphism. If a c /3 is a subbundle, then thefibers of the quotient bundle are taken to be the vectorspaces fP/aP and the quotient bundle is denoted by /3/a.

The short exact sequence 0 -, a f - (3 9 - µ 0 is saidto split if there is a monomorphism h : µ - /3 such that gh =idµ. Working through fibers, this is equivalent to theexistence of an onto map (epimorphism) k: /3 -+ a such that kf= id,. Then the Whitney sum of bundles a, µ over M can bedefined as the bundle a a A whose fiber over p e M is a

Pe

µP. If 0, r are charts for a, µ respectively over U, a chartB for a e µ over U is BP = OP e rP : aP a µP - R' 9 R" as wehave defined before.

The natural exact sequences of vector spaces0 - aP f, -, aP a µP 9P - µP -, 0

can fit together to form a split exact sequence0- a f- a® /L 9- A -+ 0.

Now let us apply this short exact sequence to the tangentbundle of a vector bundle. Let a = (B,M,,r) be a Cr+1 vectorbundle, then each fiber aP is a vector space with origin atp. Thus we can identify aP with the tangent space of aP,i.e., (aP)P. Hence a is a subbundle of TBIM, the tangentbundle of the bundle space B restricted to M, in a naturalway. Take note that TBIM is of Cr. Since M c B is asubmanifold, TM is a Cr subbundle of TBIM. Thus we have ashort exact sequence 0 -+ a -+ TB I M dr - TM -+ 0 which issplit by di: TM -, TBIM. Thus we have the following:

Theorem 2.9.1 Let a = (B,M,,r) be a Cr+1 vector bundle,0 -< r - . The short exact sequence of Cr vector bundles

0 -, a - TBIM - TM -+ 0is naturally split by di:TM -, TBIM. Thus there is a naturalCr isomorphism nQ : TBIM = a e TM. And particularly, a cTBIM is a natural subbundle.

Here we provide the natural split of vectorbundles. Recall the construction of orthogonal complements

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and the normal bundles, one can see that it is easy toestablish the existence of such constructions. In fact, wehave the following useful result for a vector bundle:

Theorem 2.9.2 Every short exact sequence of Cr vectorbundle splits, for 0 <_ r _< , provided the base space isparacompact. [Hirsch 1976].

Whitney sums and restricting a bundle to a subset of thebase space are methods to construct new vector bundles outof old. A more general construction is the induced bundlemethod. Let a =(B,M,7r) be a vector bundle. Mo be anarbitrary topological space, f: Mo M be any map, then onecan construct the induced bundle (or pullback) f*a over Mo.The bundle space Bo of f'a is the subset Bo c M xB consistingof all pairs (p,b) with f(p) = 7r(b). The projection map 7ro :

Bo - M. is defined by 7ro(p,b) = p. Thus one has the followingcommutative diagram

B0 f. I B

Vo 7r

Mo - f q Mwhere f'(p,b) = b. The vector space structure in 7ro"gy(p) is

defined by t,(p,bl) + t2(p,b2) = (p,t1b, + t2b2). Thus f'carries each vector space (f*a) P isomorphically onto thevector space (a) f(p). We leave it to the reader to show thatf*a is a fiber bundle by showing that f*a is locallytrivial.

Now if a is a smooth vector bundle and f a smooth map,then it can be shown that Bo is a smooth submanifold ofMoxB, and hence f*a is also a smooth vector bundle.

The above commutative diagram suggests the concept of abundle map. Let a and n be vector bundles, a bundle may fromn to a is a continuous map g: B(n) R21 B(a) which carrieseach vector space nP isomorphically onto one of the vectorspace aq.

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Lemma 2.9.3 If g: B(n) - B(a) is a bundle map, and ifg': M(n) - M(a) is the cooresponding map of base spaces,then n is isomorphic to the induced bundle g'*a.

Before we state the classifying theorem of vectorbundles, we must first introduce the concept of theuniversal bundle over a Grassmann manifold. Let G",k be theset of k-dim linear subspaces of R" (k-planes through theorigin). Since any element of the orthogonal group O(n)carries k-plane into a k-plane, and in fact, O(n) istransitive on G",k. If Rk is a fixed k-plane and Rn-k is itsorthogonal complement, then the subgroup of O(n) mapping Rkonto itself splits up into the direct product ofO(k)xO(n-k) of two orthogonal subgroups whereby the firstleaves

Rn-k

and the second leaves Rk pointwise fixedrespectively. Thus we may identify G",k = O(n)/O(k)xO(n-k).The set G"k with this structure as an analytic manifold iscalled the Grassmann manifold of k-planes in n-space.

Let a",k be a vector bundle over the Grassmann manifoldG",k, and the fiber of a"k over the k-plane F c R" is the setof pairs (F,p) where p c F; this makes sense because F is ak-dim subspace of R". Furthermore, F is trivially asubbundle of the vector bundle (G",kxR",G",k,lr) . Thus an ,k canbe made into an analytic k-dim vector bundle in a naturalway. We call a",k the universal bundle over G .,k' As we shallsee, it is also called the classifying bundle for n-dimvector bundle.

The following two theorems will help us in understandingthe construction of classifying bundles as well as preparingus for the classification theorem.

Theorem 2.9.4 Let a be a k-dim C' vector bundle over amanifold M, where 0 <_ r <- w . Let U c M be a neighborhood ofa closed set A c M. Assume f: alU -+ UxR" to be a one-to-oneCr map of vector bundles over idu. If n >- k + dim M, thenthere is a one-to-one Cr map (Cr monomorphism) a -+ MxR" overidM which agrees with f over some neighborhood of A in U.

Theorem 2.9.5 Let a be a Cr k-plane bundle over a m-manifold M, 0 <- r -< -. Then there is a Cr m-plane bundle n

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over M such that a a n z, MxR" (n > k + m).Let us define a vector bundle map

a -+ On, k1 1

M s - Gn,k.

To p c M, g assigns the k-plane g(p) = f(aP) E Gn,k where f:

a - MxR" is a monomorphism over idM. The map g: M -+ Gn,k canbe shown to have the property that g'pnk Z a. The pullbackis called the classifying map for a. From the previous twotheorems, we are ready to state the classification theorem!

Theorem 2.9.6 If n >_ k+m then every Cr k-plane bundle aover a m-manifold M has a classifying map fQ: M - Gn,k, whenn > k+m, the homotopy class of fQ is unique, and if n isanother k-plane bundle over M then fQ z f, iff a = Ti.

This theorem is of great importance because it convertsthe theory of vector bundles into a branch of homotopytheory. One can use what one knows about maps to studyvector bundles. We shall not go into this, for those readerswho are interested in this development, one can consult[Steenrod 1951, Husemoller 1975, Spanier 1966]. As anexample, we have the following theorem:

Theorem 2.9.7 Every Cr vector bundle a over a C°manifold M has a compatible C` bundle structure, and such astructure is unique up to e isomorphism.

Remark: This reminds us of the Whitney theorem about theC° structure of manifolds. Thus from now on it is notnecessary to specify the differentiability class of a vectorbundle either. Although these last three theorems are statedfor manifolds, they are also true (ignoring thedifferentiability) for vector bundles over simplicial or CWcomplexes of finite dimension [Steenrod 1951, Spanier 1966].

Now we are ready to introduce briefly the concept of atubular neighborhood and its properties.

Let M be a submanifold of N. A tubular neighborhood of Mis a pair (f, a) where a = (B,M,rr) a vector bundle over M

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and f: B - N is an imbedding such that : (i) f I M = idM where

M is identified with the zero section of B; (ii) f(B) is anopen neighborhood of M in N. Loosely speaking, we can referthe open set V = f(B) as a tubular neighborhood of M. It isunderstood that associated to V is a particular retractiong: V - M making (V,M,g) a vector bundle whose zero sectionis the inclusion M - V. A slightly more general concept isthe partial tubular neighborhood of M. This is a triple (f,a, U) where a = (B,M,a) is a vector bundle over M, U c B isa neighborhood of the zero section and f: U - N is animbedding such that fIM = idM and f(U) is open in N. Apartial tubular neighborhood (f, a, U) contains a tubularneighborhood in the sense that there is a tubularneighborhood (h, a) of M in N such that h = f in aneighborhood of M.

Theorem 2.9.8 Let M c R" be a submanifold withoutboundary. Then M has a tubular neighborhood in R" [Hirsch1976, Golubitsky & Guillemin 1973].

Theorem 2.9.9 Let M c N be a submanifold, and aM = aN =0. Then M has a tubular neighborhood in N (Hirsch 1976].

It is useful to be able to slide one tubularneighborhood of a manifold onto another one, and to mapfibers linearly onto fibers. Such sliding is a special caseof the concept of isotopy. Here we give a more restrictiveversion of isotopy.

If M, N are manifolds, an isotopy of M in N is ahomotopy f: M x I - N by f(p, t) = ft(p) such that therelated map f' : M x I - N x I, where (p, t) R (ft(p), t) isan imbedding. It is clear that "f is isotopic to g" istransitive.

Theorem 2.9.10 Let M c N be a submanifold, and aM = aN0. Then any two tubular neighborhoods of M in N are

isotopic [Hirsch 1976].It is clear that the boundary of a manifold cannot have

a tubular neighborhood, nonetheless, it has a kind of "half-tubular" neighborhood called a collar. A collar on M is animbedding f: aM x [0,-) - M such that f(p,0) = p.

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Theorem 2.9.11 aM has a collar. (Note, M can be a C°manifold) [Brown 1962, Hirsch 1976].

When a submanifold M c N whose boundary is nicelyplaced, N is called a neat submanifold. More precisely, wecall M a neat submanifold of N if aM = M n aN and M iscovered by charts (0j,Uj) of N such that M n U, = Oi"' (R10)where m = dim M. A neat imbedding is the one whose image isa neat submanifold.

For example:

M1 is neatM2, M3 are not.

If M c N is a submanifold and aM = 0, then M is neat iffM n aN = 0. In general, M is neat iff aM = M n aN and ifboth M and N are at least C1, M is not tangent to aN at anypoint p e aM; i.e., MP ¢ (aN)P.

Theorem 2.9.12 Let M c N be a closed neat submanifold,then aN has a collar which restricts to a collar on aM in M[Hirsch 1976].

Theorem 2.9.13 Let M c N be a neat submanifold, then Mhas a tubular neighborhood in N. Moreover, every tubularneighborhood of aM in aN is the intersection with aN of atubular neighborhood for M in N.

Finally, we have the important theorem on the existenceof tubular neighborhoods.

Theorem 2.9.14 Let M be a submanifold of N. Then thereexists a tubular neighborhood of M in N [Golubitsky andGuillemin 1973].

In this section, we have illustrated the concept andtechniques to "thickening" a submanifold. This concept willbe very useful for the discussion of convergence of orbitsto a periodic orbit in stability analysis. Although in the

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discussion we may not explicitly invoke the notion oftubular neighborhoods, nonetheless, the reader can feel such"construction". The simplest situation is discussing thePoincare return map. The mapping cylinder is the tubularneighborhood.

So far in this chapter, we have discussed finitedimensional manifolds and their topological and geometricproperties. In the next chapter, we shall give some briefdiscussions of infinite dimensional manifolds and globalanalysis which will be useful for our subsequent discussionsof dynamical systems and structural stabilities.

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Chapter 3 Introduction to Global Analysis and InfiniteDimensional Manifolds

3.1 What is global analysis?First recall that linear analysis is the study of

topological vector spaces, such as real, complex or vectorvalued functions on R" or on some domain in R", and linearmaps usually are differential (or integro-differential)operators. This may be viewed as "local" linear analysis. Togeneralize to "global" linear analysis, an arbitrarydifferentiable manifold M replaces the domain in R", andtopological vector spaces of cross-sections ofdifferentiable vector bundles over M are considered. Again,the linear maps are defined by linear differential orintegro- differential operators. Roughly speaking, thequestions here are relating analytic invariants of theoperators with topological invariants of M and the givenvector bundles, and it is the proper setting, for example,for Hodge's theory of harmonic forms, the Atiyah-Singerindex theorem, and the Atiyah-Bott fixed point formula.

What about "global non-linear analysis"? Instead ofdifferentiable vector bundles over a differentiable manifoldM, we consider more general differentiable fiber bundlesover M; instead of topological vector spaces of sections ofthe vector bundle, we consider differentiable manifolds ofsections of the general fiber bundle, and we take non-lineardifferential operators which define differential mapsbetween such manifolds of sections. This seems to be theproper arena for a variety of subjects, such as the theoryof non-linear differential operators and the calculus ofvariations (in particular, Morse theory andLucternik-Schnirelman theory), and the generaltransversality theorem, just to name a few. For an earlierreview, see [Eells 1966; Kahn 1980; Palais 1968; Berger1977].

The underlying technique which runs through all the

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nonlinear analyses is the idea of "linearization", i.e.,approximating a nonlinear map "locally" by a linear map.Usually the sets where the map is defined and into which itmaps have natural infinite dimensional manifold structures,and the map is differentiable with respect to such manifoldstructures. Moreover, the linearization of the maps near agiven point is just its differential at that point. Thusabstract nonlinear analysis turns out to be the study ofinfinite dimensional manifolds and differentiable maps onthem.

An analogous situation in differential geometry is thatfor a given differentail manifold, an open chart can befound such that in a neighborhood of a point, all the localproperties of the manifold can be represented in a Cartesiancoordinate system. For arbitrarily small neighborhoods, themanifold can be considered "Euclidean". Everything is"trivialized", which corresponds to "linearalization". Yetthe different pieces at different points have differentgeometrical and local properties. The means to piecetogether is through the overlapping region UanU9. Here, thetwo pieces have to agree on all the local properties in theoverlapping region. This reminds us that in calculus, thedefinition of continuity of a function at a point is that,not only limits on either side of the point exist, but theyhave to agree. The matching of the overlapping region overthe entire manifold is the spirit of "linearization" withoutdistroying the properties (or more plainly, losing someinformation) of the manifold.

3.2 Jet bundlesIn the following, we shall define jet bundles and

display their utility. But before we start the propermathematical definition, it is helpful to remind ourselvesan analogy in calculus. If f(x) and g(x) are C° andanalytic, one can easily show that f(x) = g(x) up to k-thorder iff the Taylor series of f and g are identical up to

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the k-th order at every point of x. The slightly moregeneralized case is the power series of a complex variable.Indeed, one could generalize to a function ofmultivariables. In the following, we go one step further, wegeneralize the idea of Taylor series to the setting of maps

over manifolds.Let M and N be Co manifolds, and let f,g : M - N be C°

maps, f is said to agree with g up to order k at p e M ifthere are coordinate charts at p e M and f(p) = g(p) c N

such that they have the same Taylor expansion up to andincluding order k. One can convince oneself that theagreement of f and g up to order k (denoted by f - g) iscoordinate independent. In fact, -k is an equivalencerelation, and the equivalence class of maps which agree withf to order k at p is called the k-jet of f at p and denotedby jPkf. Let xa be local coordinates around p e M and yµ belocal coordinates around f(p) a N, then jPkf is specified by

xa, Yµ = fµ(P) , Yaµ = aaf"(P) i Yµab = aabfµ(P) .... , Yµa ...a = as.. a

f"(p), where fµ(p) is the coordinate presentation of f, and ka . a denote partial derivatives

as = a/axa, ... , as .. a = aklaxal ... axa#and Latin indices a,b, ... a,, .. , ak range from 1 to dim M andGreek indices µ, a,... range from 1,..,dim N. Conversely,any collection of numbers xa, y", yaµ,.., YUa a, where xa andyµ are the corresponding coordinate charts andkyµabl yµa,...akare symmetric in their lower indices, determining a uniqueequivalence class. Putting this formally, one has thefollowing:

Let M and N be smooth manifolds, p e M. Suppose f,g: M -+N are smooth maps with f(p) = g(p) = q. (i) f has firstorder contact (i.e., agrees to first order) with g at p if(df)P = (dg) P as mapping of MP Nq. (ii) By induction, f hask-th (k is a positive integer) order contact with g at p if(df):TM - TN has (k-l)-th order contact with (dg) at everypoint in MP. This is written as f -k g at p. (iii) LetJk(M,N)Pq denote the set of equivalence classes under "-k atp" of mappings f: M -+ N where f(p) = q. (iv) Let Jk(M,N),

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the k-let bundle of M and N, be the disjoint union ofJk(M,N)p,q, i.e., Jk(M,N) =U(p,q)EMXN Jk(M,N)p,q. Any element h E Jk(M,N) is a k-let ofmappings (or just k-let) from M to N. (v) Let h e Jk(M,N) bea k- jet, then there exist p in M and q in N such that h eJk(M,N)p,q. Then p is the source of h and q is the target ofh. The mapping a : Jk(M,N) M, defined by h - source of h,is the source map and the mapping p : Jk(M,N) -. N, defined

by h - target of h, is the target map.For a given smooth map f:M - N and for every p e M,

there is a canonically defined mapping jkf : M Jk(M,N)

called the k- iet of f defined by jkf(p) = equivalence classof f in Jk(M,N) P,f(P). Thus, jkf at p is a local lifting from Mto Jk(M,N)pf(p), i.e., jkf(p) is just an invariant way ofdescribing the Taylor expansion of f at p up to order k. Onecan also show that jkf is a smooth mapping.

Since J°(M,N) = M x N, so f has _° contact with g at piff f(p) = g(p), and j°f(p) = (p,f(p)) is just the graph off!

Since any given map determines a k-jet at each point ofits domain, thus if f:M - N is smooth, the k-jet extensionof f is a map jkf: U - Jk(M,N) by x - j,kf, where U is thedomain of f. Clearly, the k-jet extension of f is across-section of the source map a, i.e., a.jkf = ide.

A map from a jet bundle Jk(M,N) to another smoothmanifold P induces maps of higher jet bundles and they arecalled prolongations. More precisely, let M, N and P aresmooth manifolds and let 0: Jk(M,N) - P be smooth. The 1-thprolongation of 0 is the unique map pLO: Jk't(M,N) J((M,P)

such that the following diagram commutes:

P(0 jL

Jk+t (M, N) J((M,P) E ------ Pk+tf kf) 0 .\

M ------- ------ - - - -4 Jk(M,N)jkf

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In appropriate local coordinates on M, N, and P, roughlyspeaking, the prolongations amount to the taking of totalderivatives of 0. We shall not go into any details. Forthis, one may want to consult [Pirani, Robinson and Shadwick1979]. Jet bundles not only are important in discussingglobal analysis and nonlinear differential operators onwhich we shall embark briefly, but they are also importantin describing higher order vector bundles on smoothmanifolds [see e.g., Yano and Ishihara 1973].

Over the past couple of decades, a class oftransformations discovered by Backlund over a century agohas emerged as an important tool in the study of a widerange of nonlinear evolution equations in theoreticalphysics and continuum mechanics. Backlund transformationsare significant because the invariance under Backlundtransformations can be utilized to generate an infinitesequence of solutions of certain nonlinear evolutionequations by purely algebraic procedures. Current work onBacklund transformations are mainly concerned with extendingtheir applications to other nonlinear partial differentialequations of physical interest and with encompassing theknown results in a unified theory so that generalizations tohigher dimensions can be made. Jet-bundle formulation ofBacklund transformations seems to provide an appropriategeometric setting for the main concerns of Backlundtransformations described above and their connection withthe inverse scattering formalism, prolongation structures,and symmetries. For those readers who are interested inthese areas of research, please consult Pirani, Robinson andSchadwick [1979], Rogers and Schadwick [1982], Hermann[1976, 1977], Estabrook and Walquist [1975].Lemma 3.2.1 Let U c R" be open, and p e U. Let f,g: U -+ R"

be smooth mappings. Then f _k g at p if f (Daf,) (p) = (D°g1) (p)for Ial <_ k, 1 < i <- m where a = (a1,..,a) are n-tuples ofnon-negative integers, a l = al + a2 +....+ a", D° =al°l/axa, .. ax,,'", and f; and g, are the coordinate functionsdetermined by f and g respectively.

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This lemma sometimes is used to define the equivalencerelation "-k". As we have pointed out earlier, the followingcorollary follows immediately.

Corollary 3.2.2 f and g : U Rm such that f -k g at piff the Taylor expansions of f and g up to and includingorder k are identical at p.

The following lemma which concerns the equivalencerelation between a pair of composite maps is intuitivelyclear and can be proven by induction.

Lemma 3.2.3 Let U c R" and V c Rm are open subsets. Letf1, f2 : U - V and g1,g2: V Rt be smooth maps such thatg1 f1 and g2 f2 make sense. Let p e U and suppose f1 -k f2 at p

and g, -k g2 at f1 (p) = f2(p) = q. Then g, - f1 _k g2 f2 at p.Theorem 3.2.4 Let X, Y, Z and W be smooth manifolds.

(i) Let h: Y - Z be smooth, then h induces a map h.: Jk(X,Y)- Jk(X,Z) defined by: let a e Jk(X,Y)pq and let f: X - Yrepresent a. Then h* (a) = the equivalence class of in

Jk(X,Z)p.h(q). (ii) Let a: Z - W be smooth. Then a.-h. =as mappings of Jk(X,Y) _ Jk(X,W) and (idy). = ids (X,Y) Thus ifh is a diffeomorphism, h* is a bijection (one-to-one, onto).(iii) Let g : Z - X be a diffeomorphism, then g induces amap g*: Jk(X,Y) Jk(Z,Y) defined as follows: let a eJk(X,Y)Pq and let f:X - Y represent a. Then g*(a)= theequivalence class of in Jk(Z,Y)91(p) q' (iv) Let b: W - Z

be a diffeomorphism. Then the induced composite mappingsb'-g' = are mappings of Jk(X,Y) - Jk(W,Y) and (id')' _idak(X y) so that g* is also a bijection.

Let A"k be the vector space of polynomials inn-variables of degree <- k with their constant term equal tozero. Choose the coefficients of the polynomials ascoordinates of A"k, then A"k is isomorphic to some Euclideanspace and thus a smooth manifold. Let B",mk = ® ;_1m A"k. Heres is the Whitney sum. Then B"mk is also a smooth manifold.

Theorem 3.2.5 Let X and Y be smooth manifolds and dim Xn, dim Y = m.

(i) Jk(X,Y) is a smooth manifold, and dim Jk(X,Y) = m + n +dim(B"mk).

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(ii) The source map a : Jk(X,Y) X and the target map Q :

Jk(X,y) -+ Y, and a x Q : Jk(X,Y) -+ X x Y are submersions.(iii) If h : Y - Z is smooth, then h, : Jk(X,Y) Jk(X,Z) issmooth. If g : Z - X is a diffeomorphism, then g* : Jk(X,Y)

-+ Jk(Z,Y) is a diffeomorphism.(iv) If h : X - Y is smooth, then Jkh : X - Jk(X,Y) issmooth. [Golubitsky & Guillemin 1973, p.40].

Note that in general Jk(X,Y) is not a vector bundlebecause there is no natural addition in Jk(X,Y)pq. But if Y= R1", then Jk (X, R'") is indeed a vector bundle over X x R'where the addition of jets is given by the addition offunctions representing these jets in Jk(X,R10)pq.

Although Jk(X,Y) is not a vector bundle in general, itis more than just a manifold, it is a fiber bundle. In fact,if k > 1 and ignoring all derivatives above the 1-th yieldsthe natural projection Irk ( : Jk(X,Y) - JL(X,Y) by jkxf j1xf.Thus, Jk(X,Y) is a fiber bundle over Jt(X,Y) with projectionrk,L. In particular, J°(X,Y) may be identified with X x Y andJ'(X,Y) may be thought of as a vector bundle over X x Y.Furthermore, lr,,, is understood as the identity map ofJL (X, Y).

As we have seen in this section, jet bundle is theproper language to describe relations between smooth mapsamong smooth manifolds. Thus, it will be very important indiscussing stable maps, their singularities, andbifurcations. Of course, it will also be very important todescribe global stability properties. Indeed, it is one ofthe mathematical techniques frequently used in much currentliterature on the theoretical dynamical systems. There is nosingle volume which discusses jet bundles and theirappliations. Jet bundles can be found in many booksdiscussing global analysis. The following books cover somediscussions of jet bundles: Pirani, Robinson and Schadwick[1979], Golubitsky and Guillemin [1973], and of course,Palais [1968].

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3.3 Whitney C" topologyAs we have mentioned earlier, in order that any physical

fields, be they scalar, vector, or tensor fields, havephysical significance, they must have some form of stabilityunder "perturbation". In other words, "nearby" fields shouldhave the same properties and physical significance. But inorder to give a precise meaning of "nearby", one has todefine a topology on the set of fields. There are variousways in which this can be defined depending on whether onerequires a "nearby" field to be nearby in just its values(CO topology) or also in its derivatives up to the kth order(Ck topology) and whether one requires it to be nearbyeverywhere (open topology) or only on compact sets (compactopen topology).

Let M and N be smooth manifolds and C°(M,N) be the setof all smooth mappings from M to N. For a fixed non-negativeinteger k, let U be a subset of Jk(M,N). Let 0(U) denote theset (f a C°(M, N) I jkf (M) c U) . Also note that 0 (U) nO (V) =9(UnV). This family of sets (A(U)) forms a basis for theWhitney Ck topology. Let Wk denote the set of open subsetsof C°(M,N) in the Whitney Ck topology. Thus the Whitney Cmtopology on Ck(M,N) is the topology whose basis is the unionof all Wk's where k= 0,..,., i.e., W = Uk=o Wk. This basis isa well-defined one because Wk C W1 for k _< 1.

In usual literature, the Whitney Ck topology is alsocalled the strong or fine topology on Ck(M,N) sometimesdenoted by Csk(M,N), and the compact-open topology on Ck(M,N)is also called the weak topology on Ck(M,N), denoted byCwk(M,N). If M is compact, the weak topology is the same asthe strong topology. Thus the strong topology is moreuseful. But it is only fair to point out that weak topologyhas very nice features: it has a countable basis (we shallpoint this out for strong topology with compact M), acomplete metric, and for compact M, Cwk(M,R") is a Banachspace. Moreover, the Whitney imbedding theorem (Sect.2.4)can be stated in relation to the weak topology: Let M be acompact Cr manifold, 2 <_ r 5 - . Then imbeddings are dense

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in C.r(M,R4) if q > 2m, while immersions are dense if q >- 2m,

m = dim M.Let us describe a neighborhood basis in the Whitney Ck

topology for a function f in C°(M,N). Since all manifoldsare metrizable [Kelley 1955], one can choose a metric onJk(M,N) compatible with the Ck topology. Let B6(f) = (g eC°(M,N)I for all p e M, d(jkf(p),jkg(p)) < 6(p)), where 6: M- R' is C°. Thus one can consider B6(f) as the set of smoothmappings of M - N whose partial derivatives up to the k-thorder are 6-close to f's. Let U be an open neighborhood of fin C°(M,N), one can show that B6(f) c U, and indeed, B6(f) isopen for every S.

If M is compact, C°(M,N) satisfies the first axiom ofcountability. This is because one can find a countableneighborhood basis of f by B6(f) where 6,(p) = 1/n for all pe M, thus 6 is bounded below for large n. One can easilyprove that: a sequence of functions fn in C(M,N) convergesto f (in the Ck topology) iff jkf,, converges uniformly tojkf. Clearly, in the local situation, fn and all of thepartial derivatives of fn of order <- k converge uniformly tof's.

For noncompact manifolds, the concept of the convergenceof fn to f is stronger than uniform convergence and the weaktopology does not control the behavior of a map at"infinity" very well. Thus the statement for compactmanifolds has to be modified. Since the manifolds areparacompact, then it is locally compact (Sect.2.4). Thus onecan extend the result of a compact manifold to a compactsubset of the noncompact manifold. More precisely; thesequence of mappings fn converge to f (in the Ck topology)iff there is a compact subset C of M such that jkfnconverges uniformly to jkf on C and all but a finite numberof the fn's equal to f outside C. In fact, for a noncompactM, C°(M,N) in the Whitney Ck topology does not satisfy thefirst axiom of countability. Nonetheless, the Whitney(strong) Ck_ topology is very important for differentialtopology for the fact that many interesting and important

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subsets are open. For example: let Immr(M,N) be the set ofCr immersions of M in N, Imbr(M,N) be the set of Crimbeddings of M in N, Diffr(M,N) be the set of Crdiffeomorphisms from M onto N, and Propr(M,N) be the set ofproper Cr maps M N where f:M -' N is proper if f"1 takes

compact sets to compact sets. Then we have the followingimportant theorem.

Theorem 3.3.1 Immr(M,N) and Imbr(M,N) are open inCsr(M,N) (Whitney Ck toplogy) for r >_ 1; Propr(M,N) is openin CSr(M,N) for r >_ 0; if M and N are Cr manifolds withoutboundary, then Diffr(M,N) is open in Csr(M,N) for r 1; andif M and N are manifolds without boundary and f:M -+ N is ahomeomorphism, then f has a neighborhood of onto maps inCS (M, N) .

A very basic approximation theorem for a manifoldwithout boundary is:

Theorem 3.3.2 Let M and N be Cs manifold withoutboundary, 1 <_ s _< w . Then CS(M,N) is dense in CSr (M,N), 0 <_

r 5 s.For its many applications in differential topology, see

e.g., [Hirsch 1976, Chapter 2, section 2]. For example, fromthe above theorem and the openness theorem and a theorem ondiffeomorphisms, one has:

Theorem 3.3.3 Let 1 <_ r 5 , every Cr manifold is Crdiffeomorphic to a CO manifold. (This is the theorem due toWhitney we quoted at the begining of Chapter 2).

The following definitions and propositions are oftechnical nature and they are useful in describing thegeneralized transversality theorem, or in proving someresults. The reader may skip the following five propositionswithout loss of continuity.

Let X be a topological space, then: (i) A subset Y of Xis residual if it is the countable intersection of opendense subsets of X. (ii) X is a Baire space if everyresidual set is dense.

Proposition 3.3.4 Let M and N be smooth manifolds, thenC°(M,N) is a Baire space in the Whitney Co topology.

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[Goluoitsky & Guillemin 1973, p. 44].Let us list some properties of C°(M,N) which will be

used later.Proposition 3.3.5 Let M and N be smooth manifolds. Then

the mapping jk: C°(M,N) -+ C°(M,Jk(M,N)) defined by f - jkf iscontinuous in the Whitney CO topology.

Proposition 3.3.6 Let M, N and P be smooth manifoldsand let 0: N - P be smooth. Then the induced map 0.: C°(M,N)-, C°(M,P) defined by f - O -f is a continuous mapping in the

Whitney e topology.Proposition 3.3.7 Let M, N and P be smooth manifolds.

Then C°(M,N)xC°(M,P) is homeomorphic to C°(M,NxP) in the COtopology by the standard identification (f,g) - f x g wheref e C°(M,N) , g e CO(M,P) , p e M and (f x g) (p) _(f(p),g(p))

Proposition 3.3.8 Let M, N and P be smooth manifolds,and in addition, M is compact, then the mapping ofC°(M,N)xC°(N,P) - C°(M,P) defined by (f,g) - is

continuous, where f e C°(M,N), g c C°(N,P).Note that this proposition will not be true if M is not

compact. Nonetheless, if we replace C°(M,N) by its opensubset of proper mappings f of M into N (recall, f is properif f-1 maps compact sets to compact sets), then the sameconclusion is valid. Note also that the induced mapping f*:C°(N,P) - C°(M,P) defined by g - is (not) continuous iff is (not) a proper mapping. In particular, if M is an opensubset of N and f is just the inclusion map, then therestriction map of C°(N,P) - C°(M,P) given by g - gIM is notcontinuous.

We now state Thom's transversality theorem in a moregeneral form and show that Theorem 2.4.27 is a corollary.

Theorem 3.3.9 (Thom's transversality theorem) Let M andN be smooth manifolds and W a submanifold of Jk(M,N). LetT. = (f a C°(M,N)I jkf X W). Then Tw is a residual subset ofC°(M,N) in the Whitney e topology.

Since J°(M,N) = M x N and j°f(p) = (p,f(p)), and thetarget map (3: M x N - N is onto, so fl-'(W) is a submanifold

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of M x N. If j°f ,r p1(W) at p then clearly f J W at p. [If

j°f (p) it f-t (W) then, f (p) e W. But if j°f (p) e r1 (W) and(fi-1 (W)) (P f(P)) + (dj°f) MP = (MxN) (P.f(P)). Then apply dQ oneobtains Wf(P) + (df)MP = Nf(P). Thus f ,rj W at p] . Since theset of all maps which are transverse to W contains the set(f e C°(M,N)Ij°f A Q"'(W)), which is dense by Theorem 3.3.9.Thus it is established that Theorem 2.4.27 is a special caseof Theorem 3.3.9. Note that usually Thom's transversalitytheorem is stated as a combination of Theorems 2.4.26 and2.4.27. Sometimes Theorems 2.4.26 and 2.4.27 are called theelementary transversality theorem.

Let us describe an "obvious" generalization. Let M and Nbe smooth manifolds and define Ms = Mx...xM (s times) andX(S)= ((x1, ..xs) E ms I x j + x, for 1 S i <_ j <_ s). Let a bethe source map, i.e., a: Jk(M,N) - M. Now define as: Jk(M,N)s- Ms in an obvious way. Then Jsk(M,N) = (as)-'(Xwsw) is thes-fold k-jet bundle. A multilet bundle is some s-fold k-jetbundle. And X(s) is a manifold since it is an open subset ofMs. Thus Jsk(M,N) is an open subset of Jk(M,N)s and is also asmooth manifold. Now let f: M - N be smooth. Then definejskf: X(s) - Jsk(M,N) in the natural way, i.e., jskf (x1, .. , xs) _

Theorem 3.3.10 (multijet transversality theorem) Let Mand N be smooth manifolds and W a submanifold of Jsk(M,N).Let T. = (f a C'(M,N) jskf A W). Then T. is a residual subsetof C'(M,N). Furthermore, if W is compact, then Tw is open [G& G 1973, p.57].

3.4 Infinite dimensional manifoldsDifferentiable manifolds are natural generalization of

Euclidean spaces by building up from Euclidean spaces andgluing together by some overlapping maps and charts.Infinite dimensional manifolds can also be viewed in asimilar way. For instance, starting with some fixed,infinite dimensional linear vector spaces, such as a Hilbertspace or a Banach space, one can form a manifold by gluing

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together open sets by nice overlapping functions. The theoryof infinite dimensional manifolds can provide the properframework for certain analyses and various connectionsbetween geometry and analysis, particularly the operatortheory. But the need for infinite dimensional manifolds alsoarises in another vein. For instance, let M and N be twodifferentiable manifolds, there is great interest in thespace of continuous maps from M to N and its subspaces ofvarious degrees of differentiability for these maps. Weshall state that under certain conditions, this space ofcontinuous maps is an infinite dimensional manifold.

First, we shall review some definitions and propertiesof Banach and Hilbert spaces, then an infinite dimensionalmanifold on a given topological vector space will be definedand facts recalled. The spaces of maps between manifoldswill be defined and some results stated.

Let V be a vector space over the field F (either R orC). V is a topological vector space if there is a topologyon V such that 0: V x V - V; ( O(v1,v2) = vI + v2) and0: F x V - V; ( O(a,v) = av) are continuous.

Let V be a vector space, a norm on V is a function VR, lixil, for each x e V, such that:

(i) IIxII ? 0, IIxII = 0 iff x = 0;(ii) llaxII = lal jlxii, for x e V, a e F;(iii) lix + yfi < jjxii + Ilyll , for any x, y e V.With the given norm, one can define a metric on V bye(x,y) = lix - yll. Then one can define a topology on V by

taking the open ball, B,(x)= (y e Vle(x,y) < e ), as abasis.

A topological vector space whose topology arises fromthe norm, such as the one given above, is a Banach spaceprovided that it is complete with respect to the norm or themetric (this is the Cauchy completion).

Let Bi and B. be Banach spaces and let f be a linear mapf: B, - B2. Suppose f is bounded, i.e., there is k > 0 suchthat Iif(x)11 5 kilxll for all x e B,. Then one can show that fis continuous.

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If we define IIfli = g.l.b.(kI IIf(x)II <_ kIIxII for all x eB1), then this norm makes the space of linear maps of Banachspaces, into a Banach space Lin(B1,B2).

Let U1 c B11 U2 c B2 are open subsets of Banach spaces,and let g: U, -. U2 be continuous, g is differentiable at x eU, with derivative g' e Lin(B1,B2) if for all h e B, ofsufficiently small norm e(h) = g(x + h) - g(x) - g'(x)hsatisfies limJhJ_O II a (h)/IIhII II = 0. It is not difficult tocheck the usual linearity and the chain rule.

Next we shall introduce the notion of an infinitedimensional manifold modeled after some given topologicalvector space, rather than just usual R", and discuss somebasic properties.

Let V be a topological vector space (usually a Banachspace). A T2 space M is called a manifold modeled on V, iffor every p e M there is some open subset OP of M, with p eO. and a homeomorphism 0P of OP onto an open subset of V.

Of course there are many Op and 0P for each p e M. Thisdefinition is very similar to an earlier definition of afinite dimensional manifold except that 0P is only requiredto be a homeomorphism onto an open subset of V, not onto allof V. In fact, in general we shall not be able to specifythe range of 0p. It is clear that every finite dimensionalmanifold is a manifold modeled on R" for some n.

If V is a topological vector space, then any open subsetof V is a manifold modeled on V. If M" is a finitedimensional manifold and B a Banach space, then M"xB is amanifold modeled on the (Banach) space R"xB.

It is more convenient to work with norms in order todefine an infinite dimensional manifold. For the time being,we shall restrict our attention to manifolds modeled on aBanach space and it will be called a Banach manifold.

Let B be a given Banach space, M is a smooth manifoldmodeled on B if M is a manifold modeled on B such that if OPn Oq + 0, then the composite map

OP (OP n Oq) (0 %0eme" 09))., OP n Oq m4 - Oq (Op n Oq)is a smooth map in the sense that it has continuous

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derivatives of any order (obtained from iterating thedefinition of the differentiable at a given point).

Let M and N are manifolds modeled on Banach spaces. Letf:M - N be a continuous map. f is smooth (or differentiable)if for each p e OP c M with homeomorphism 0P O, -, OP(OP) and

q e Oq c N with oq: Oq - Oq(Oq) , where f (p) = q,T

then in asufficiently small neighborhood of p, 0P(OP) OP- M fy N Of-

Oq(Oq) is smooth (or differentiable).Some of the notions for finite dimensional manifolds can

easily be generalized to our current interest, nonetheless,occasionally some care has to be exercised.

Submanifold is defined by requiring that each point hasa neighborhood homeomorphic to O,xO2 where O, and 02 are openin B, and B2 respectively. And the big manifold is modeledon B1xB2, and the manifold is described locally in terms ofOtx(pt). One can easily check that such a submanifold is amanifold.

A diffeomorphism of infinite dimensional manifolds is asmooth map with a smooth, two sided inverse. An imbedding 0:M N is a smooth one-to-one map that is a diffeomorphism ofM onto a submanifold of N. An immersion is a smooth mapwhich is locally an imbedding.

For a given smooth manifold M modeled on a Banach spaceB, let us choose a coordinate neighborhood OP, for each p eM, endowed with a homeomorphic 0P onto an open set in B,i.e., 0P: O, - OP(OP) c B. Consider the sets 0PxB and definean equivalence relation in their union by specifying that if(u,vI) a 0PxB, (u,v2) a OgxB, then (u,vl) is equivalent to(u,v2) iff (oq ' ¢P ,) ' (v1) = v2 where the prime refers to thederivative taken at u. The quotient space is defined to bethe tangent bundle TM and the projection map r : TM - M isdefined by ,r((u,v)) = u. And as before, the fiber is thetangent space at p, i. e. , 7r"I (p) = M.

As before, the differential of a smooth map betweensmooth manifolds can be defined. Then it is possible tocharacterize immersion in terms of the tangent space byrequiring that at every point the differentials of the

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immersion (df)P: MP - Nf(P) have a left inverse. Bygeneralizing the inverse function theorem to Banach spacesone can prove the equivalence of the two notions ofimmersion.

Since a separable Hilbert manifold, a manifold modeledon L2, is a fortiori a paracompact space, hence partition ofunity exists from a theorem in point-set topology. It can beshown that such a manifold has a smooth partition of unity[Kahn 1980, p.216].

Theorem 3.4.1 Let M be a separable manifold modeled onL2, and let (B.) be an open covering, where BQ(x) is an openball of radius a > 0 about x e L2. Then there is acountable, locally finite open covering (On), a refinementof (B.), and there exists a smooth partition of unitysubordinate to this covering {O,,).

Let M be a smooth m-dim Riemannian manifold, i.e., forany p e M, and vectors u, v e MP we have a continuous,symmetric, positive definite inner product defined by (u,v) P

ut(g(p))v and the norm is defined by HOP = uu).Let X be any compact Hausdorff space, M be a Riemannian

manifold. We set F(X,M) = (fl f: X - M, f continuous). Ametric on F(X,M) can be introduced by

d(f,g) = l.u.b.XEx a(f(x),g(x)), f,g a F(X,M) wherea(a,b) is the greatest lower bound of the length of allsmooth curves in M from a to b. It is easy to show that dmakes F(X,M) into a metric space.

For a given f e F(X,M), one may define a tangent spaceto F(X,M) at that given map, denoted by F(X,M)f, by letting0: TM - M be the projection map of the tangent bundle of M.Then F(X,M)f is the set of all f': X TM such that ,n f' _f. Obviously one can make F(X,M)f into a linear spacebecause if we set (f1' + f2')(x) = ft'(x) + f2'(x), where f1',f2 : X - TM and x e X, then ir- (f1' + f2') (x) = f (x) andlikewise for of'. One can also introduce a norm into F(X,M)fby setting IIf' Ilf = l.u.b.XEx IIf' (x) JIf(X). Note that IIf' (x) IIf(X)is the norm in Mf(X) defined by the Riemannian metric. It isthen straight forward to show that F(X,M)f is a complete

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metric space with respect to the norm. That is, F(X,M)f is areal Banach

Theoremspace.

3.4.2 If X is a compact Hausdorff space and Mis a smooth Riemannian manifold, then F(X,M) is a smoothmanifold modeled on a real Banach space - F(X,M)f,

independent of the choice of f [Eells 1958, see also Eells1966, Kahn 1980, p.218].

If here X and M are compact manifolds, as before,F(X,M)f be the vector space of all smooth liftings of f eF(X,M), i.e., f' a F(X,M)f is smooth and f': X- TM suchthat f where r: TM - M is the canonical projection.Earlier, we have pointed out that such space F(X,M)f is acomplete linear vector space, thus a Frechet space. (A

Frechet space is a topological vector space which ismetrizable and complete). Thus it is not surprising to notethat:

Theorem 3.4.3 The group of diffeomorphisms of M,Diff(M), is a locally Frdchet C' group [Leslie 1967].

For the relations between the homeomorphism group andthe diffeomorphism group of a smooth manifold and theirhomotopic type, one should consult [Burghelea and Lashof1974a,b].

In addition to the manifold and differentiablestructures for spaces of differentiable maps, one can alsoconstruct such structures to a more general class, thespaces of sections of fiber bundles.

Let a be a smooth vector bundle over a compact smoothmanifold M, then define S(a) to be the set of all sections sof a such that s e S(n) for some open vector subbundle n ofa, i.e., S(a) = Un S(n) where the union is over all openvector subbundles n of a. Then one can show that S(a) notonly is a Banach manifold, but also has a uniquedifferentiable structure [see Palais 1968].

We have only scratched the surface of this evolving andvery interesting area. Interested readers are urged toconsult Lang [1962], Eells [1966], Burghelea and Kuiper[1969], Eells [1958], Eells and Elworthy [1970], Marsden,

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Ebin, and Fischer [1972], and some papers in Anderson[1972], Palais [1965, 1966a,b, 1971].

Although most results in finite dimensional manifoldscan easily be extended to infinite dimensional manifolds,nonetheless, there are a few surprises. It has beenestablished that every separable, metrizable C°-manifold canbe C°-imbedded as an subset of its model [Henderson 1969].In other words, any reasonable Hilbert manifold isequivalent to an open subset of Lz space. This is contraryto the case of finite dimensional manifold!

Let M be a e manfold modeled on any infinitedimensional Hilbert space E, Kuiper [1965] asserts that M isparallelizable. Eells and Elworthy [1970] observe that thereis a diffeomorphism of M onto MxE. They also observe that ifM and N are two e manifolds modeled on E and if there is ahomotopy equivalence 0: M - N, then 0 is homotopic to adiffeomorphism of M onto N. As a corollary, thedifferentiable structure on M is unique. Again, theseresults are different from the finite dimensional cases aswe have pointed out earlier in Sect.2.4 [see also Milnor1956].

3.5 Differential operatorsMost of the basics of linear differential operators may

be generalized to manifolds by piecing together differentialoperators on Euclidean spaces, but it turns out much moreelegant and convenient to give a general definition in termsof vector bundles. We shall adopt this approach to avoidcumbersome details involving coordinate charts at thebeginning. After that, we shall discuss various importantcases and examples of differential operators. Finally, weshall briefly discuss the important notions of ellipticity,the symbol of an operator, linearization of nonlinearoperators, and the analytic index of operators. Referencesfor further reading will be provided.

Let (B,,M,rl) and (B2,M,,r2) be two smooth vector bundlesover an m-dim compact smooth manifold M, and let C°(M,B1) be

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the vector space of sections of the bundle (B,,M,7r1), here i= 1, 2. A linear differential operator is a linear map ofvector spaces P: C' (M, B1) - C' (M, B2) such that supp P (s) csupp s, where supp s = closure of (p e M I s(p) + 0).

Note that this definition is very elegant and simple,nonetheless, some work is needed so that we can "visualize"that these operators are locally generated bydifferentiation. In the following, we shall specify thesmooth manifold M and its vector bundles, the linear map,and sections to make the definition "visualizable".

If M = Rm and B1, B2 are trivial 1-dim vector bundles(line bundles) over M, and if a = (ai,...,am) is an indexset, then for a smooth f one can define a lineardifferential operator D° by D°f = ak°If/ax1°l... axma'.If P is any polynomial over the ring C'(M,R) in m variableszi,... , zm, then if we substitute a/ax; for z,, then theresulting polynomial gives a linear differential operator P: C'(M,R) - C'(M,R) by P(f) = P(a/ax1,..., a/axm)(f). If svanishes in an open set, every term of P(s) and P(s) itselfvanishes on that open set. Clearly supp P(s) c supp s.

Let B1, B2 and M be as before, and let P: C'(M,B1)C'(M,B2) be a linear differential operator. Then we say thatP has order k at the point p e M if k is the largestnon-negative integer such that there is some s e C'(M,B1)and some smooth function f defined in an open neighborhoodof p and vanishing at p such that P(fks)(p) + 0. The orderof P is the maximum of the orders of P at all points of M.

It is easy to check that this notion of order for Pagrees with the usual definition of order for a lineardifferential operator defined on Euclidean space. Here aword of caution is called for. Recall in our definition ofdifferential operator and its order, we assume that M beingcompact. If it happens that M is noncompact, then the orderof a linear operator may not exist. For example, M = R1 andchoose 0; a C (R) = C' (R1, R1) with support suppo1 Q [ i, i+l ]and ¢1(i+1/2) > 0. Set P(f)(p) = E1Oi(p)d'f(p)/dx'. Clearly,with such construction, P is a linear differential operator,

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but the order of P is not defined.Fortunately, the next theorem asserts that a linear

differential operator locally will have an order, andconsequently a linear differential operator defined on acompact manifold will have a finite order.

Theorem 3.5.1 (local theorem) Let 0 c R" be open andlet P: C°(0,0) - C°(O,r)) be a linear differential operatorfrom the sections of the trivial s-dim vector bundle 4 overO to those of the trivial t-dim vector bundle n over 0. Let01 c 0 have a compact closure contained in 0. Let V(§,tf) bethe vector space of linear maps from Rs to Rt. Then there isan m >- 0 such that for every multi-index a, jal <- in, there

are e maps gQ: 0, - such that for any f e CO(01,0), pe 01, (Pf) (p) = Elalcm g,(p) (D°f) (p) [Kahn 1980, p.194; Peetre1960].

The next theorem has the essential idea and it is aglobal theorem for differential operators defined on amanifold.

Theorem 3.5.2 (global theorem) Let M be a smoothmanifold, and let B, and B2 be two vector bundles over M,with dim B, = a,. Let P: C (M, B1) -. C (M, B2) be a lineardifferential operator. Take p e M. then there is acoordinate neighborhood of p, U, over which both bundles aretrivial and a positive integer m so that in U, (Pf)(p) =

g,(p) (D°f) (p) for smooth maps g,,: U - V(B1,B2) [Kahn1980, p.196].

In the last section we have mentioned that the space ofsections of a vector bundle is a Banach manifold and has aunique differentiable structure. We shall say a little bitmore and state the theorem due to Hormander [1964].

Let (B,M,7r) be a smooth vector bundle, si a C`(M,B) be a

sequence of e sections of B, and a given fixed section s.We say that s, converges to s locally uniformly if for any pe M there is a coordinate neighborhood U of p over which Bis trivial such that in U, s; and D°s; converge uniformly tos and Das, respectively. For two given smooth vector bundlesB1 and B2, a linear map L: C° (M, B1) -+ C° (M, B2) is weakly

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continuous if whenever si converges to s locally uniformly,then L(s1) converges to L(s) uniformly over K c M.

The term weakly refers to the fact that we have not yettopologized the vector spaces C°(M,B,), so it does not makesense to ask whether L is continuous. Nonetheless, clearlyany classically defined linear differential operator isweakly continuous.

Theorem 3.5.3 Let Bt and B2 be two smooth vectorbundles over M, let L: C°(M,B,) - C°(M,B2) be a weaklycontinuous linear map. The necessary and sufficientcondition for L to be a linear differential operator oforder <- m is that: for any s e C°(M,B1), p e M, and f asmooth function of M, then there exist a function g in xfrom R to Rdim B, , g (x) (p) = e"xf(P) [L(elxfs) (p) ] such that ineach coordinate it is a polynomial of degree <_ in.

For the proof we refer the reader to (Hormander 1964;Kahn 1980, p.198).

We wish to define the symbol and ellipticity of a lineardifferential operator before we discuss nonlineardifferential operators. Locally, the symbol of a linearpartial differential operator of order m defined overEuclidean space R" can be thought of by: (a) ignoring theterms of order less than in, and (b) in each term, replacinga/ax; by 0, thus obtaining a form in the variables 'i withsmooth functions as coefficients. For higher derivatives,they are written as a power of ,, i.e., replace a'/ax,k by1ik. If the symbol is definite, i.e., all the coefficientsare positive, or in other words, the symbol vanishes onlywhen all the variables are set equal to zero, then we saythat the linear differential operator is elliptic.

For instance, the 3-dim Laplacian a a2/ax2 + a2/ay2 +a2 /az2 has symbol '2 + 022 + 132 and clearly it is elliptic.The linear operator L = aa2/axe + bat/aye - ca/ax + da/ayhas symbol at12 + b022 and it is elliptic. But the waveoperator a2 /ax2 + a2 /ay2 + a2 /az2 - C-2 a2 /at' has symbol 12+ 422 + 032 - C-2$42 is not elliptic. The term elliptic comesfrom the theory of quadratic forms as it relates to conic

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sections. Thus, there are also linear differential operatorsof the types parabolic and hyperbolic. For instance, thewave equation is hyperbolic, and the diffusion equation isparabolic.

In order to treat the symbol in a global setting, onecannot just consider a single form, we need the followingmachinery:

Given a smooth m-dim manifold M and two smooth vectorbundles over M, (B1 , M, 7r1) , (i = 1, 2) . Let T'M be thecotangent bundle of M with projection p. Let P: C(M,B1) -+C°(M,B2) be a linear differential operator of order k. Letus now consider the induced bundle p"1(B,) = ((u,v) e T'MxB,I

a1 (v) = p (u)) . Let a e M, fl a Ma'. Let f be a smooth functionin a neighborhood of a where f(a) = 0 and df(a) = fl. For e en1-1 (a) , let s be a smooth section of (B1, M, 7r1) such thats(a) = e. Now set aP(fl,e) = P(fks) (a). Since fl a Ma* and e e7r11(a), clearly (fl,e) a p-1(BO .

We call the map aP

: p'1 (B1) - B2 the symbol of thedifferential operator P. For each a e M and fl a Ma', a

Pis a

map from ,r1-' (a) to 7r2"1 (a) . A linear differential operator Pis elliptic if for each nonzero fl a Ma', a e M, the map aP isone- to-one.

As a simple example, consider the two-dimensionalLaplacian in the plane e = a=/c3x2 + a1/ay2 in a trivialone-dimensional vector bundle over R2. Then 7r-1(B) is a

trivial one-dimensional vector bundle over R2xR2 = R4. For agiven 1-form a in R2, a = adx + bdy, choose f(x,y)= ax + by.Let the section be s(x,y) = ((x,y),m), where m is avariable. Then

(a' /ax2 + 132 /ay2 ) ((ax + by) 2 m) = 2a2 m + 2b2 m.If a and b are not both zero, this is clearly a one-to-onemap in terms of the variable m, thus a is elliptic.

As another example, let M be a smooth n-dim manifold andlet Ak(T*M) be the bundle of k-forms. We have a lineardifferential operator of order 1, d : Dk(M) - Dk'1(M) where

Dk(M) = C°(M,Ak(T*M)) are smooth differential k-forms. When i= 0, it is easy to show that the symbol ad of d is elliptic.

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Let f be a smooth function of M, a c M8*and (df) (a) = a.

Let s c D°(M) = C°(M) and s(a) = e. Then in a localcoordinate chart,

ad(a,e) = d(fs) (a) = ((3fs(a)/axj)dX1f(a) (c3s(a)/c3x,)dX1.

Since we may choose any s such that s(a) = e, we choose s(x)= e identically, i.e., as(a)/ax, = 0. Thus ad(a,e) = ea, andd : D°(M) -+ D'(M) is elliptic.

With the same information as provided earlier, it isknown [Ch.l of Palais 1965] that the vector spaces for anelliptic differential operator P of positive order on acompact manifold

Ker P = {f a C°(M,B1)I P(f) = 0) andCoker P = C°(M,B2)/ (g a C°(M,B2)I g = Pf) are both

finite dimensional. The analytic index of P is an integerie(P) = dim (Ker P) - dim (Coker P).

It is known that this index is invariant under deformationof P, and thus suggests that there might be a topologicaldescription of ia. This has led to the theory of thetopological index and the theorem of Atiyah and Singer whichasserts that these two indices are the same.

For the definition of the topological index see [Ch. 1,3, 4 of Palais 1965], analytic index see [Ch. 1, 5 of Palais1965], both analytic and topological indeces on unitball-bundle or unit sphere bundle of M see [Ch.15 of Palais1965], and the index theorem see [App.I of Palais 1965],applications see (Ch.19 of Palais 1965) and the topologicalindex of elliptic operators see [App.2 of Palais 1965]. Avery good review of differential operators on vector bundlesis Ch. 4 of [Palais 19651. A good source of reference on thesubject of Atiyah-Singer index theorem is the book edited byPalais [1965]. For those readers who want to go to thesource, the series of papers by Atiyah and Singer [1963,1968] are recommended.

So far we have defined linear differential operators,the symbol of a linear operator, the ellipticity andanalytic index of a linear differential operator. We shall

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briefly introduce the notion of nonlinear differentialoperators, the linearization and the symbol of suchoperators, and the index of a nonlinear elliptic operator.

Recall that, let ir: B -+ M be a C° fiber bundle over M.Given bo e B with ir(bo) = po and local sections sl, s2 of Bdefined near po with s1(pc) = s2(po) = bo. By choosing a chartat po c M, a local trivialization of B near po, and a chartnear bo in the fiber ir"i (po) , we can define the kth orderTaylor expansions of s, and sz at p0. If the Taylorexpansions are the same for one set of choices, they will bethe same for any other (i.e., all in the same equivalenceclass), then we say that s, and s2 have the same k-jet at po.Recall again, this set of equivalence classes is denoted byJk(B)b and the equivalence class of s is denoted by jk(S)P.Let ok (B) = Ub,B Jk (B) b and let iro : Jo (B) - B maps Jok (B) b tob. It can be shown that iro has the structure of a C° fiberbundle over B whose fiber at bo is, ek .1 LS10(MP. , (BPC) b) , allpolynomial maps of degree less then or equal to k from M1,into (BP.)b,. We define a CC fiber bundle irk: Jk(B) -+ M whosetotal space is ok(B) and the projection is irk = i.e.,irk(jk(S)PO) = p0. If 0 <_ n <- k, we can generalize the fiberbundle by defining ir,k : Jnk(B) J"(B) and whose total spaceis Jk(B) and the projection is given by irflk(jk(S)PV) = j'(s)PO

. We then have a natural map, the k-let extension map, jk:C°(B) - C°(Jk(B)), defined by jk(s)(p) = jk(s)P. If is avector bundle neighborhood of s e C°(B) then Jk(O) is avector bundle, and in fact it is a vector bundleneighborhood of jk(S) a C°(Jk(B)). Furthermore, jk: C°(B) -C°(Jk(B)) restricts to a linear map of C°(f) to C°(Jk(I)).

Equipped with the knowledge of proper relationships, werecall that if 0 and if are vector bundles over M, then alinear differential operator of order k from C°(s) to C°(r)is a linear map P: C°(f) -+ C°(n) and it can be factored as acomposition C°(f) jo_ C°(Jk(,)) t*,., C°(i7) where f: Jk(§) - 17 isa C° vector bundle morphism over M. By analogy, it is clearhow to define a non-linear differential operator of order k.

Let Bi and B2 be CC fiber bundles over M. A mapping D:

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C'(B1) - C(B2) is called a non-linear differential operatorof order k from C'(B1) to C(B2 if it can be factored asC'(B1) it_ C'(Jk(B1)) f+-' C'(B2) where f: Jk(B1) - B2 is a C'fiber bundle morphism over M. Let us denote the set of alldifferential operators of order k from C'(B1) to C'(B2) (orfrom B1 to B2 for simplicity) by Dk(B,,B2) .

Theorem 3.5.4 If 1 <_ k, D1(B1,B2) c Dk(B,,B2) we canconstruct higher order differential operators from the lowerorder ones by repeating the jet extension map (or byinduction method through jet extension map) as indicated inthe following lemma.

Lemma 3.5.5 Let jk: e(B) - C'(Jk(B)) and jr': C'(Jk(B))

+ C'(JL(Jk(B))) be jet extension maps. Then is a

differential operator of order k+l from B to Jt(Jk(B)).Using the definition of Dk(B,,B2) and the above lemma, we

can prove the following:Theorem 3.5.6 If P1 E Dk(B,,B2), and P2 E Di(B2,B3), then

P2P1 E Dk+l (Bl , B3) .Now we shall see what a nonlinear differential operator

really looks like in local coordinates. Let us choosecoordinates xl,...,xm in a neighborhood U of po in M,coordinates yl,...,y,, in a neighborhood V of bo in BP andusing a local trivialization to identify a neighborhood ofba in B with U x V so that It restricted to U x V is aprojection on the first component. A section s of B over Uwith s(p) E V for p e U is given by a map p - (p,s(p)) of Uinto U x V. Let s, (p) = y1(s(p)), then jk(s) in coordinateform is given by a map p (p,s(p),y;a(jk(s)(p))) whereyia(jk(s) (p) ) = D°S1 (p) , and as usual,

D° ° a ll/ax,a' ... c3xma^', where 0 <- 1 a 1 -< k.Now let ,r' : tf - M be a Co vector bundle over M and let P

E Dk(B,n), say P = f*jk where f: Jk(B) - n is a C' fiberbundle morphism. Let vl,...,vr be a basis of Co sections of t7over U. If we restrict f to the part of Jo(B) over U x V,it is given by fj (j = 1,..,r) with coordinates

(XJ, ... ,Xm,y1.. ,Yn,Y1a) by (x,Y,Yja) _ Erj=1 f (x,Y,Y;a)vj (X) . Then

the explicit expression for Ps is135

Ps (X) = Erj=1 fj(X,S1(X),...ISn(x), Dga(X)vj(X)We shall call the ordered r-tuple of functions fj(x,y,y,)the parametric expressions for the operator P near b,relative to the choices made.

By utilizing the linear structure in n, it is possibleto single out certain vector subspaces of the vector spaceDk(B,n) of kth order differential operators from B to nwhich are polynomial functions of certain derivatives. Theseclasses play an important role in nonlinear analysis,particularly in calculus of variations.

Let B be a CO fiber bundle over M, t a C° vector bundleover M and P e Dk(B,f). Let w and 1 are integers with w > 0,and 0 <- u < k. We say that P is a polynomial of weight <5 wwith respect to derivatives of order > u (denoted by P eDkw;u(B,f)) if for each parametric representation of P asabove, and each of the functions fj(x,y,y,a) can be writtenas a sum of functions of the form G (x, y, y') y, fi' ... yiII9Y whereall Ir1 <- u, all IQ;1 > u, and Ip,I +... + 1pg1I S w. Weabbreviate Dk";O(B,f) to Dk"(B,f) and elements of this spaceare referred as polynomial differential operators of order kand weight <- w.

As a special case, elements of Dkk;k-l(B,I), i.e., the fjof a parametric representation are linear in derivatives oforder k, are called cxuasi-linear differential operators oforder k from B to t.

It is evident from the definition that D0(B,.) cDk;u(B,@) for all k and u < k. Nonetheless, it is notimmediately clear that Dkw;u(B,f) form a subspace of Dk(B,4')with positive dimension if w > 0. The next theorem shows howto construct lots of operators in Dkw;u(B,+).

Theorem 3.5.7 Let B be a e fiber bundle over M, t a C°°vector bundle over M and P e Dk(B,$). In order that P eDk";u(B,,), it is sufficient that for each b0 e B there existat least one parametric expression for P near b0 satisfyingthe conditions in the above definition of polynomialdifferential operators of weight <_ w.

It should be noted that in order to prove the theorem,

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it suffices to assume that B is a vector bundle. This is notsurprising. After all, Theorem 3.5.7 is a local statement.With a choice of coordinates, there are many formulas andresults of polynomial differential operators. Since some ofthem will involve some results in functional analysis, whichwe have neither the time nor the space to discuss, we shallonly recommend a few references for further reading [Palais1965, Ch.4; Palais 1968; Berger 1977).

Let V be a function which associates to each C° vectorbundle 4 over a compact n-dim e manifold a complete,normable, topological vector space V(4) which includesC°(0) .

Axiom A (for V) Let M and N be compact n-dim C°manifolds and let 0: M - N be a diffeomorphism of M into N.If f is a vector bundle over N, then s - is a continuouslinear map of V(0) into V(O*0).

In usual examples, this map is onto. If M c N and 0 isinclusion, then the Axiom says that restriction iscontinuous from V(4) to V($/M) and the onto-ness expressesthe possibility of extending s e V(4/M) to an element ofV(§).

Axiom B If 0 is a vector bundle over a compact C° m-dimmanifold M then V(4) c C°(4) and the inclusion map iscontinuous. Furthermore, if ^ is another vector bundle overM and f: 0 - n is a CO fiber preserving map, then f,: C°(4)C°(n) restricts to a continuous map V(f): V(0) - V(n).

Let 4 be a e vector bundle over a compact C° m-manifoldM, and let V(k) (I) = (S E Ck($) I jk(s) E V(Jk(4)) ). Thus jk isa continuous linear injection of VM(4) into the Banachspace V(Jk(0)) and Vck)(4) become a normable topologicalvector space if we topologize it by requiring that jk be ahomeomorphism into. Let Vk(4) to be the completion of VMMso that jk extends to a continuous linear isomorphism ofVk($) onto a closed linear subspace of V(Jk(4)).

With these axioms stated and notations defined, we areready to discuss briefly the linearization and the symbol ofa differential operator.

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Let Bi and B2 be e fiber bundles over a compact n-dim emanifold M and let P: C'(B1) -+ C(B2) be an element ofDk(B,,B2), let us say P = f,jk where f: Jk(B1) -, B2 is a fiberbundle morphism over M. Let V satisfy Axiom A and B so thatP extends to a C' map of Vk (Bl) into V (B2) . If s e C (B1) ,then dPs is a linear map of (Vk(Bl))S into (V(B2))PS. It canbe shown that there exists A(P)s a Diffk((Bl)S,(B2)ps), the

"linearization of P at s", such that A(P)s: C((B1)s) -C'((B2)PS) extends to dPs. Furthermore, A(P), depends only onP but not on V. The linear differential operator A(P)s has asymbol ak(A(P)S). This symbol is a function on the cotangentbundle of M, TA(M), and for (v,x) c T*(M), a(A(P)s)(v,x) is alinear map of the fiber of the tangent space of (B1)x ats(x)1, ((Bl)x)s(x), into the the fiber of ((B2)x)Pscx)Furthermore, ak(A(P)S)(v,x) depends only on jk(s)(x) and v,thus we can denote it by ak(P)(jk(s)(x),(v,x)). In otherwords, ak(P) is a function defined on the total space of thefiber product Jk (B,) xNT* (M) , which we denote by T`k (Bl) . Thereare also two natural maps ,r: T*k(Bl) - B, and f': T*k(Bl) -a B2defined by V(jk(s)(x),(v,x)) = s(x) and f'(jk(s)(x),(v,x)) _fik(s)(x) = Ps(x). But these give rise to two C' vectorbundles over T*k(Bl), i.e., n (Tf(Bi)) and f(Tf(B2)) andtheir fibers at (jk(s) (x) , (v,x) ) are ( (Bl)x)s(x) and ( (B2)x)Ps(x)respectively. Thus a (P) is an element of Hom (,r* (Tf (Bi)) ,f''(Tf(B2))), and this vector bundle homomorphism over T*k(Bl)is called the (k-th order) symbol of the differentialoperator P.

Theorem 3.5.8 Given s e C'(Bl) , djh(S)f is a e vectorbundle homomorphism of Jk(Bi)s) into (B2)ps and hence definesan element A(P)s a Diffk((Bt)s,(B2)PS) called thelinearization of P at s. If I is a vector bundleneighborhood of s in B, (so we may identify (B1)s with 0),then for a e C'(0), A(P)Sa(x) depends only on jk(s)(x) andjk(a)(x). Moreover, A(P)s(a)(x) = d/dtIt,a(P(s+ta)(x)).

Corollary 3.5.9 Let s e C'(B1), 'j a vector bundleneighborhood of s in B1 and 02 a vector bundle neighborhoodof Ps in B2. Then for a e C'(01), (P(s+ta) - Ps)/t converges

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in the C° topology to A(P)s(a) as t - 0.We shall now derive an expression for A(P)s in local

coordinates. We can suppose M = D", and we can replace B,and B2 with vector bundle neighborhoods of s and Ps, whichwe can identify with MXRm and MxRq respectively. Then asection s of B, is given by n real functions of x =(xl,...,X") a D", s(x) = (si(X),...,sm(x)) and similarly Ps isgiven by q-real functions of x, Ps(x) =((Ps)1(x),...,(Ps)q(x)) and when we say P e Dk(Bl,B2) we meanthat there exist q of C° functions fi (x,y,,y1°) (i =

and a range over n-multi-indices with jal <- k) such that(Ps),(x) = fj(x,s1(x),Pas1(x)). Recall that the f,s definedearlier are called the parametric representation of P. NowA(P)s a Diffk(B,,B2). Similarly, there are q C°-functionsgj(x,y,,yj°) such that if a = (ai,..,am) is a C°-section of B,then (P)s(a) (x) = (g1(x,a;(x),Pma;(x)), ...,gq(x,a,(x),P°a,(x))). Since A(P)s is a linear differentialoperator, gj(x,y,,y,°) are functions linear in (y,,y,°) . Thatis, there exists C°-functions of x, Al and Aja,i such that

gj(x,Y;,y,a) = E, A,j(x)y, + EQ; Aja,i(x)Y;a.The problem is to express these A;J and Aja,i in terms of thef, and s. As we shall see in the following theorem, theanswer is A, i(x) = (af/ay,)(x,s;(x),P°sj(x)) and

AJa,i = (af;/ay,a) (x, s; (x) , mss; (x) )More precisely, we have the following theorem.

Theorem 3.5.10 Let P e Diffk(B,,B2) be givenparametrically by P(s) = (f1(x,s1(x),P°s1(x)), ...,fq(x,sj(x),Pasj(x))), then A(P)s is given parametrically by

A(P)s(a) (x) =where gq(x,yj,yja) = Ej (aft/ayj) (X,s, (x) ,Past (X) )yj

+ Za; (aft/ay ) (x, s, (x) ,pas; (x) ) Yja-

Let us recall the definition of the symbol of a lineardifferential operator. Let 01 and tZ be C° vector bundlesover M and let P e Diffk(01,02). If v is a cotangent vectorof M at x, the symbol of P at (v,x) is a linear mapak(P) (v,x) of (t,)X into (12)x defined as: choosing any C°function g on M such that g(x) = 0, and dgx = v and given e

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e ($1) x choose any C' section f of 01 such that f(x) = e.

Then ak(P) (v,x)e = (k!)-1P(gkf) (x). It is immediate from thedefinition of the symbol that if P = F.jk where F is avector bundle homomorphism of Jk($1) into $2, then ak(P)(v,x)depends on F only through its value at x. Indeed,ak(P)(v,x)e = Fx(jk(gkf)(x)/k!) where g and f as before. Letus now restate the definition of the symbol of P E Dk(Bl,B2)in a more compact form.

Let B1 and B2 be CO fiber bundle over M and let P EDk(B,,B2), say P = where f: Jk(B1) - B2 is a Co fiberbundle morphism. Let Tk*(B,) be the fiber bundle over M whichis the fiber product of Jk(B1) and T*(M) and one defines themaps 7 and f' of Tk` (Bl) into B, and B2 respectively by7r(jk(S)x'(v,x)) = s(x) and f'(jk(S),(v,x) = f(jk(s)x) = PS (x)The symbol of P, ak(P), is then defined to be the element ofHom(7r*Tf(B1),f'"Tf(B2)) given by ak(P)(jk(s)x,(v,x)) =ak(Ak(P)S)(v,x) where Ak(P)S is the linearization of P at s.(See Theorem 3.5.8 for the definition of A(P)S).

The next theorem describes how to compute ak(P) in localcoordinates.

Theorem 3.5.11 Let M = D", Bt = MXRm, B2 = MxRq and P eDk(B,,B2) be given by

Ps (x) = (f1 (x,s,(x),Pas, (x)),...Ifq(x,si(x),P°s1(x)

where fj (x,y,,y,a) are C'-function of x = (x1, ... ,x") , y1 _(y1, ... ,ym) and y'a (i = 1, .. ,m and a ranges over alln-multi- indices with 1 <- Jal <- k). Since each fiber ofTf (B1) and Tf (B2) is canonically isomorphic to Rm and Rqrespectively, ak(P)(jk(s)x,(v,x)) is given by a (q x m)-matrix alJ(x,jk(s)x,(v,x)). If v = E v;dx; then this matrix isgiven explicitly by:

a ;i (x,jk(S)x, (v,x) ) = EJaJ=k va(af,/ay;a) (x,s, (x) Pas, (x)where va = via ... v"a

Let 01 and 02 be C' vector bundles over M and let P =f. jk a Diffk(0l,42) where f: Jk(&l) - 02 is a C' vector bundlemorphism. It is natural to ask what is the connectionbetween ak(P) a Hom(p*li,p*4,2) , where p: T*(M) - M is thenatural projection, and ak(P) a Hom(7r*Tf ($,) ,,r*Tf ($2)) , where

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7r: Tk*($1) -+ 0, is the natural map.Now if g: T' *(01) = Jk (,,) xMT* (M) - T*(M) is the natural

projection, then g*p*(I,) = 7r*Tf(&,) and g*p*(02) = f*Tf(02), sowe can regard ak(P) as an element of Hom(g*p*(§,),g*p*(§2)when P is considered as a nonlinear operator. Thecomposition with g maps

Hom(p*+1,p*02)

intoHom(g*p*(01) ,g p*(I2)) where "ak(P) = ak(P) g" expresses therelation between the nonlinear ak(P), the left hand side,and the linear ak(P), the right hand side, through thecomposition of g. The whole point is that if P is linear,then A(P), = P for all s e C°(01), so A(P)S is independent ofs, and ck(P) (jr(s) , (v.x) ) = ak(A(P)S) (v, x) = ak(P) (v, x) doesnot depend on jk(s), i.e., ak(P) is lifted from T*(M).

Let us now introduce the notion of an ellipticdifferential operator.

An element P e Dk(B,,B2) is called an ellipticdifferential operator of order

11

k from B1 to B2 if for all

(jk(S) , (V,x) ) c Tk*(Bl) with v + o, ak(P) (jk(S) , (V,x) )T((B7)x)s(x) - T((B2)x)Ps(x) is a linear isomorphism. We willdenote the set of all such elliptic differential operatorsby Elpk(B,,B2). Note that Elpk(B,,B2) is nonempty only if B,and B2 have the same fiber dimension.

Recall that if +1 and 02 are e vector bundles over Mthen the set Ellk(tl,42) of k-th order linear ellipticdifferential operators from t, to 412 is the subset of P eDiffk(fl,02) such that ak(P)(v,x) is a linear isomorphism of($yx onto (12)x for all (v,x) a Mx* with v + 0. Obviously,

Ellk(01,02) Q Elpk(011§2).Theorem 3.5.12 If P

OEElpk (B, , B2) then A(P) S e

Ellk((Bt)s,(B2)vs) for all s e C°(B1). Conversely, if eachelement of Jk(B1) is the k-jet of a global section of B1,then P e Elpk(B,,B2) provided each linearization of P iselliptic.

This theorem gives some conditions for the linearizationof nonlinear elliptic differential operators. In theremainder of this section, we shall give a definition andsome properties of the analytic index of a nonlinear

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elliptic operator.For a given e fiber bundles B1 and B2 over a compact C°°

manifold M without boundary, we will associate to each P eElpr(Bi,B2) an element ia(P) E K(C°(B1)) called the analyticindex of P. If we choose a "base point" so for C°°(B1) then wehave a canonical augmentation map dim: K(C°(B1)) - Z anddim(ia(P)) turns out to be the usual analytic index of thelinearized elliptic operator A(P), : C°((B1)s) - Cm((B2)P5) ,i.e., dim (Ker (A (P) s )) - dim (coKer (A (P) s )). Note that, in

general, when C(B1) is not homotopically trivial, ia(P) cancarry more information about P than does its "dimension".

Let X and Y be C' Hilbert manifolds and let f: X -+ Y bea C' map. We say f is a Fredholm may if df: TX - f"TY is aFredholm bundle morphism over X and in this case we defineind(f) a K(X) by ind(f) = ind(df). More generally if j: f2-+X is a continuous map we say f is a Fredholm map relative toj if j*TX y j*f*TY is a Fredholm bundle morphism (i.e.,if dfi(w): TXj(w) -+ TYfJ(w) is a Fredholm operator for each w ef2), and we define ind(f,j) a K(fl) by ind(f,j) >_

Note that if f: X - Y is a Fredholm map then clearly f isalso a Fredholm map relative to any j: f! - X and thenind(f,j) = j*ind(f) where j*: K(X) - K(f2) is the functorialK(j)

In order to define K for an arbitrary space X, the basicrequirement is that K should be a functor from spaces andhomotopic classes of maps to abelian groups. For compactspaces, such a functor is naturally isomorphic to theGrothendieck group of vector bundles and is representable,i.e., K(X) should be naturally equivalent to [X,C], thehomotopy classes of maps of X into a homotopy abelianH-space C. This uniquely determines C up to homotopy type,i.e., C = Z x BG where BG means the classifying space of G,and G is either lin O(n) or lin U(n) depending on whether wemean Ko or K,. By a theorem proved by Janich [1965], andAtiyah independently, there is a nice choice of C for ourpurpose. Let H be a separable, infinite dimensional Hilbertspace and let Fred(H) be the space of Fredholm operators on

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H, i.e., bounded linear maps. Let T: H - H be such a mapsuch that Ker T and coker T are finite dimensional. T(M) isthen automatically closed in H so we can take cokerT = T(H)J-= Ker(T*). Fred(H) is topologized as a subspace of the spaceof all bounded operators on H with the usual norm 1ITh =sup(IITxII I Jlxii = 1). If X is a topological space, then acontinuous map f: X - Fred(H) is called admissible if Ker(f)= ((v,x) e H x X1 v e Ker f(x)) and coker(f) = {(v,x) e H xXI v e Im f(x)1) are vector bundles over X under the obviousprojection. Then it is known that if X is a compact spaceand r e [X,Fred(H)] then r has an admissible representativeg and the element ind(r) = [Ker(g)] - [Coker(g)] of theGrothendieck group K(X) of vector bundles over X is welldefined and independent of the choice of admissible g, andind: [X,Fred(H)] - K(X) is a bijection. Furthermore, Fred(H)is homotopy abelian H-space under usual operatorcomposition, making [X,Fred(H)] an abelian group, and ind isa group isomorphism.

Now let X be a paracompact space and let Bi and B2 beHilbert space bundles over X with GL(H), the general lineargroup of Hilbert space with the norm topology as structuralgroup. A Hilbert bundle morphism f: Bi - B2 is a Fredholmbundle morphism if fx: (BM)X -+ (B2)x is a Fredholm map foreach x e X. In this case, by a theorem of Kuiper whichstates that GL(H) is constructable, thus there exist bundleisomorphisms g: X x H = B1 and h: B2 = X x H. Then x hxfxgxis a map hfg: X - Fred(H). By the contractability of GL(H),it follows that g and h are well determined up to homotopyand hence the homotopy class of hfg is a well determinedelement of [X,Fred(H)] = K(X) and we denote by ind(f).

Now we are ready to define the index of a nonlinearelliptic differential operator.

Definition and Theorem 3.5.13 Let M be a compact n-dimC° manifold without boundary, let B1 and B2 are C° fiberbundles over M, and let P e Elpr(Bi,B2). Let k > n/2+r, thenP: C°(B1) - C°(B2) extends to a C° map of Hilbert manifoldsPck): Lz k(Bl) - L2 k-r(B2) . Then PM is a Fredholm map relative

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to the inclusion map rk: C° (B1) - L2 k (Bl) and moreoverind(P(k) ,I'k) a K(C°(B,)) is independent of k and hence definesan element ie(P) a K(C°(Bl)) called the analytic index of P.

For more detailed definitions of topological andanalytic index of operators, the index theorem for manifoldswith boundary and other applications of the index theorem,see articles in Palais [1965].

This chapter, in particular, discusses the subjectswhich are not extensively utilized in the subsequentchapters, at least not explicitly. Nonetheless, some of theconcepts and even terminology do find their way to our laterdiscussions. This chapter is included, and indeed islectured, to prepare the students with some concepts andunderstanding about global analysis in general, and sometechniques important to the global theory of dynamicalsystems. In particular, the reader may find it useful whenthey venture to theoretically oriented research literature.

We have reviewed a broad range of mathematical conceptsand techniques useful for our discussion, we can now turn toour main interest, namely, the nonlinear dynamical systemsand their structural stability.

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Chapter 4 General Theory of Dynamical Systems

4.1 IntroductionIn order to place our discussion of dynamical systems in

proper perspective, let us discuss briefly three aspects ofthe theory of dynamical systems. For lack of a betterdescription, we shall refer to them as the local, theglobal, and the abstract theories.

The local theory is concerned with the application ofgeometrical and topological methods to the qualitative studyof differential equations. The general setting is a set ofdifferential equations in R" and one is interested inasking: "What does the omega-limit set look like?"; "Whathappens in the neighborhood of a fixed point?"; "Is itstable?", etc.

The object for the study of the global theory is the setof vector fields on a manifold. One is interested incharacterizing the structurally stable vector fields and instudying the "orbit picture" of the flow associated with agiven vector field.

The setting of the abstract theory is a generaltransformation group but the notions studied are thosearising in the qualititive study of differential equations.One can show that many of the results for differentialequations are valid in a much broader domain.

In current literature, the abstract theory is known astopological dynamics, the global theory is known as smoothdynamical systems, and the local theory is known asqualitative theory of diffferential equations. Topologicaldynamics deals with continuous actions of any topologicalgroup G on a topological space X. Smooth dynamical systemsare smooth actions of the group R or Z on a differentiablemanifold M. We shall begin by illustrating a few fundamentaldefinitions with some simple examples. Most of thesedefinitions and examples are also common to the qualitativetheory of differential equations. Indeed, the latter theoryprovides the proper intuition and phenomena for the

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development of dynamical systems.Let G be either the additive topological group R of real

numbers or the additive topological group Z of integers. Adynamical system on a topological space X is a continuousmap 0: G x X X such that for all x e X, for all g, h e G,o(g+h,x) = O(g, O(h,x)), and 0(0,x) = x. (4.1-1)

The space X is called the phase space of 0. If X is adifferentiable manifold and ¢ is a Cr map, r >- 0, then wecall 0 a Cr dynamical system.

For instance, for any X the trivial dynamical system isdefined by O(t,x) = x. For X = R', O(t,x) = etx defines a C"dynamical system on X.

Let 0 be a dynamical system on X. Given t e G, we definethe partial map Ot: X - X by Ot(x) = 0(t,x). If G = R, Ot issometimes called the time map of m. Likewise, given x e X,we define the partial map Ox: G -, X by Ox(t) = 0(t,x). Notethat if 0 is Cr, then so are Ot and Ox. Then Eq.(4.1-1) canbe written as O9Oh, and 0° = id. (4.1-2)Sometimes for brevity we denote O(t,x) by when underthe context there is no confusion. With this convention,Eqs. (4.1-1) and (4.1-2) become:

(g + and x. (4.1-3)

Proposition 4.1.1 For all t e G, Ot is a homeomorphism.If 0 is Cr then 0 is a Cr diffeomorphism.

Exercise Prove this by the definition of homeomorphismand diffeomorphism.

Note that, if G = R, then the dynamical system 0 iscalled a flow on X, or an one-parameter group ofhomeomorphism of X.

Let 0 be a dynamical system on X. We define a relationon X by putting x - y iff there exists t e G such that q,t(x)= y.

Proposition 4.1.2 The relation - is an equivalencerelation.

The equivalence classes of - are called orbits of 0 (orof the homeomorphism 0' in the case G = Z). For each x e X,the equivalence class containing x is called the orbit

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through x. It is the image of the partial map Ox: G - X. Wesometimes denote it by Thus Prop. 4.1.2 implies thattwo orbits either coincide or are disjoint. We denote thequotient space X/- by X/O and call it the orbit space of 0.The quotient map, which takes x to its equivalence class, isdenoted by y0: X - X/O or just y: X - X/O. As usual, we giveX/O the finest topology with respect to which y iscontinuous. (That is, a subset U of X/O is open in X/O iffy"1(U) is open in X). Now let us look at some examples of

dynamical systems.Earlier we have pointed out that if G = R, the trivial

dynamical system on any X is the point x e X, and the orbitis the set of single element {x}. For the nontrivial flow

etx, there are three orbits, namely, the origin, thepositive and negative half lines.

0

The arrows on the orbits indicate the orientations inducedby the flow. If, however, for all t,x e R we put x +

t, then this flow has only one orbit, R itself.For any 0 e R, put [x + 6t] where [x] is a fixed

value of S1. If 0 = 0, we have the trivial flow on S1,otherwise we have the single orbit S1. If we imbed S' in theplane by the standard imbedding [x] - (cos 2rx, sin 2rx),then the rotation is counter-clockwise if 0 is positive, andclockwise if 0 is negative. We call 0 the angular speed ofthe flow.

For flows on R2, it is sometimes more convenient toidentify R2 with the complex line C because the two aretopologically indistinguishable. The simple non-trivial flowis for all t e R, and for all (x,y) a R=, put(xet,yet). The origin is the only point orbit, and all otherorbits are open rays radiating from the origin.Fig.4.1-1(a). If we change the formula slightly to(xet,ye"t), the phase portrait is radically changed, this isbecause the new flow has only two orbits beginning at the

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origin. It is associated with a saddle point of the flow..

Fig.4.1-1(b).

LFig.4.1-1(a) Fig.4.1-1(b)

For all t e R and z = x + iy a C, put t z = ze't. Then theorigin is a point orbit and the other orbits are all circleswith center at the origin. But if t z = ze('-')t, then theorigin is a point orbit, and all other orbits spiral intowards it. See Fig.4.1-2.

Fig.4.1-2There are several ways to construct new dynamical

systems from the given ones. The most simplest and directone is the product. Let 0: G x X - X and g: G x Y - Y bedynamical systems. The product of the two systems is adynamical system on X x Y defined for all g e G and (x,y) e

X x Y by For example, let 0 be the flowetx and µ be the rotation flow t- [x]= [x + At] for 0 e

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R. Then 0 x µ is a flow on the circular cylinder R x S1. Thecircle (o)xS' is an orbit, and all the other orbits spiralaway from it.

SI

0

Fig.4.1-3Let o: R x X - X be any flow. For any t e R, Ot: X- X

is a homeomorphism which generates a discrete dynamicalsystem µ: Z x X - X, then we say Ot (or µ) is imbedded in 0.For instance, any rotation of S' is imbedded in anynon-trivial rotation flow on S1. It is not true that everyhomeomorphism of every topological space X can be imbeddedin a flow on X.

In general, every flow on X yields a homeomorphism of X(in fact, many homeomorphisms), but the reverse is notusually true. Nonetheless, we can associate with a givenhomeomorphism f a flow with similar properties provided weallow the flow to be on a larger space Y. Thus we have thespace X imbedded in Y and the homeomorphism of X imbedded ina flow 0 on Y. The way to construct the larger space Y andthe flow 0 is known as suspension. It should be noted thatthis is different from the suspension discussed in algebraictopology (see, for instance, Bourgin [1963]; Greenberg(1967); Eilenberg and Steenrod [1952]). We shall not getinto this any further, except to say that when reading theliterature, be aware of the difference of meaning of terms.It is fairly easy to distinguish them under very differentcontext.

Let f: X -+ X be a homeomorphism generating a discretedynamical system µ. Let - be the equivalence relationdefined on R x X by (u,x) - (v,y) iff u = v + m for some m e

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Z and y = fm(x). Then there is a flow 0: R x Y - Y on Y (R

x X)/- defined by O(t,[u,x]) = [u+t,x], where [u,x] denotesthe equivalence class of (u,x) e R x X. The flow 0 is calledthe suspension of the homeomorphism f (or of µ). Clearly,for any u c R, the restriction of 0 to any cross-section[u,X] with the obvious identification [u,X] = X coincideswith f.

If f: R - R is defined by f(x) = -x, then the suspensionis a flow on the open Mobius band, and all its orbits aretopologically circles.

Y = Rz/-

We shall discuss rational and irrational flows on atorus T2 = S1 x S' and we shall show that all rational flowsare essentially the same, but there are infinitely manydifferent types of irrational flow. It should also bepointed out here that the phase portrait of a product flowis not uniquely determined by the phase portraits of itsfactors. This is because the phase portraits of rational andirrational flows are completely different topologically, butthey both come from factors with identical phase portraits.We shall discuss this point in detail later.

Let 0: G x X - X be a dynamical system on X, let a: G -G be a continuous automorphism of the additive group G, andlet h: X -+ Y be a homeomorphism. Then µ: G x Y - Y, µ =

(axh)-', is a dynamical system on Y. Then µ is thedynamical system induced from 0 by the pair (a,h) (or by h,

if a = id).The simplest example is that if h = id: X - X, and a =

-id: R - R, we then obtain 0-, the reverse flow of 0, by

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0_'(t, x) = 0(-t,x). That is, in O points moving along theorbits of 0 at the same speed but in opposite direction.

The construction of induced systems is not particularlyinteresting, but the generalization to quotient systems doesproduce new systems.

Let 0: G x X - X be a dynamical system on X and let - bean equivalence relation on X such that for all t e G and allx, y e X, O(t,x) - 0(t,y) iff x - y. Then 0 induces adynamical system µ, called the quotient system on thequotient space X/- by µ(t,(x]) = [O(t,x)] where t e R and[x] is the equivalence class of x e X.

If f and g are commuting homeomorphisms of X, i.e., fg =gf, then f takes orbits of g onto orbits of g, thus inducesa homeomorphism of the orbit space of g. An example of aquotient system 0, which is the discrete dynamical systemgenerated by f and - is the equivalence relation givingorbits of g as equivalence classes. Similarly, if 0 and µare commuting flow on X, i.e., Osµt = µtgs for all s, t e R,then 0 induces a quotient flow on the orbit space X/µ.

Let us define an equivalence relation - on R" by x - yif f x- y e Z". Then the quotient space R"/_ is the n-dimtorus T" = S1xS1x...xS1, the Cartesian product of n copies ofthe circle S1. Let f be a linear automorphism of R" whosematrix A, with respect to the standard basis of R", is inGL"(Z). That is, A has integer entries and detA = ±1. Then fmaps Z" onto itself and thus f and f1 preserve theequivalence relation. Thus f induces a homeomorphism (infact, a diffeomorphism) of T. The induced homeomorphism ofT" is called a hyperbolic toral automorphism. For anyhyperbolic toral automorphism g: T" - T", its periodic andnon-periodic point sets are dense in T. Note that, for apoint x e T" which is periodic if gr(x) = x for some r > 0.We shall see that hyperbolic toral automorphisms are thesimplest examples of Anosov diffeomorphisms on compactmanifolds. Originally, the hyperbolic toral automorphismswere counter-examples to the conjecture that structurallystable diffeomorphisms have finite periodic sets.

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Theorem 4.1.3 The periodic set of the hyperbolic totalautomorphism g: T" - T" is precisely Q"/_, where Q is therational numbers.

Remark: One can verify that induced and quotientsystems are dynamical systems.

Let 0 be a dynamical system on a topological space X.For each x c X, the subset Gx = (g a G, O(g,x) = x) is asubgroup of G and is called an isotropy subgroup (orstabilizer) of x (or of 0 at x).

Proposition 4.1.4 If X is a Ti space, then for all x eX, Gx is a closed subgroup of G.

Proposition 4.1.5 If X is T2 and G/Gx is compact, thenthe orbit is homeomorphic to G/Gx.

Proposition 4.1.6 Every orbit of every flow isconnected.

The definitions and propositions in this section arecommon to the theory of dynamical systems in general. Thenext section will introduce various equivalence relationsand conjugacy, which are essential to the introduction oflimiting sets.

4.2 Equivalence relationsTo classify dynamical systems is one of the center

themes and is of special interest to the subject. One beginsby placing certain equivalence relations upon the set of alldynamical systems. Such equivalence relations should benatural in the sense that the systems have qualitativeresemblance. Equipped with the equivalence relation, one canform the equivalence classes and be able to distinguish themby means of algebraic or topological invariants (quantitiesthat are associated with all systems and are equal for allsystems in the same equivalence class). A goodclassification scheme requires a careful choice of"intrinsic" equivalence relations with tractable invariants.We shall consider several "obvious" equivalence relations.

In order to appreciate the difficulties involved in the

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classification problem, one only has to look at thesituation when the manifold is the circle S1. See, forinstance, (Irwin 1980, Section 2-II; Nitecki 1971, Chapter

1)

Let f: X - X, g: Y - Y be homeomorphisms of topologicalspaces X and Y. A topological conjugacy from f to g is ahomeomorphism h: X - Y such that That is, thefollowing diagram commutes:

X f ' X

h 1 1 h

Y 9 YIf such a homeomorphism h exists, then the homeomorphisms fand g are said to be topologically conjugate. Clearly,topological conjugacy is an equivalence relation. It is easyto show that a topological conjugacy maps orbits ontoorbits, periodic points to periodic points, and it alsopreserves periods.

In our discussion, we are mainly concerned withdifferentiable manifolds M and N, and f and g arediffeomorphisms. It seems natural to require the map h to bea diffeomorphism. This gives the notion of differentiableconjugacy. It is a stronger relation than topologicalconjugacy, and in general, there are many more equivalenceclasses with respect to it. Nonetheless, with differentiableconjugacy, we do find stable diffeomorphisms, ones whichstay in the same equivalence class when slightly perturbed,which are very rare. Moreover, we also have to classify asnon-equivalent diffeomorphisms which most people would feelare qualitatively the same such as the contractions x - x/2,and x -+ x/3 of the real line. For these reasons, topologicalconjugacy remains as the basic equivalence relation evenwhen we are dealing with a differentiable category.

Let 0 and µ be flows on topological spaces X and Yrespectively. h: X -+ Y is a flow map from 0 to u if it iscontinuous and if there exists an increasing continuous

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homomorphism a: R - R such that the diagram

R x X I y Xaxh 1 1 h

R x Y Y

commutes. Recall that a is just multiplication by a positiveconstant. If h is a homeomorphism, we then call (a,h) or h,in the case a = id, a flow equivalence from m to u. We saythat 0 is flow equivalent to u if such a pair of (a,h)exists. In this case, p is the flow induced on Y from 0 by(a,h).

The definition of flow equivalence seems to be verynatural. But it is rather too strong for the qualitativetheory of flows. For instance, it preserves the ratios ofperiods of closed orbits, but flows may differ in thisrespact and yet have a very similar appearance. We nowdefine a weaker equivalence relation which is regarded as abasic notion. We call h: X - Y a topological equivalencefrom m to u if it is a homeomorphism, which maps each orbitof ¢ onto an orbit of µ, and it preserves the orientation oforbits. For instance,

h

We have remarked earlier that it is undesirable tostrengthen "topological" to "smooth" in the definition ofconjugcy of homeomorphisms. The arguments apply equally wellto the definition of equivalence of flows.

We say that homeomorphisms f and g of a topological

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space X are topologically equivalent if there is ahomeomorphism h of X which maps orbits of f onto orbits ofg. This relation is different from topological conjugacy,but oftenly they are the same. One can also show that flowequivalence implies topological equivalence and both areequivalence relations.

In some situations, we need to deal with topologicalconjugacy, topological equivalence and flow equivalence in alocal form. It is often possible to modify the definitionsof such relations by paraphrasing them as "in someneighborhood of a given point". Even if the "local"definition does not make perfect sense, it often pointstoward a sensible concept.

Let U and V are open subsets of X and Y respectively andlet f: X - X and g: Y -+ Y be homeomorphisms. Somewhatabusing the notation, we say that flU is topologicallyconjugate to gdV if there is a homeomorphism h: U v f(U) -+ V

u g(V) such that h(U) = V and, for all x e U, hf(x) = gh(x).If p e X and g c Y, we say that f is topologically conjugateat p to g at a if there exists open neighborhood U of p andV of q such that flu is topologically conjugate to glv andby a conjugacy h which takes p to q.

Let 0: D - X be a continuous map where X is atopological space and D is a neighborhood of (0) x X in R xX. We write as before, for O(t,x) and DX = (t a R:

(t,x) e D). We say that is a partial flow on X if, for all xE X,

(i) DX is an interval,

(ii) O- x = x(iii) for all t e DX with s e Dt,X, (s+t) x = s (t x) ,(iv) for all t e DX, Dt,X = (s-t : s e Dx).

That is, 0 is a flow not defined for all time. (iv) implies

m is maximal and it cannot be extended.Proposition 4.2.1 Let 0: D - X be a partial flow on X.

Then (i) D is open in R x X, (ii) if DX = (a,b) with b < oo

0 cannot be extended to a continuous map of D U ((b,x)) intoX, (iii) if x is a fixed point or periodic point of 0, then

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Dx = R.It is clear that flow equivalence and topological

equivalence for partial flows can be defined in astraightforward way. If µ is a partial flow on a topologicalspace Y, then a flow equivalence from m to u is a pair (a,h)where h: X -+ Y is a homeomorphism, a: R R is amultiplication by a positive constant, andfor all (t,x) a D. And a topological equivalence from 0 to uis a homeomorphism h: X - Y that maps all orbits of 0onto orbits of µ and preserves their orientation.

Proposition 4.2.2 Flow equivalence and topologicalequivalence are equivalence relations on the set of allpartial flow on topological spaces.

Let 0 be any flow (or partial flow) on X and let U be anopen subset of X. Once again, we are abusing the notationand defining a map oIU: D - U by ((plU)(t,x) = 0(t,x) where D

= uxeuDxx(x) of R x U, and call olU the restriction of theflow 0 to the subset U. Notice again, this is an abuse ofnotation, since U is not even in the domain of 0. One caneasily check that olU is a partial flow on U. If µ is a flowon a topological space Y and V is an open subspace of Y,then we say that olu is flow or topologically equilvalent toµ1V if they are equivalent as partial flows. If p e X and ge Y, we say that 0 is equivalent at p to µ at q, if thereexists open neighborhoods U of p and V of q and anequivalence from OIU to /IV taking p to q.

Proposition 4.2.3 Flow equivalence and topologicalequivalence are equivalent relations on ((O,p): 0 is a flow,p e phase space of 0). If okp is flow equivalent to g1q,then 01p is topologically equivalent to Ajq.

4.3 Limit sets and non-wandering setsLet 0 and µ be flows on topological spaces X and Y.

Suppose h is a topological equivalence from 0 to g, then hmaps the closure t in X of each orbit r of 0 onto theclosure h(I') of h(t) in Y. Thus, h maps the set F/F onto the

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set h(r)/h(r).We want to study the orbit with very large positive and

negative t. Let It = [t,oo) . The fl-set fl(x) of x e X withrespect to the flow 0 is defined by fl(x) = ox(It).Intuitively, fl(x) is the subset of X thatt - -. It follows from the definition of flow that for all te R, fl (t, x) = fl (x) . Thus we may define the fl-set, fl (r) ofany orbit r of 0 by fl(r) = fl(x) for any x e r. Note that ifr is a fixed point or periodic orbit then ox(It) = r for allx c r and t e R, and fl(r) = r. Thus fl(r) is not necessarilypart of I'/r. Similarly, the a-set of a point x e X isdefined by a(x) = nt,R Ox(Jt), where Jt = (- w,t], and thea-set a(r) of an orbit r is denoted by a(r) = a(x) for any xc r. Again, intuitively, the a-set is the subset that

t - --. Since the fl-set and a-set are"symmetric" in time, we shall confine our attention tofl-set, and the corresponding results for a-sets are exactlyanalogous.

Let a vector field X E X(M) and let 0 be the flow of X.The orbit of X through p e M is the set r(p) = (o(t,p)l t eR, p e M). If X(p) = 0, the orbit reduces to p, then we saythat p is a singularity of X. Otherwise, the map a: R - M,a(t) = 0(t,p) is an immersion. If a is not injective, thereexists (3 > 0 such that a(p) = a(0) = p and a(t) + p for 0 <t < P. In this case, the orbit of p is diffeomorphic to S'and we say that it is a closed orbit with period P. If theorbit is not singular or closed, it is called regular. Thusa regular orbit is the image of an injective immersion ofthe line.

The fl-limit set of p e M, fl(p) = (q a M, for which thereexists a sequence t" -+ u with 0(t",p) -+ q). Simiarly, wedefine the a-limit set of p, a(p) = (q a MI O(t,,p) - q forsome sequence t" -+ -00 ). Thus, a-limit of p for the vectorfield X is R-limit of p for the vector field -X. Also, fl(p)= fl(p') if p' belongs to the orbit of p. Intuitively, a(p)is where the orbit of p is "born" and fl(p) is where it"dies".

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Example 1. Let us consider the system of differentialequations defined in R2 (in polar coordinates):

dr/dt = r(1 - r), dO/dt = 1.It can easily be shown that the solutions are unique and allsolutions are defined on R2. Thus, the system defines adynamical system. The orbits are shown in the following.They consist of (i) a critical point, namely the origin 0,(ii) a periodic orbit r coinciding with the unit circle,(iii) spiraling orbits through each point p = (r,6) with r0, r + 1. For points p with 0 < r < 1, the n-limit set of pis the unit circle and the a-limit set of p is the (0). Forpoints p with r > 1, n(p) is the unit circle and a(p) is anempty set.

Example 2. Consider the unit sphere S2 R3 with centerat the origin and use the standard coordinates (x,y,z) inR3. Let pn = (0,0,1) and ps (0,0,-1) be north and southpoles of S2 respectively. Let the vector field X on S2 be X= (-xz, -yz, x'+y=). Clearly, X is C°, and the singularitiesof X are pn and ps. Since X is a tangent to the meridians ofS2 and points upwards, n(p) = pn and a(p) = ps if p 6 S2 -

(Pn,Ps).Pn

PS

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Theorem 4.3.1 (Poincare-Bendixson) Let X e X(S2) with afinite number of singularities, and let p e S' and fl(p) bethe fl- limit set of p. Then one of the followingpossibilities holds:

(i) fl(p) is a singularity,(ii) fl(p) is a closed orbit,(iii) fl(p) consists of singularities pl,...,p, and

regular orbits such that if r c fl(p), then a(r) = pi, andn(r) = pj .

Example Let X be a vector field on S2 as in thefollowing:

Pn

Ps

Both pn and ps are singularities and the equator is a closedorbit. The other orbits are born at a pole and die at theequator.

Proposition 4.3.2 Let r be an orbit of 0 and let o-(r)and fl- (r) denote the a-set and fl-set of r as an orbit of '.Thena' (r) = fl (r) and n' (r) = a (r) .

Proposition 4.3.3 Let h: X - Y be a topologcialequivalence from 0 to g. Then for each orbit r of 0, h mapsfl(r) onto fl(h(r)), the fl-set of the orbit h(r) of µ.

Proposition 4.3.4 Let r be an orbit of 0. Then fl(r) isa closed subset of the topological space X, and fl(r) c F. Infact, r = r u a(r) u fl(r). Moreover, if 6 is another orbitof 0 such that r c fZ (d) , then fl (r) c n(6).

A fl-limit cycle (or a-limit cycle) is a closed orbit rsuch that r belongs to fl-set (or a-set) of x for some x

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which does not belong to r. Note that, limit cycles enjoycertain properties not shared by other closed orbits,namely, if r is an fl-limit cycle, then there exists x not onr such that limt, d(0,(x),r) = 0. Geometrically, this meansthat some trajectory spirals toward r as t -, w. For ana-limit cycle, replace ao by --.

We list some other results of limit cycles for futurereference.(A) Let r be an fl-limit cycle. If r = il(x), x is not on r,

then x has a neighborhood V such that r = fl(y) for all y eV. In other words, the set (yJ r = fl(y)) - r is open.(B) A nonempty compact set K that is positively ornegatively invariant contains either a limit cycle or anequilibrium.(C) Let r be a closed orbit and suppose that the domain W ofthe dynamical system includes the whole open region Uenclosed by r. Then U contains either an equilibrium or alimit cycle. Indeed, one can state a much sharper one.(D) Let r be a closed orbit enclosing an open set Ucontained in the domain W of the dynamical system. Then Ucontains an equilibrium.(E) Let H be a first integral of a planar C' dynamicalsystem. If H is not constant on any open set, then there areno limit cycle.

Any union of orbits of a dynamical system is called aninvariant set of the system. We then have the following:

Proposition 4.3.5 Any fl-set of 0 is an invariant set of

0.Consequently, if fl(r) is a single point p, then p is a

fixed point of 0.We may have noticed in linear examples that an orbit may

have an empty fl-set, and this seems to be associated withthe orbit "going to infinity". Thus, it is reasonable tosuppose that if we introduce some compactness conditions, wecan ensure non- emptiness of the fl-sets. Indeed, we have thefollowing propositions:

Proposition 4.3.6 If X is compact and Hausdorff, then,

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for any orbit r, n(r) is non-empty, compact and connected.A complimentary statement is the following theorem: If X

is locally compact, then a n-limit set is connected wheneverit is compact. Furthermore, whenever a n-limit set is notcompact, then none of its components is compact.

We have noticed earlier that if r is a fixed point or aperiodic orbit, then n(r) = r. In fact, one can show thatunder very general conditions, some r are periodic. We nowwant to investigate this relation more closely.

Proposition 4.3.7 Let X be compact and Hausdorff, thenan orbit r of q is closed in X iff n(r) = r. (That is,

closed orbits are periodic or fixed points).Theorem 4.3.8 Let X be Hausdorff. Then an orbit r of 0

is compact iff it is a fixed point or a periodic orbit.Corollary 4.3.9 Let X be compact and Hausdorff, then

the following three statements on an orbit r of 0 areequivalent:

(i) r is a closed subset of X,(ii) r is a fixed point or a periodic orbit,(iii) n(r) = r.This Corollary can be generalized to non-compact but

locally compact X.A minimal set of a dynamical system is a non-empty,

closed invariant set that does not contain any closedinvariant proper subset. By using the Zorn's lemma, one canprove that if, for any orbit r of 0, P is compact, then itcontains a minimal set.

For discrete dynamical systems, the theory of a- andn-limit sets can also be developed similarly. If f is ahomeomorphism of a topological space X, then the n-set n(x)of x e X with respect to f is defined by n(x) = nncN ( fr(x):

r >- n). Again, the a-set a(x) of x is the n-set of x withrespect to f-1. All results of a- and n-sets have analoguesform except that n-sets of homeomorphisms need not beconnected.

When classification of dynamical systems becomesdifficult, there are new equivalence relations with respect

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to which classification can be made easier. One of the mostimportant concepts is concerned with an invariant set calledthe non-wandering set which was defined by Birkhoff [1927].The intuitive idea is the following. If one compares phaseportraits of dynamical systems, such as the following two:

It seems that certain parts are qualitatively more important(or attract more attention) than others. For instance, if wewere asked to pick out the more significant features of theleft hand picture, we would inevitably begin by mentioningthe fixed points and closed orbits. In general, qualitativefeatures in a phase portrait of a dynamical system can betraced to sets of points that exhibit some form ofrecurrence. Of course, the strongest form of recurrence isperiodicity, where a point resumes its original positionarbitrarily often, but there are weaker forms which are alsoimportant. We shall define such recursiveness and some ofits properties.

Note that, for two-dimensional flows, all the possiblenonwandering sets fall into three classes: (i) fixed points,(ii) closed orbits, and (iii) the unions of fixed points andthe orbits connecting them [Andronov et al 1966]. The thirdsets are referred to as heteroclinic orbits when theyconnect distinct points, and homoclinic orbits when theyconnect a point to itself. Closed paths formed ofheteroclinic orbits are called homoclinic cycles. It is

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worthwhile to note that the fixed points contained in suchcycles must all be saddle points, if they are hyperbolic,this is because sinks and sources necessarily have wanderingpoints in their neighborhoods.

A set A c X is said to be positively recursive withrespect to a set B c X if for T e R there is a t > T and anx c B such that a A. Negative recursiveness can bedefined by using t < T. A set A is self positively recursiveif it is positively recursive with respect to itself.

A point x e X is positively Poisson stable if everyneighborhood of x is positively recursive with respect to(x }

Theorem 4.3.10 Let x e X. The following statements areequivalent:

(i) x is positively Poisson stable,(ii) given a neighborhood U of x and a T > 0,

t

e n(x),

(iv) r+ (x) = n(x), where r+ (x) _ (O(t,x)I 0 <_ t < a),

(v) r(x) c n(x),(vi) for every e > 0, there is a t >_ 1 such that e

B(x,e), where B(x,e) is an open ball centered at x withradius e. Note also that if x is positively Poisson stable,then so is t e R.

The following alternative definition of Poissonstability is customary in the literature and is clearlysuggested by the above theorem.

A point x e X is positively or negatively Poisson stablewhenever x e n(x) or x e a(x) respectively. It is Poissonstable if it is both positively and negatively Poissonstable.

Theorem 4.3.11 F+(x) = n(x) iff s is a periodic point.Remark: Note that, if r+(x) = n(x) then the point x is

indeed Poisson stable. The following example shows thatthere exist points which are Poisson stable but notperiodic.

Example: Consider a dynamical system defined on a torus

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by the differential systemdx/dt = f(x,y), dy/dt = af(x,y)

where f(x,y) = f(x+l,y+l) = f(x+l,y) = f(x,y+l), and f(x,y)> 0 if x and y are not both zero, f(0,0) = 0. Let a > 0 beirrational. It is clear that there is a fixed point p =(0,0). There is only one orbit ri such that a(r1) = (p), andexactly one orbit r2 such that fl(r2) = (p). For any otherorbit r, a(r) = fl(r) = the torus. Moreover, fl(rl) = a(r2) _the torus. It is also clear that points on r1 arepositively, but not negatively, Poisson stable. Likewise,points on r2 are negatively, but not positively, Poissonstable. All other points are Poisson stable. But, no pointexcept the fixed point p is periodic.

Pr

a

P

The following theorem sheds some light on positivelyPoisson stable points x when r+ (x) + fl(x).

Theorem 4.3.12 Let X be a complete metric space. Let xc X be positively Poisson stable, and let it not be aperiodic point. Then the set fl(x) - r(x) is dense in n(x),i.e.,

n(x) - r(x) = fl(x) = r(x) .Corollary 4.3.13 If X is complete, then r(x) is

periodic iff r(x) = fl(x).

This corollary is closely related to Corollary 4.3.9.Now let us introduce the definition of a non-wandering

point. A point x e X is non-wandering if every neighborhoodU of x is self-positively recursive.

We shall state a few theorems showing the connectionbetween Poisson stable points and non-wandering points.

Lemma 4.3.14 Let x e X. Every y c fl(x) isnon-wandering.

Theorem 4.3.15 Let P c X such that every x e P iseither positively or negatively Poisson stable. Then every x

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e P is non- wandering.Theorem 4.3.16 Let X be complete. Let every x e X be

non- wandering, then the set of Poisson stable points P isdense in X.

Again, a closely related theorem is the following:Theorem 4.3.17 For any dynamical system 0 on X, the set

of all non-wandering points of 0, 0(0), is a closedinvariant subset of X, and is non-empty if X is compact.Furthermore, topological conjugacies and equivalencespreserve the set of non-wandering points.

Before we introduced the notion of non-wandering points,we were seeking some new (or additional) equivalencerelations with respect to which classification might be madeeasier. The new equivalence relation are calledn-equivalence (for flow) and n-conjugacy (forhomeomorphisms). They are just the old ones, topologicalequivalence and conjugacy, restricted to n-sets. Thus ifmin(o) denotes the restriction of the flow 0 to n(0),defined by (OIn(0))(t,x) = O(t,x) for all (t,x) e R x n(0),then 0 is n- equivalent to u iff 01 n(0) is topologicallyequivalent to ,In(µ). Similarly, homeomorphisms f and g aren-conjugate iff their restrictions fln(f) and gin(g) aretopologically conjugate. From the last theorem, topologicalequivalence (or conjugacy) is stronger than n-equivalence(or conjugacy).

Earlier, we touched upon the concept of a minimal set.In the following, we shall characterize a minimal set andits existence theorems.

Theorem 4.3.18 A non-empty set A c X is minimal iffr(x) = A for every x e A.

Theorem 4.3.19 If A c X is minimal and the interior ofA is non-empty, then A = Int(A).

Theorem 4.3.20 Let A c X be non-empty and compact. Thenthe following statements are equivalent:

(i) A is minimal,(ii) r(x) = A for every x e A,(iii) r=(x) = A for every x e A,

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(iv) n(x) = A for every x e A,(v) a(x) = A for every x e A.A fixed point and a periodic orbit are examples of

compact minimal sets. From the above theorem, every point ina compact minimal set is Poisson stable. The example for anirrational flow on T2 indicates that the closure of aPoisson stable orbit need not be a minimal set. This isbecause the closure of every Poisson stable orbit except thefixed point is the whole torus, which is not minimal, for itcontains a fixed point.

Birkhoff [1927] discovered an intrinsic property ofmotions in a compact minimal set, which is usually calledthe property of recurrence.

For any x c X, the motion 0x is recurrent if for each e> 0 there exists a T = T(e) > 0, such that r(x) cB([t-T,t+T],x,e) for all t e R.

Since every motion 0Y with y c r(x) is also recurrent ifOx is recurrent, thus we shall speak of the orbit r(x) beingrecurrent. Moreover, a point x e X is recurrent if 0x isrecurrent. Note also that every recurrent motion is Poissonstable.

Theorem 4.3.21 Every orbit in a compact minimal set isrecurrent. Thus every compact minimal set is the closure ofa recurrent orbit.

Theorem 4.3.22 If r(x) is recurrent and r(x) iscompact, then r(x) is also minimal.

Corollary 4.3.23 If X is complete, then r(x) of anyrecurrent orbit is a compact minimal set.

So far our discussions were centered on compact minimalsets, not much is known about the properties of non-compactminimal sets. It has been established that all minimal setsin R2 consist of single orbit with empty limit sets [Bhatia& Szego 1967]. Nonetheless, usually compact minimal setscontain more than one orbit.Lemma 4.3.24 There exists non-compact minimal sets which

contain more than one orbit.Consider the dynamical system of irrational flow on a

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torus discussed earlier restricting the system to thecomplement of the fixed point in that example. The resultingspace X is non- compact, but for each x e x, r(x) = X, sothat X is minimal. This proves the lemma. Note that, in theabove construction, 0, are not recurrent. This shows thatTheorem 4.3.21 is not necessarily true for non-compactminimal sets.

For any x e X, the motion ox is positively Lagrangestable if r x) is compact. If r-(x) is compact, then thenotion ox is negatively Lagrange stable. It is Lagrangestable if r(x) is compact.

Remark: If X = R", then the above statements areequivalent to the sets r=(x), r(x) being bounded. One canalso show that (i) If X is locally compact, then a motion oxis positively Lagrange stable iff n(x) is a non-emptycompact set; (ii) If ox is positively Lagrange stable, thenn(x) is compact and connected; (iii) If ox is positivelyLagrange stable, then n(x)) - 0 as t - co.

Theorem 4.3.25 Every non-empty compact invariant setcontains a compact minimal set.

Theorem 4.3.26 The space X contains a compact minimalset iff there is an x e X such that either r+(x) or r-(x) iscompact.

It is worthwhile to note that the only recurrent motinsin R2 are the periodic ones. As a consequence, all compactminimal sets in R2 are the orbits of periodic points.Indeed, Hajek [1968] shows that all positively Poissonstable points in R2 are periodic. Moreover, the onlynoncompact minimal sets in R2 consist of a single orbit withempty n-limit and a-limit sets [Bhatia and Szego 1967]. ThenTheorem 4.3.18 implies that all minimal sets in R2 haveempty interiors. This theorem also poses an important andinteresting problem, i.e., which phase spaces (or manifolds)can be minimal.

A special case of recurrence, namely almost periodicity,is deferred to next chapter because it is intimatelyconnected to the notion of stability of motion. It is

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worthwhile to note that the concepts of recursiveness can begeneralized to non-metric topological spaces, whereas almostperiodicity requires a uniformity on the space.

4.4 Velocity fields, integrals, and ordinary differentialequationsIn this section, we shall discuss the existence and

uniqueness of a flow whose velocity is a given vector field.By using a chart, the local problem is equivalent to theexistence and uniqueness of solutions (integral curves) of asystem of ordinary differential equations.

Recall that, if M is a differentiable manifold, a vectorfield on M is a map X: M - TM associated with each point p eM a vector X(p) in the tangent space MP. We can think of MPas the space of all possible velocities of a particle movingalong paths on M at p. Also recall that if X is a givenvector field on M. we call any flow 0 on M an integral flowof X if X is the velocity vector field of 0. We also saythat X is integrable if such a flow exist.

Theorem 4.4.1 Let 0 be a flow on M such that, for all pc M, the map 0P: R - M is differentiable. Then the velocityof 0 at any point is independent of time. Thus 0 has a welldefined velocity vector field.

An integral curve of X is at least a C' map r: I M,

where I is any real interval such that r'(t) = Xr(t) for allt e I. A local integral of X is a map 0: I x U M, where Iis an interval neighborhood of 0 and U is a non-empty opensubset of M, such that for all p e U, ¢P: I - U is anintegral curve of X at p. For all p e U, we call 0 a localintegral at p, and say that X is integrable at o if such alocal integral exists. If it does and is at least C', thenthe diagram

T(IxU)

Y t

IxU

TO, TMt X

M

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commutes. If I can be extended to R, and U to M, then thelocal integral is a flow on M. We shall come back to thisshortly.

Theorem 4.4.2 Let X be a vector field on M, and let h:M - N be a C' diffeomorphism. If r is an integral curve ofX, then h is an integral curve of the induced vector fieldh,(X) = (Th)Xh-1 on N from X by h. If 0: IxU M is a localintegral of X at p, then ho(id x (hIU)"1) is a localintegral of h,(X) at h(X).

It is intuitively clear that the following theoremfollows.

Theorem 4.4.3 Any integral curve of a Ck vector fieldis Ck+1.

Let V be an open subset of a Banach space B, and let Xbe a Ck vector field on V (k ? 0). Also suppose that 0: IxU- V is a local integral of X. We now express the conditionthat r be an integral curve of X in terms of localrepresentatives with respect to natural charts. Let (U, µ)be a chart of M and suppose the image of r is contained inU. Then the local representative of r with respect to theidentity of R and (U, µ) is r,, = µ r, while the localrepresentative of the curve r' with respect to the identityof R and natural chart (TMIU,Tg) is given by (r')µ(t) =Tg- r' (t) = Tj Tr(t) = T(µ r) (t) = (r.) ' (t) by the compositemapping theorem. Also, the local representative of x -r withrespect to the identity of R and the natural chart Tg isTµ X r = Tµ X A-1. µ r = x r,, where X,, is the localrepresentative of X. Thus r is an integral curve of X iffx- r = r,, iff xµ rµ = rµ' , iff rµ is an integral curve of XThis condition takes a simple and usual form if µ(U) c R.Then we have X,(p) = (p;X,(p),...,X,(p)) where p e µ(U) c R",(Xi(p)) are the components of Xµ, rµ(t) = (r,(t),...,r,(t)rµ' (t) _ (r (t) ; F1 ' (t) , ... , rr,' (t)) , and Xµ' r,, = rµ' iffrµ'(t) = X,(r(t)) for i = 1,...,n and all t e I. Thus, r isan integral curve of X iff the local representatives satisfythe system of first-order ordinary diffeerential equations

r,' (t) =

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r,'(t) = x,(r,(t),...,r"(t)).

Note that t does not appear explicitly on the right. Such asystem of equations (a local dynamical system) is called anautonomous system. It includes regular equations of higherorder and the Hamiltonian equations of motion as specialcases.

We should also point out that any m-th order ordinarydifferential equation in standard form

dmy/dtm = h (t, y, dy/dt, .. , dm-ly/dtm'1) ,can be reduced to the form dx/dt = g(t,x), where x =(x,,..-,x.) a Rm, and g is a vector valued function. Bysubstituting x, = y, x2 = dy/dt, ..., xm = dm"ly/dtm-1 , andg(t,x) = (x2,x3,.... x,, h(t,x,,.... xm)).

Example: In Chapter 1 we introduced the pendulumequation d'O/dt' = -g sing and it was reduced to dO/dt = f2,dfl/dt = -g sinO by the substitution f2 = dO/dt.

Example: In classical mechanics, with a conservativeforce field, Lagrange's equations of motion can be obtainedfrom the Euler-Lagrange equation

d/dt ((3L/aqj) - caL/aq1 = 0,where q1 (1 <- i <- n) are the "generalized coordinates", theLagrangian L = T - V, where T is the kinetic energy and V isthe potential energy. In terms of the generalizedcoordinates, T = 1/2 E,_,n m;(dq,/dt)' , then the equations ofmotion becomes

mi dq1' /dt' = - aV/aq1.With the substitution m,dq,/dt = pi, the "generalizedmomentum", converts the equations of motion to dp,/dt =-aV/aq,. The generalized coordinates q, and its timederivatives dqj/dt form the coordinates of the tangentbundle of the configuration space M; while the generalizedcoordinates and generalized momenta form the coordinates ofthe cotangent bundle T*M of the configuration space. In theT*M, the equations of motion are the Hamilton's equations

dq1/dt = aH/ap,, and dp,/dt = - aH/aq;,170

where the Hamiltonian H = T + V is also the total energy ofthe system.

Example: (Van der Pol's equation) This equation modelsthe electronic oscillators in electric engineering.

e9

E

Consider the vacuum tube circuit represented by the integro-differential equationL di/dt + Ri + C-1 f UL idt - M die/dt = E = Eosin flit.Assume ie ke9(1- e92/3VS2) where k is the transconductanceof the tube, VS is the saturation voltage, i.e., asufficiently high grid voltage beyond which the current iedoes not change appreciably. Neglecting the grid current andanode reaction, let

x = eg/Vs = (VsC) -1 f o idt, a = Mk/LC - R/L,b = Mk/3LC, flog = 1/LC, and A = EJVSLC = Bflo2 .

Then the integro-differential equation becomes adifferential equation of the form:

x" - ax' + b(dx3/dt) + floe x = A sin flit.If the driving force function A sin at is set equal to zero(i.e., A = 0), then it simplifies to:

x" - (a - 3bx2) x' + flog x = 0.For simplicity, one can write the van der Pol equation inthe following "normalized" form:

x" - ax' (1 - x2) + x = 0,171

where a is a positive constant. We can transform this secondorder equation into a pair of first order equations as thefollowing,

x' = y, y' = -x + ay( 1 - XI).The phase portrait of this vector field on R2 is

topologically equivalent to the following figure.

It has a unique closed orbit which is the f2-set of all theorbits except the fixed point [Hirsch & Smale 1974]. Thesystem is said to be auto-oscillatory since all (except one)solutions tend to become periodic as time increases.

Let p be a point of a Banach space B, let X be a vectorfield on some neighborhood U of p. We want to find outwhether for some neighborhood V of p there are uniqueintegral curves at each point of V. In order to proveuniqueness, we need something stronger than continuity andwhich is the Lipschitz condition. First we shall define aLipschitz may.

Let P and Q are non-empty metric spaces with distancefunction d. Let k be any positive number. A map f: P Q isLipschitz (with constant k) if, for all p,p'E P,

d(f(p),f(p')) 5 kd(p,p')Clearly, any Lipschitz map is continuous (in fact, uniformlycontinuous). A map f is locally Lipschitz if every p e P hasa neighborhood on which f is Lipschitz.

Clearly, Lipschitz condition implies continuity and itis satisfied by any C' map on U. Thus, Lipschitz conditionis about half way between continuity and differentiability.

As before, let p be a point of a Banach space B, X be avector field on some neighborhood U of p, and V be someother neighborhood of p. Let d' and d are the distancefunctions of U and V respectively.

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Theorem 4.4.4 (Picard) Let f: U - B be Lipschitz withconstant k, and let I be the interval [-a,a], where a <(d'-d)/If10 if d'< w, and Ifl. is the Banach norm. Then foreach p e V, there exists a unique integral curve 0P: I - Uof f at p. Let 0: IxV U send (t,p) to 0P(t). Then for allt e I, Ot: V - U is uniformly Lipschitz (in t), and Cr if fis Cr.

Theorem 4.4.5 Same assumptions as in the above theorem.If d'< ao , then the map 0 is Lipschitz. And it is locallyLipschitz.

This theorem provide an answer as to the dependence of 0on t and x together. It says that 0 is locally Lipschitz,and 0 is as smooth as f. Then the main theorem on thesmoothness of local integrals follows easily.

Theorem 4.4.6 If the map f of Theorem 4.4.4 is Cr (r1), then the local integral 0 is also Cr. Furthermore,D1(D2)rO exists and equals to (D2)r(fO): I x B - LS(B,B).

Since any C' vector field is locally Lipschitz, we havethe following useful corrollary:

Corollary 4.4.7 Any Cr vector field (r ? 1) on amanifold has a Cr local integral at each point of themanifold.

So far we have been dealing with local integrabilitycondition and uniqueness. The main tool for extending localintegrability condition to global one is the uniqueness ofintegral curves proved earlier. The next theorem is theglobal uniqueness theorem.

Theorem 4.4.8 Let X be a locally Lipschitz vector fieldon a manifold M, and let a: I -+ M, and /3: I -, M be integral

curves of X. If for some to a I, a (to) = /3 (to) , then a = /3Recall that a point p e M is a singular point of a

vector field X if X(p) = 0. One can easily show that allintegral curves of a locally Lipschitz vector field X at pare constant functions if p is a singular point of X.

Earlier we commented that if the velocity of a flow isindependent of time, we may consider such a flow as a vectorfield. Conversely, the following theorem states that a local

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integral of a vector field is locally a flow.Theorem 4.4.9 Let 0: IxU - M be a local integral of a

locally Lipschitz vector field on a manifold M. Then for allp e U and all s, t e I such that s+t e I and 0(t,p) a U,

O(s,O(t,p)) = O(s+t,p)It follows immediately:Corollary 4.4.10 If 0: IxM - M is a local integral of a

locally Lipschitz vector field on a manifold M, then it is aflow on M.

The next theorem shows the uniqueness of an integralflow in a Banach space.

Theorem 4.4.11 Any Lipschitz vector field X on a Banachspace B has a unique integral flow 0. If X is Cr then ¢ isalso Cr.

The next theorem shows that the local integrals are assmooth as the vector field we integrate and this iscompletely general.

Theorem 4.4.12 Any local integral of a Cr (r >_ 1)

vector field on a manifold M is Cr. Likewise, any integralof a locally Lipschitz vector field is locally Lipschitz.

Theorem 4.4.13 Any locally Lipschitz vector field X ona compact manifold M has a unique integral flow 0, which islocally Lipschitz. If X is Cr (r >_ 1) then 0 is also Cr.

By virtue of the fact that velocity fields and integralflows are so closely related, the qualitative theories ofsmooth vector fields and smooth flows can be thought of asone and the same subject. Since it is sometimes easier todescribe the qualitative theory in terms of vector fields,and at other times in terms of flow, we shall feel free touse whichever terminology or viewpoints we find moreconvenient in any given situation.

4.5 Dispersive systemsIn Section 4.3 we were interested in the systems of

Poisson stable points or non-wandering points, andrecursiveness. In this section, we shall discuss dynamical

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systems marked by the absence of recursiveness. We shallbriefly describe dispersive dynamical systems andinstability, and we shall also study the theory ofparallelizable dynamical systems.

First we shall introduce the definition of positive(negative) prolongation of x e X. For any x e X, thepositive (negative) prolongation of x, D=(x) = {y e X: thereis a sequence (x1,) in X and a sequence (tn) in R= such that

t -x - x and x y). The positive (negative) prolongationalnn n

limit set of x, J!(x) = (y e X: there is a sequence (x)

in

X and a sequence { tn)n

in R= such that xn - x, to - ± a, andXntn - Y) .

It is clear that for any x c X, P*(x) is a subset ofD1 (x) ,fl (x) is a subset of J+ (x) , a(x) is a subset of J' (x) .These inclusions can be illustrated by considering thesystem, dx/dt = -x, dy/dt = y. This is a dynamical systemwith a saddle point at the origin. We will leave to thereader to graph the orbits and to establish the aboveinclusions.

Let x e X, and the partial map 0x is positively Lagrangeunstable if the closure of the positive semi-orbits, P'(x),is not compact. It is called negatively Lagrange unstable ifthe closure of r (x) is not compact. And it is calledLagrange unstable if it is both positively and negativelyLagrange unstable. In Section 4.3 we have defined Poissonstable and wandering points. Now we are ready to define someof the concepts of dispersiveness of a dynamical system.

A dynamical system is, Lagrange unstable if for each x eX the partial map 0x is Lagrange unstable, Poisson unstableif each x is Poisson unstable, completely unstable if everyx is wandering, dispersive if for every pair of points x, ye X there exist neighborhoods Ux of x and UY of y such thatUx is not positively recursive with respect to UY.

As an example, let us consider a dynamical system in R2,whose phase portrait is Fig.4.5.1. The unit circle containsa fixed point p and an orbit r such that for each point q er we have fl(q) = a(q) = (p). All orbits in the interior of

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the unit circle (= (p) U t) have the same property as r. Allorbits in the exterior of the unit circle spiral to the unitcircle as t - -, so that for each point q in the exterior ofthe unit circle we have n(q) = (p) U r, and a(q) = 0. Noticethat if we consider the dynamical system obtained from thisone by deleting the fixed point p, now the dynamical systemis defined on R2 - (p), then this new system is Lagrangeunstable and Poisson unstable, but it is not completelyunstable because for each q e r we have J+(q) = r, i.e., q e

J+ (q)

Fig. 4.5.1

It should be noted that for dynamical systems defined bydifferential equations in R2, the concepts of Lagrangeinstability and wandering are equivalent. This may be easilyproven by using the Poincare-Bendixson theorem for planarsystems. Note that the above example has the fixed pointremoved, i.e., in R2 - (p).

We next consider an example of a dispersive system whichturns out to be not parallelizable when we introduce thisconcept shortly. Let the dynamical system in R2 be,

dx/dt = f(x,y), dy/dt = 0,where f(x,y) is continuous, and f(x,y) = 0 whenever thepoints (x,y) are of the form (n,1/n), and n is a positive

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integer. For simplicity, we assume that f(x,y) > 0 for allother points. The phase portrait is shown in Fig.4.5.2. Letus now consider a new system which is obtained from theabove system by removing the sets

I, = ((x,y)l x <_ n, y = 1/n), n = 1, 2, 3, ...

from R2. This system is dispersive.

(2,'12 )

(3,1/3)

0XI

Fig. 4.5.2

In the following, we shall state several resultscharacterize dispersive systems.

Theorem 4.5.1 A dynamical system is dispersive iff foreach x e X, J* (x) = 0.

From this theorem, the above example is clearlydispersive. A more useful criterion is the followingtheorem.

Theorem 4.5.2 A dynamical system is dispersive iff foreach x e X, D+(x) = r +(x) and there are no fixed points orperiodic orbits.

A dynamical system is called oarallelizable if thereexists a subset S of X and a homeomorphism h: X - S x R suchthat SR = X and h(xt) = (x,t) for every x c S and t e R.Notice the analogue of this notion of parallelizability fordynamical systems and the notion for fiber bundles. Indeed,a notion of cross section is in order. A subset S of X is

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called a section of the dynamical system if for each x c Xthere is a unique r(x) such that xr(x) a S. Note that notevery dynamical system has a section. In fact, any dynamicalsystem has a section iff it has no fixed points or periodicorbits. This is not surprising, because from the fiberbundle view point, the existence of a section implies thatfor each r(x), continuous or not, each section ishomeomorphic to each other. Note also, in general r(x) isnot continuous, but the existence of a section S withcontinuous r(x) implies certain properties of the dynamicalsystem.

Lemma 4.5.3 If S is a section of the dynamical systemwith r(x) continuous on X, then: (i) S is closed in X, (ii)

S is connected, arcwise connected, simply connected iff X isconnected, arcwise connected, simply connected respectively,(iii) if a subset K of S is closed in S, then Kt is closedin X for every t e R, (iv) if a subset K of S is open in S,then KI is open in X, where I is any open interval in R.

Theorem 4.5.4 A dynamical system is parallelizable iffit has a section S with r(x) continuous on X.

This theorem shows that the dynamical system of thesecond example is not parallelizable. The next theorem is avery important one.

Theorem 4.5.5 A dynamical system on a locally compactseparable metric space X is parallelizable iff it isdispersive.

The proof of this theorem depends on certain propertiesof sections. We shall not go into any detail here. Itsuffices to say that the concept of tubular neighborhoods isessential to its development. For more details, the readeris directed to Bhatia and Szego [1970], Ch. 4.

4.6 Linear systemsNaively, one might hope to obtain a complete

classification of all dynamical systems on Euclidean spaceR", starting with the simplest types and gradually building

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up some understanding of more complicated ones. The simplestdiffeomorphisms of R" are translations, and it is easy toshow that two such maps are linearly conjugate (topologicalconjugate by a linear automorphism of R") providing theconstant vector is non-zero. Furthermore, any two non-zeroconstant vector fields on R" are linearly flow equivalent.The next simplest systems are linear discrete systemsgenerated by linear automorphisms of R' and vector fields onR" with linear principal part. Even more, the classificationproblem is far from trivial, nor is it solved.

The difficulties one encounters even with linearproblems underline the complexity and richness of the theoryof dynamical systems. And indeed this is a major part of itsattraction. Meanwhile, these difficulties also indicate theneed for a modest approach to the problem. We only attemptto classify a "suitably large" class of dynamical systemsinstead of the complete solution.

For linear systems on R", it is easy to give a precisedefinition of what a "suitably large" class means. The setL(R") of linear endomorphisms of R" is a Banach space withthe norm

ILI= sup(IL(x)I: x e R", IxI=1).

A "suitably large" subset of linear vector fields on R" isthe one both open and dense in L(R").

Let X be a topological space and _ is an equivalencerelation on X. We say that a point x e X is stable withrespect to - if x is an interior point of its - class. Thestable set E of - is the set of all stable points in X.Clearly, E is an open subset of X. If X has a countablebasis, then E contains points of only countably manyequivalence classes. Nonetheless, E may fail to be dense inX. In the present context, X = L(R") (or GL(R")), - is

topological equivalence (conjugacy), and E is called the setof hyperbolic linear vector fields (automorphisms). In sucha case, E is dense and easily classifiable.

The basic idea of calculus is to approximate a functionlocally by a linear function. Similarly, when discussing the

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local properties of a dynamical system we study its "linearapproximation", which is a linear system. The importantquestion is whether this linear system is a goodapproximation in the sense that it is locally qualitativelythe same as the original system. And this obviously leads tothe stability (in the broader sense) of the linear system.The importance of considering the hyperbolic linear vectorfields (automorphisms) is that they are stable not onlyunder small linear perturbations but also under small smooth(but not necessarily linear) perturbations. We shall beginwith a review of linear systems on R", which provide abackground of concrete examples against which the moregeneral results on Banach spaces may be tested and placed inperspective.

Let T be any linear endomorphism of R", then we maythink of it as a vector field on R. The correspondingordinary differential equation is x'= T(x) where x e R. Theintegral flow can be written down immediately.

Since L(R") is a Banach space, then the infinite seriesexp(tT) = id + tT + t2T2/2 +....+ t"T"/n! +....,

t e R, converges for all t and T and the integral flow 0 ofT is O(t,x) = exp(tT)(x). We shall call a flow 0 linear ifpt is a linear automorphism varying smoothly with t. Thusis linear iff its velocity field is linear.

As an example, if T = a(id) for some a e R,exp(tT)= id + at(id) +....+ (at)"id"/n! + ...

= (1 + at +...+ (at)"/n! +..)id = eaLid.A notion of linear equivalence for linear flow can be

defined by letting 0 and µ be linear flows with velocities Sand T e L(R") respectively. Then µ is linearly ecruivalentto 0, µ -c 0, if for some a a R, a > 0, and linearautomorphism h e GL(R") of R", the following diagramcommutes:

R X R"axh 4

R X R"

m y Rn

1 h

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- Rnµ

That is, ho(t,x) = µ(at,h(x)) for all (t,x) e R x R", orequivalently, from the standpoint of velocity fields, µ _L 0iff S = a(h"'Th). Thus the problems of classifying linearflows up to linear equivalence is the same as classifyingL(R") up to similarity (linear conjugacy).

The fundamentally important result by Kuiper (1975]states that if 0 and µ are linear flows given by linearendomorphisms S and T of R" whose eigenvalues all have zeroreal part then 0 is topologically equivalent to µ iff 0 islinearly equivalent to A. More generally, two linear flowsand µ on R" with decompositions U. + U_ + Uo and V. + V_ + Volwhere U+(V+), U_(V_), U0(V0) are subspaces corresponding topositive, negative, zero real part of the eigenvaluesrespectively, are topologically equivalent iff OIU0 islinearly equivalent to µ I Vo, dim U+ = dim V+, and dim U_ _dim V-.

Let us recall again the linear flows in R and R2. (i)

Linear flows in R. Any linear endomorphism of the real lineR is of the form x - ax, a e R. The map is hyperbolic iff a

0. The integral flow is xeat and there are exactlythree topological equivalence classes:

0 0 . . . , 1 0

a< 0 a= 0 a> 0

(ii) Linear flows on R2. The real Jordan form for a real 2x2matrix is one of the following three types:

(a) r a 01 (b) [a 1] (c) [a -b 10 µ0 a, b a

where a, g, a, b e R and b > 0. One can show that a linearvector field on R2 is hyperbolic iff (a) µa + 0, (b) a + 0,

(c) a + 0.

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X0] [1 0]1

0 -1 0 -1 0 1

There are also five topological equivalence classes of non-hyperbolic linear flows [see, e.g., Irwin 1980, p.87].

As we have pointed out earlier, discrete lineardynamical systems on R" are determined by linearautomorphisms of R. There are two equivalence relations,topological conjugacy and linear conjugacy, which aresimilar to the usual sense of linear algebra. Nonetheless,the relation between linear and topological conjugacy iseven more difficult to pin down than the relation betweenlinear and topological equivalence of flows. Here we shallrestrict ourselves by noting that there is an open densesubset of GL(R"), (where S & T e GL(R"), S is similar to Tif for some P e GL(R"), T = PSP-1, i.e., GL(R") is the set oflinear similarity mappings), analogous to GL(R") in the flowcase, HL(R"). Elements of HL(R") are called hyperboliclinear automorphisms of R". For any T e GL(R") iff none ofits eigenvalues lies on the unit circle S' in C. It shouldbe noted that the term "hyperbolic" must carry a dualmeaning, this is because T is a hyperbolic vector field iffexp T is a hyperbolic automorphism. Nonetheless, the meaningshould be clear from the context.

The simplest example is the automorphisms of R. Sinceany element T of GL(R) is of the form T(x) = ax, where a isa non- zero real number. Thus there are six topological

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conjugacy classes, (i) a < -1, (ii) a = -1, (iii) -1 < a <0, (iv) 0 < a < 1, (v) a = 1, (vi) a > 1. Clearly, bydefinition, except (ii) and (v) are hyperbolic.

The map T: R2 - R2 by T(x,y) = (x/2,2y) is in HL(R2).Its spectrum is the set (1/2,2) of eigenvalues of T andneither of these points is on S1. The following figuredemonstrates the effect of T. Note that the hyperbola andtheir asymptotes are invariant submanifolds of T. The x-axisis stable in the sense that positive iterates of T take itspoints into bounded sequences (they converge to the origin).The y-axis is unstable (meaning stable with respect to T"1).A direct sum decomposition into stable and unstable summandsis typical of hyperbolic linear automorphisms.

Y

E

CjOne can show that if T E GL(R") and if a e R is

sufficiently near, but not equal to, 1, then aT ishyperbolic. From this, one can deduce that HL(R") is densein GL(R") .

The next is the set of automorphisms in R2. Theclassification is already quite complicated. We shall referto Irwin (1980, p.88-9].

To embark on studying linear dynamical systems on aBanach space B, we take note of the analogy of the set ofeigenvalues of a linear endomorphism T, which is called the

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spectrum of T, a(T). Recall that the set of linearendomorphisms of B, L(B), is a Banach space with the normdefined by IT1= sup(IT(x)l: x c B, lxi <- 1). And thespectrum radius r(T) is r(T) = sup{l i : µ e a(T)}. It is ameasure of the eventual size of T under repeated iteration,i.e., it equals to lim,_ lT"l11". One may also choose a normof B such that T has norm precisely r(T) with respect to thecorresponding norm of L(B). When r(T) < 1, we call T alinear contraction, in the sense of metric space theory. IfT is an automorphism, and

T"1

is a contraction, then T iscalled expansion. Note that T is an automorphism iff 0 $a(T), and that r(T-1) _ (inf (i Ll : µ e a(T) ))''. In fact,a(T-) = (,u-1: µ e a(T) }.

Analogous to the finite dimensional case, let GL(B) bethe open subset of L(B) consisting of all linearautomorphisms of B. Let T e GL(B), and if a(T) n S' isempty, then we call T a hyperbolic linear automorphism. Theset of all hyperbolic linear automorphisms of B is denotedby HL(B). It is straightforward to show that if T e HL(B), 0e B is the only fixed point of T.

Now let D be a contour, symmetric about the real axis,if B is a real Banach space, such that a(T) n D is empty.Furthermore, suppose 0 $ a(T) and that D separates a(T) intotwo subsets as(T) and au(T), where the subscripts s and ustand for stable and unstable respectively, as we shall seelater. Then by a corollary of Dunford's spectral mappingtheorem, B splits uniquely as a direct sum Bs + Bu ofT-invariant closed subspaces such that, if Ts: Bs - BS andTu: Bu - Bu are the restrictions of T, then a(TS) = as(T) anda(Tu) = au(T). By the spectral decomposition theorem, onecan show that T. is a contraction and Tu is an expansion.Let c be any number less than one but greater than thelarger of the spectral radii of Ts and Tu". Then one canprove:

Theorem 4.6.1 If T E HL(B) then there is an equivalentnorm 11 11 such that: (i) for all x = xs + xu a B, lixIl =max(IIx5Il,jIxuIj); (ii) max{IlTsll, JIT,,-'Ii} = a <_ c. Thus, with

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respect to the new norm II II, T. is a metric contraction, andT,, is a metric expansion.

The stable summand B of B with respect to T ischaracterized as (x a B: T"(x) is bounded as n - oD ), andequivalently as (x a B: T"(x) y 0 as n - oD ). B. is also

called the stable manifold of 0 with respect to T, and TS iscalled the stable summand of T. Similar characterizationsand definitions hold for the unstable summand B, with T'"replacing T. We say that T e HL(B) is isomorphic to T'eHL(B') if there exists a topological linear isomorphism fromB to B' taking BS(T) onto Bs'(T') and Bu(T) onto Bu'(T').Equivalently, T is isomorphic to T' iff there are linearisomorphisms of Bs (T) onto Bs' (T') and Bu (T) onto Bu' (T') .Thus there are exactly n+1 isomorphism classes in HL(R").

Theorem 4.6.2 HL(B) is open in GL(B), and hence inL(B).

Theorem 4.6.3 Any T c HL(B) is stable with respect toisomorphism.

Theorem 4.6.4 Let T and T' belong to the same pathcomponent of GL(B) and also have spectral radii < 1. Then Tand T' are topologically conjugate.

Corollary 4.6.5 Two hyperbolic linear homeomorphisms ofR" are topologically conjugate iff

(i) there are isomorphic,(ii) their stable components are either both

orientation preserving or both orientation reversing, and(iii) their unstable components are either both

orientation preserving or both orientation reversing.Thus, there are exactly 4n topological conjugacy classes

of hyperbolic linear automorphisms of R" (n ? 1). The maintheorem is:

Theorem 4.6.6 For any Banach space B, any T E HL(B) isstable in L(B) with respect to topological conjugacy.

Corollary 4.6.7 Let B be finite dimensional, then HL(B)is the stable set of GL(B) with respect to topologicalconjugacy. Moreover, HL(B) is an open dense subset of GL(B).

Let us now turn to study linear vector fields. We are

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interested in determining which vector fields are stable inL(B) with respect to topological equivalence. We call T eL(B) a hyperbolic linear vector field if its spectrum a(T)does not intersect the imaginary axis of C, and we denotethe set of all hyperbolic linear vector fields on B byHLV(B).

Proposition 4.6.8 A linear vector field T is hyperboliciff for some non-zero real t, exp(tT) is a hyperbolicisomorphism. Moreover, T has integral flow 0: RxB - B givenby 0(t,x) = exp(tT)(x).

As before, we shall define stable summands and stablemanifolds at 0. Let T e HLV(B), as(T) = (µ a a(T): Re p < 0)and au(T) = (µ a a(T): Re µ > 0). Also let D be a contourconsisting of the line segment [-ir, ir] and the semi-circle{re'° : 7/2 -< 0 <- 37/2) where r is large enough such that Dencloses as. As before, the spectral decomposition theoremgives a T-invariant direct sum splitting B = Bs ® Bu suchthat a(Ts) = as(T) and a(Tu) = au (T), where Ts a L(Bs) and Tue L(Bu) are the restrictions of T. We call Ts and Bs stablesummand and stable manifold at 0 respectively. Likewise, Tuand Bu are unstable summand and unstable manifold at 0respectively. One can easily show that for all t > 0, thestable summand of the hyperbolic automorphism exp(tT) isexp(tTs) and similarly for unstable summands.

One can show that the map exp:L(B) - L(B) is continuous,and since HL(B) is open in L(B), Proposition 4.5.8 impliesthat HLV(B) is also open in L(B). Moreover, we can defineisomorphism for hyperbolic vector fields just as forhyperbolic automorphisms. Since the stable and unstablesummands of B are the same with respect to T 6 HLV(B) aswith respect to exp T e HL(B) as noted before, thus T eHLV(B) is stable respect to isomorphism follows immediatelyfrom the corresponding result for HL(B). We then have thefollowing theorem.

Theorem 4.6.9 Let T e HLV(B), then T = Ts e Tu, wherea(T.) = as(T) and a(Tu) = au (T). The set HLV(B) is open inL(B), and its elements are stable with respect to

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isomorphism.As in the case of hyperbolic linear automorphism, we may

choose a norm on B such that the stable and unstablesummands of exp(tT) are respectively a metric contractionand a metric expansion.

Theorem 4.6.10 Let T e HLV(B) have spectrum a(T) _as(T). Then there exists a norm 1111 on B equivalent to thegiven one such that, for any non-zero x e B, the map from Rto (0, -) taking t to Ilexp(tT)(x)II is strictly decreasingand surjective. There also exists b > 0 such that for all t> 0, IIeXp(tT) II <- a"bt.

Corollary 4.6.11 If 0: RxB B is the integral flow ofT e HLV(B), then 0 maps RxS1 homeomorphically onto B/(o).

Theorem 4.6.12 Let T, T'e HL(B) have spectra in Re z <0. Then T and T' are flow equivalent.

Corollary 4.6.13 If T and T'e HLV(B) are isomorphic,then they are flow equivalent, thus topologicallyequivalent.

Thus we have the following important result:Theorem 4.6.14 Any T c HLV(B) is stable in L(B) with

respect to flow equivalence, thus with respect totopological equivalence.

The above discussion will provide the basis forcomparing the vector field or flow in question with theknown linear vector field or flow. The stable and unstabledecomposition and summonds are the basis for the discussionof stability.

4.7 LinearizationAs we have pointed out earlier we can hardly expect to

make much progress in the global theory of dynamical systemson a smooth manifold M, if we ignore completely how systemsbehave locally. We shall attempt to classify dynamicalsystems locally, i.e., linearly. But even here, as we havepointed out at the begining of section 5, there aredifficulties with linear systems, thus we can only expect

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partial success.As we have seen earlier, linear systems exhibit a

diverse variety of behavior near the fixed point zero. Inorder to focus on our classification scheme, we shall onlydiscuss the class of fixed points which often occurs to bethe sort that one "usually comes across". For linearsystems, we are able to give this vague notion a precisetopological meaning in terms of the spaces of all linearsystems on a given Banach space B.

Let 0 be a given dynamical system on a smooth manifold Mwhich has a fixed point p. Since we are only interested inlocal structure, we can assume M = B and p = 0, after takinga chart at p. Suppose now that by altering q slightly near 0we end up with another fixed point near 0, with a localphase portrait resembling that of 0 at 0. Intuitively, thelinear approximation of 0 at 0 (we shall make sense of thisterm later) must have a phase portrait near 0 resemblingthat of 0 near 0 as most all "nearby" linear systems. Withthe stability theorems of the last section, it is clear thatwe should consider fixed points whose linear approximationsare hyperbolic systems.

A fixed point p of a diffeomorphism f of M is hyperbolicif the tangent map TPf: M - MP is a hyperbolic linearautomorphism. Corollary 4.5.5 supplies a classification ofhyperbolic linear automorphisms. We shall extend this resultto a classification of hyperbolic fixed points using theHartman theorem [Hartman 1964; Moser 1969] which states thatany hyperbolic fixed point p of f is topologically conjugateto the fixed point 0 of TPf. An analogous situation existswith flows. A fixed point p of a local integral 0: IxU - Mof a vector field X on M (or equivalently a zero of X) ishyperbolic if for some t + 0 in I, p is a hyperbolic fixedpoint of Ot [Grobman 1959, 1962].

Proposition 4.7.1 The point p is a hyperbolic zero of avector field X iff for any chart µ at p, the differential ofthe induced vector field (Tp)Xg-1

at µ(p) is a hyperbolicvector field.

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An equivalent but more sophisticated approach is toconsider TX: TM - T(TM), which is a section of the vectorbundle TIrM: T(TM) - TM, where 7M: TM - M. But TnM may beidentified with the tangent bundle on TM, where ATM: T(TM) -TM by the canonical involution, and TX then becomes a vectorfield on TM. With such identification, T

PX maps MP into

T(MP). Thus TPX is a linear vector field on MP. Recall that

this linear vector field is the Hessian of X at p. Thus:Proposition 4.7.2 p is a hyperbolic zero of X iff the

Hessian of X at p is a hyperbolic linear vector field.We shall state later that if p is a hyperbolic fixed

point of a vector field X then it is flow equivalent to thezero of the Hessian of X at p.

Recall that a point p of M is a regular point of adynamical system on M if it is not a fixed point of thesystem. The following two theorems show that regular pointsare uninteresting from local viewpoint.

Theorem 4.7.3 If p is a regular point of adiffeomorphism f: M - M and g is a translation of the modelspace B of M by a non- zero vector X0, then fIp istopologically conjugate to glO.

Theorem 4.7.4 Let p be a regular point of a C' flow 0on M. Let x° be any non-zero vector of the model space B ofM, and let Abe the flow on B defined by µ(t,x) = x + tx°.Then oIp is flow equivalent to µIO.

It should be noted that the equivalence in the above twotheorems are as smooth as the system under consideration.

Let T be a hyperbolic linear automorphism of B withskewness a < 1 with respect to the norm I I on B, i.e.,max(IT,I, IT,,-'I) = a < 1. The main piller of the theory ofthe stability of T with respect to topological conjugacyunder smooth small perturbations is the theorem of Hartman[1964].

Lemma 4.7.5 Let n : B - B be Lipschitz with constant k< 1-a. Then T+n has a unique fixed point.

Theorem 4.7.6 (Hartman's linearization theorem) Letn e C°(B) be Lipschitz with constant k < min(1-a, ITS-'I"').

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Then T+n is topologically conjugate to T.A more detailed version can be stated as:Theorem 4.7.7 Let n, g e C°(B) be Lipschitz with

constant k < min{1-a, ITS-'"'}. Then there exists a unique ge C°(B) such that (T+1)(id+g) = (id+g) (T+µ). Furthermore,id+g is a homeomorphism, and thus is a topological conjugacyfrom T+µ to T+n.

Note that, the conjugacy id+g in the above theoremcannot always be as smooth as T+n or T+µ. The mainapplication of Hartman's theorem is to compare adiffeomorphism near a hyperbolic fixed point with its linearapproximation at the point.

Corollary 4.7.8 A hyperbolic fixed point p of a C'diffeomorphism f: M - M is topologically conjugate to thefixed point 0 of TPf.

Corollary 4.7.9 There are 4n topological conjugacyclasses of hyperbolic fixed points that occur onn-dimensional manifolds.

In relation to our main interest, we shall say a fewwords concerning stability. We call a fixed point p of adiffeomorphism f: M - M structurally stable if for eachsufficiently small neighborhood U of p in M there is aneighborhood V of f in Diff(M) such that, for all g e V, ghas a unique fixed point q in U and q is topologicallyconjugate to p. In fact, one can prove that fixed points arestructurally stable if (and in finite dimensions, only if)they are hyperbolic. The following is an outline of theproof.

Let f: U - f(U) be a diffeomorphism of open subsets of Bwith a fixed point at 0 is hyperbolic, and that Df(O) hasskewness a with respect to the norm I I on B. Choose k,where 0 < k < 1-a, so small that for all T e L(B) with IT -Df(O)I <- k, T is hyperbolic and topologically conjugate toDf(O). Let Bb be a closed ball in B with center 0 and radiusb (< 1) such that IDf(x) - Df(O)I _< k/2 for all x e Bb.

Since if n: Bb - B is Lipschitz with constant k and if I17IO-< b(1-a) then T+n: Bb - B has a unique fixed point. One can

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prove that for all C' maps g: U - B with Ig - f1 <_ kb/2, g

has a unique fixed point p in Bb. It then easily followsthat g1p is topologically conjugate to fIO, and hence 0 isstructurally stable under Cl-small perturbations of f.Conversely, one can prove that if B is finite dimensionaland if 0 is structurally stable under C1-small perturbationof f then 0 is hyperbolic.

Of the statement we have just outlined, the proof is avery important one. It forms a basis for our discussion ofstructural stability in Chapter 6.

Let us now discuss Hartman's linearization theorem inthe context of flows. It was discovered independently byGrobman [1959,1962].

Theorem 4.7.10 Let T e HL(B). For all Lipschitz maps 'e C°(B) with sufficiently small Lipschitz constant, there isa flow equivalence from T to T+n.

Let us recall the definition of the Hessian of a map ata critical point. Let M be C', f is C' on M. f has acritical point at p e M if dfP = 0. If p is a critical pointof f, then the Hessian of f at p, Hf, is a bilinear functionon MP defined as: if u,v a MP, X e X(M) such that if X(p) _u, then Hf(u,v) = v(Xf). Since dfP = 0, then Hf(u,v) isindependent of the choice of X and Hf is symmetric. WithHessian defined, we can state the following corollaries.

Corollary 4.7.11 A hyperbolic zero, p, of a C' vectorfield on M is flow equivalent to the zero of its Hessian at

pCorollary 4.7.12 Flow equivalence and orbit equivalence

coincide for hyperbolic zeros of C' vector fields on ann-dim. manifold M. There are precisely n+l equivalenceclasses of such points with respect to either relation.

Similar to the statement about the relationship betweenhyperbolic fixed point and structural stability, one canprove the following: Let X be a C' vector field on B with azero at p. If p is hyperbolic then it is stable with respectto flow equivalence under C1-small perturbations of X.Conversely, if B is finite dim. and if p is stable with

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respect to topological equivalence under C1 smallperturbation of X. then p is hyperbolic.

It should be emphasized that hyperbolicity of fixedpoints is not an invariant of topological or flowequivalence. In other words, it is possible for anon-hyperbolic fixed point to be topological or flowequivalent to a hyperbolic fixed point. For instance, thenon-hyperbolic zero of the vector field X(x) = x3 on R isclearly flow equivalent to the hyperbolic zero of the vectorfield W(x) = x.

There are three distinct possibilities for a fixed pointp of a flow 0 to be topologically equivalent to ahyperbolic fixed point q of a flow µ: (a) (Tgµ)s = Tqµ, (b)

(Tgµ)u = Tqµ, and (c) niether (a) nor (b). We call p (and q)(a) a sink, (b) a source, and (c) a saddle point,respectively.

sink, source, saddle.

It is clear that any sink p has the property that it isasymptotically stable in the sense of Liapunov (we shalldiscuss this in the next chapter). In other words, anypositive half orbit of a sink p, t ? 0)starting at a point x near p stays near p and eventurallyends at p. Recall in the introduction (Chapter 1), in thelanguage of differential equations, asymptotic stability isconcerned with the way an individual solution of a system

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varies as the initial conditions are varied. This isdifferent from structural stability, which deals with theway the set of all solutions varies as we change the systemitself.

It is appropriate at this juncture for us to discusssome properties of hyperbolic closed orbits and Poincaremaps. Let p be a point on a closed orbit of a Cr flow (r

1) on a manifold M. Suppose that the orbit r = of 0through p has period r. Then 07: M - M is a Crdiffeomorphism with p its fixed point. Where the tangent mapTP,P' is a linear homeomorphism of MP. It keeps the linearsubspace <X(x)> of MP pointwise fixed, where X(x) is thevelocity of 0 at x, because 0' keeps the orbit r pointwisefixed. r is said to be hyperbolic if MP has a TPO'- invariantsplitting, i.e., MP = <X(p)> a FP where TPt'IFP is hyperbolic.Thus, hyperbolicity for a closed orbit means that the linearapproximation to 0' is as hyperbolic as possible.

A clear geometrical picture can be obtained in terms ofPoincare maps. Let N be some open disk embedded as asubmanifold of M through p such that <X(x)> a NP = MP.Notice that this is exactly the condition of transversality.Indeed, we say that N is transverse to the orbit at p, andcall it a cross-section to the flow at p. We assert that forsome small open neighborhood U of p in N, there is a uniquecontinuous function p: U - R such that Q(p) = r, and0(p(y),y) a N. Here /3 is a first return function for N. Anytwo such functions agree on the intersection of theirdomains. Intuitively, fl(y) is the time it takes a pointstarting at y to move along the orbit of the flow in thepositive direction until it hits the section N again. Wedenote the point at which it hits again by f(y). Thus, f: U- N is a map defined by f(y) = O(Q(y),y).

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We call f a Poincare map for N. It is well defined up to thedomain. So far, it is not required that r to be hyperbolic.The next theorem shows that r is hyperbolic precisely whenthe Poincare map for some section N at p has a hyperbolicfixed point at p.

Theorem 4.7.13 For sufficiently small U, the firstreturn function Q is well defined and Cr, and the Poincardmap f is a well defined Cr diffeomorphism of U onto an opensubset of N. If MP has a TPO'-invariant splitting <X(p)> eFP, then T

Pf is linearly conjugate to TPO'IFP. The orbit r is

hyperbolic iff p is a hyperbolic fixed point of f.Let f as before, and f': U' - f'(U') be a diffeomorphism

of open subsets of a manifold N'. A topological conjugacyfrom f to f' is a homeomorphism h: U U f(U) - U' U f'(U')such that, for all y e U, hf(y) = f'h(y), here U denotesunion.

Proposition 4.7.14 If two diffeomorphisms f: U - f(U)and f': U' - f'(U') are topologically conjugate, then theirsuspensions are flow equivalent.

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The above result simplifies the problem of classifyingsuspension (see Sect.4.1). The main connection between theflow near a closed orbit r of a Cr flow 0 on M and thePoincare map at a cross-section N of the flow at p of r isthe following theorem.

Theorem 4.7.15 Let f: U -, f(U) be a Poincard map at pfor the cross-section N. Then there is a Cr orbit preservingdiffeomorphism h from some neighborhood of the orbit of

the suspension E(f) to some neighborhood of the orbit r ofsuch that h(p) = p.

Since if r is hyperbolic, then any Poincare map f has ahyperbolic fixed point at p. Then by Corollary 4.7.8 andProposition 4.7.14 we have

Corollary 4.7.16 If r is hyperbolic then the flow 0 istopologically equivalent at r to E(Tpf) at its unique closedorbit.

From Corollary 4.7.9 we have:Corollary 4.7.17 There are precisely 4n different

hyperbolic closed orbits that can occur in a flow on an(n+l)-dim. manifold (n >- 1).

Hyperbolic closed orbits are structurally stable, in thesense that, if r is such an orbit of a C' vector field X onM. and Y is a vector field on M that is C'-closed to X, thenfor some neighborhood U of r in M, Y has a unique closedorbit in U. and this closed orbit is topologicallyequivalent to r. We shall discuss structural stability inmore detail in Chapter 6.

Recall that in Hartman's theorem we altered a hyperboliclinear homeomorphism T by a perturbation n and found atopological conjugacy h = id+g from T to T+n. It was pointedout that h is not necessarily C' even when n is C', sincedifferentiating the conjugacy relation would place algebraicrestrictions on the first derivatives of T+n. The questionarises naturally as to whether further differentiationplaces further restrictions on higher derivatives, and evenif these algebraic restrictions are satisfied, if thesmoothness of n has any effect on h. It turns out that, in

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finite dimensions, further restrictions are the exceptionrather than the rule. We have some positive results on thesmoothness of conjugacy relations. The main theorem is dueto Sternberg [1957, 1958], for more details, see forinstance, Nelson 1969], and a couple of relevant theorems byHartman.

Theorem 4.7.18 (Sternberg) Let T e L(R") haveeigenvalues a1,...,a, (can be complex or degenerated)satisfying a1 + c1'i .... an'- for all 1 <_ i -< n and for allnon-negative integers m1...... m" with m; >- 2. Let n: U -R" be a CS map (s 1) defined on some neighborhood U of 0with n(0) = Dn(O) = 0. Then (T+n) 10 is Cr conjugate to T1O,where, for a given T, r depends only on s and tends to oo ass does.

Notice that the eigenvalue condition implies that T eHL(R") . Also, if n is C°, the maps T and T+" are C° conjugateat 0.

Theorem 4.7.19 Let T E L(R") be a contraction and n: U- R" be a C' map defined on some neighborhood U of 0, withn(0) = Dn(0) = 0. Then (T+n)IO is c' conjugate to T10.

Theorem 4.7.20 (Hartman) Let T E HL(R"), where n = 1 or2, and let n be the same as above. Then (T+n)IO isconjugate to TIO.

C'

Nitecki [1979] studied the dynamic behavior of solutionsfor systems of the form dx/dt = G(F(x)) where F is areal-valued function on R" near equilibria and periodicorbits. He has found that for n >_ 3, the behavior nearperiodic orbits is arbitrary, i.e., any differomorphism ofthe (n-l)-disk isotopic to the identity arises as thePoincare map near a periodic orbit for some choice of F andG. On the other hand, for n >_ 2 the behavior near equilibriais severely restricted. Indeed, (a) if dG/dF + 0 at anequilibrium point, the flow in a neighborhood is conjugateto that of a constant vector field multiplied by a function;(b) if an equilibrium point is isolated, it is an extremumof F, and if F satisfies a convexity condition near theequilibrium, then the flow in a neighborhood resembles that

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described in (a), except that the stable and unstable setsmay be cones instead of single orbits; and (c) when n = 2, astronger condition on F, together with a weaker condition ofG, again yields the description in (a).

Finally, let us end this chapter by giving a list offurther readings. First of all, the classic review articleby Smale [1967] is the one on the "must-read" list. Irwin[1980] is the source of most of the notes in this chapter. Amost recent introductory book, which takes the reader a longway, by Ruelle [1989] is highly recommended.

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Chapter 5 Stability Theory and Liapunov's Direct Method

5.1 IntroductionEarlier in Chapter 1, we described the general idea of

stability in the normal sense. Very early on, the stabilityconcept was specialized in mechanics to describe some typeof equilibrium of a particle (or a celestial object) orsystem (such as our solar system). For instance, consider aparticle subject to some forces and posessing an equilibriumpoint qo. The equilibrium is stable if after anysufficiently small perturbations of its position andvelocity, the particle remains arbitrarily near qo witharbitrarily small velocity. As we have discussed in Chapter1, the well known example of a pendulum, and we shall notdwell on it. It suffices to say that the lowest position(equilibrium point), associated with zero velocity, is astable equilibrium, whereas the hightest one also with zerovelocity is an unstable one.

We have also briefly mentioned that when fomulated inprecise mathematical terms, this "mechanical" definition ofstability was found very useful in many situations, yetinadequate in many others. This is why a host of otherconcepts have been introduced and each of them relates tothe "mechanical" definition and to the common sense meaningof stability. Contrary to the "mechanical" stability,Liapunov's stability has the following features:1°. it does not pertain to a material particle (or theequation), but to a general differential equation;2°. it applies to a solution, i.e., not only to anequilibrium or critical point.

In his memoire, Liapunov [1892] dealt with stability bytwo distinct methods. His 'first method' presupposes anexplicit solution known and is only applicable to somerestricted, but important situations. While this 'secondmethod' (also called 'direct method') does not require theprior knowledge of the solutions themselves. Thus,Liapunov's direct method is of great power and advantage. An

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elementary introduction to Liapunov's direct method can befound in La Salle and Lefschetz [1961].

Suppose for an autonomous system dx/dt = f(x), in whichthe system is initially in an equilibrium state, then itremains in that state. But this is only a mathematicalstatement, yet a real system is subject to perturbations andit is impossible to control its initial state exactly. Thisbegs the question of stability, that is, under an arbitrarysmall perturbation will the system remain near theequilibrium state or not? In the following, we shall makethese concepts precise, and we will discuss these questionsextensively.

Let dx/dt = f(x,t) where x and f are real n-vectors, t eR ("time"), f is defined on RxR". We assume f is smoothenough to ensure its existence, uniqueness, and continuousdependence of the solutions of the initial value problemassociated with the differential equation over RxR". Forsimplicity, we assume that all solutions to be mentionedlater exist for every t e R. Let 11.11 denote any norm on R.

A solution x1(t) of dx/dt = f(x,t) is stable at t , in

the sense of Liapunov if, for every e > 0, there is a 6 > 0such that if x(t) is any other solution with IIx(to) -

x'(to)II < b, then 11x(t) - x'(t)IJ < e for all t >_ t0.

Otherwise, x' (t) is unstable at to. Thus the stability at tois nothing but continuous dependence of the solution on xo =x(to), uniform with respect to t e [t0,oo).

We can gain some geometric insight into this stabilityconcept by considering the pendulum. As before, the set offirst order differential equations have the form: dO/dt = 0,do/dt = - gsinO. The origin of the phase space (9,n)represents the pendulum hanging vertically downward and isat rest. As we have shown before in the phase portrait, allsolutions starting near the origin form a family ofnon-intersecting closed orbits encircling the "origin".Given e > 0, consider an orbit entirely contained in thedisk B, of radius a centered at the "origin". Further chooseany other disk B6 of radius 6 contained in this chosen

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orbit. Due to the non-intersecting nature of the orbits,every solution starting in B6 at any initial time remains inBE. This demonstrates stability of the equilibrium for anyinitial time. On the other hand, any other solutioncorresponding to one of the closed orbits is unstable. Thisis because the period of the solution varies with the orbitand two points of (6,n)-plane, very close to each other at t= to, but belonging to different orbits, will appear inopposition after some time. This happens no matter how smallthe difference of periods is. Yet, it remains that theorbits are close to each other. This leads to the concept oforbital stability.

We have discussed the dissipative system, such as thedamped pendulum, whereby the stable equilibrium becomesasymptotically stable. That is, if all neighboring solutionsx(t) of x1(t) tend to x1(t) when t - oo. We have also said afew words about our Solar system and the notion of Lagrangestability. In the following we shall give variousdefinitions of stability and attractivity for our futureuse. We shall note that if we replace x by a new variable z= x - x'(t), then dx/dt = f(t,x) becomes dz/dt = g(t,z)f(t,z+x'(t)) - f(t,x'(t)), where g(t,O) = 0 for all t c R.The origin is a critical point of the transformed equationand stability of the solution x(t) of the original equationis equivalent to stability of this critical point for thetransformed equation. Of course, such a transformation isnot always possible, nor is it always rewarding.Nonetheless, for the time being, we shall concentrate onstability of critical points.

Let us consider a continuous function f: I x D - R" ,

(t,x) -+ f(t,x) where I = (r,oo) for some r e R or r = - oo,and D is a domain of R", containing the origin. Assume thatf(t,0) = 0 for all t e I so that for the differentialequation dx/dt = f(t,x), the origin is an equilibrium orcritical point. Furthermore, assume f to be smooth enough sothat through every (to,xo) a IxD, there passes one and onlyone solution of dx/dt = f(t,x). We represent this solution

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by x(t;to,xo), thus displaying its dependence on initialconditions. By definition, x(to,to,xo) = xo. For the rightmaximal interval where is defined, we writeJ`(to,xo) or simply J. We recall that Bo = (x a R": IIxII < a).

The solution x = 0 of dx/dt = f(t,x) is stable if for agiven e > 0 and t

0

e I, there is a 6 > 0 such that for allxo e B6 and t e J', one has ii x (t; to, xo) II < e. The solution isunstable if for some e > 0 and to e I and each 6 > 0 thereis an xo a B. and a t E J' such that lI x (t ; to, xo) li >- e. Thesolution is uniformly stable if given e > 0, there is a 6 =6(e) such that lix(t;to,xo) ll < e for all to a I, all 1Ix011 < 6and all t ? to.

Since J' c [to,oo), thus in principle any solution maycease to exist after a certain finite time. Nonetheless, ifBe c D, the solutions mentioned in the definitions ofstability and uniform stability may continue up to oo.

The solution x = 0 of dx/dt = f(t,x) is attractive iffor each to e I there is an n = n(to), and for each e > 0and each iixoll < n, there is a a = a(to,e,xo) > 0 such thatt+a e J' and llx(t;t(,,xo)ll < e for all t >_ to+a. The solution

is egui- attractive if for each to c I there is an n = n(to)and for each e > 0 a a = a(to,e) such that t. +a e J' and

Ilx(t;to,xo) it < e for all IIx0Ii < n and all t >- to+a. Thesolution is uniformly attractive if for some n > 0 and eache > 0 there is a a = a (E) > 0 such that to +a e J' andIlx(t;to,xo) li < e for all llxoii < n, all to 6 I and all tto a .

As remarked earlier, if B, c D, the solutions mentionedimmediately above exist over [to,oo). Thus, in the definitionof attractivity, all solutions starting from B. approach theorigin as t - oo. For equi-attractivity, they tend to 0uniformly with respect to xo e Bn whereas for uniformattractivity they tend to 0 uniformly with respect to xo eBn and to a I.

We can define attraction slightly differently. Let thephase space X be locally compact, and M a non-empty compactsubset of X. The region of weak attraction of M, Aw(M) = (x

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E X: n(x) n M + 0), the region of attraction of M, A(M) = {xc X: n(x) + 0 and n(x) a subset of M), and the region ofuniform attraction of M, Au(M) = {x c X: J'(x) + 0 and J+(x)a subset of M). Furthermore, any point x in Aw(M), A(M),Au(M) is said to be weakly attracted, attracted, oruniformly attracted to M respectively.

Proposition 5.1.1 Given M. a point x is:(i) weakly attracted to M iff there is a sequence {tn) in Rwith to - - and d(xtn,M) - 0, where d(.,.) is the metricdistance,(ii) attracted to M iff d(xt,M) - 0 as t y co,

(iii) uniformly attracted to M iff for every neighborhood Vof M there is a neighborhood U of x and a T > 0 with Ut asubset of V for t >_ T.

Theorem 5.1.2 For any given M, Au(M) is a subset ofA(M), which is a subset of Aw(M), and they all areinvariant.

A given set M is said to be: (i) a weak attractor ifAw(M) is a neighborhood of M. (ii) an attractor if A(M) is aneighborhood of M. (iii) a uniform attractor if Au(M) is aneighborhood of M. (iv) stable if every neighborhood U of Mcontains a positively invariant neighborhood V of M, (v)

asymptotically stable if it is stable and is an attractor. Aweak attractor will be called a global weak attractor ifAW(M) = X. Similarly for global attractor, global uniformattractor. An attractor is a strange attractor if itcontains a transversal homoclinic orbit. The basin ofattraction of A(M) is the set of initial points p c M suchthat O(p) approaches A(M) as t - -, where 0 is the flow.

It is important to point out at this point thatattractivity does not imply stability! For instance, anautonomous system in R2 presented in Hahn [1967]. As inFig.5.1.1, where r is a curve separating bounded andunbounded orbits. The origin is unstable, in spite of thefact that every solution tends to it as t - oo.

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y

Fig. 5.1.1

Here we offer a slightly different definition ofattraction. The origin of attraction of the origin at to isthe set A(to) = (x a D: x(t;to,xo) -+ 0, t - oo). If A(to) doesnot depend on to, we say that the region of attraction isuniform. Furthermore, if D = R" = A(to) for every to, thenthe origin is globally attractive. The origin is uniformlyglobally attractive if for any n > 0, any to e I and any xoe B,), x (t; to, xo) 0 as t oo, uniformly with respect to toand xo.

Asymtotic stability: If the solution x = 0 of dx/dt =f(t,x) is stable and attractive. If it is stable andequi-attractive, it is called equi-asymptotically stable. Ifit is uniformly stable and uniformly attractive, it iscallec uniformly asymptotically stable. If it is stable andglobally attractive, it is called globally asymptoticallystable. If it is uniformly stable and uniformly globallyattractive, then it is called uniformly globallyasymptotically stable.

Example: dx/dt = - x, x c R. The origin is uniformlyglobally asymptotically stable. Check the definitions.

Example. Consider a planar dynamical system defined by

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the differential equations,dr/dt = r(1 - r), dO/dt = sin'(0/2).

The orbits are shown in Fig.5.1.2. These consist of twofixed points p, = (0,0) and p2 = (1,0), an orbit r on theunit circle with (p2) asd the positive and the negativelimit set of all points on the unit circle. All points p,where p + p2, have n(p) = (p2). Thus (p2) is an attractor.But it is neither stable nor a uniform attractor. Note alsothat for any p = (a,0), a > 0, J+ (p) is the unit circle.

Fig. 5.1.2

Now let us state some results about attractors.Theorem 5.1.3 If M is a weak attractor, attractor, or

uniform attractor, then the coresponding set A.(M), A(M), orAu(M) is open (indeed an open neighborhood of M).

Theorem 5.1.4 If M is stable then it is positivelyinvariant. As a consequence, if M = (x), then x is a fixedpoint.

Theorem 5.1.5 A set is stable iff D+(M) = M.In view of the above theorem, we can give the following

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definition. If a given set M is unstable, the non-empty setD+(M) - M will be called the region of instability of M. IfD+(M) is not compact, then M is said to be globallyunstable.

Theorem 5.1.6 If M is stable and is a weak attractor,then M is an attractor and consequently asymptoticallystable.

Theorem 5.1.7 If M is positively invariant and auniform attractor, then M is stable. Consequently M isasymptotically stable.

Theorem 5.1.8 If M is asymptotically stable then M is auniform attractor.

Theorem 5.1.9 Let f(t,x) in dx/dt = f(t,x) beindependent of t or periodic in t. Then, stability of theorigin implies uniform stability, and asymptotic stabilityimplies unform asymptotic stability [Yoshizawa 1966].

Note that there is no such theorem for attractivity oruniform attractivity. The next few theorems determinewhether the components of a stable or asymptotically stableset inherit the same properties. We would like to point out(it is very easy to show) that in general the properties ofweak attraction, attraction, and uniform attraction are notinherited by the components.

Theorem 5.1.10 A set M is stable iff every component ofM is stable.

It should be noted that the above theorem holds even ifX is not locally compact [Bhatia 1970].

Theorem 5.1.11 Let M be asymptotically stable, and letM be a component of M. Then M* is asymptotically stable iffit is an isolated component.

The following very important theorem is a corollary tothe above theorem.

Theorem 5.1.12 Let X be locally compact and locallyconnected. Let M be asymptotically stable. Then M has afinite number of components, each of which is asymptoticallystable.

Outline of proof. A(M) is open and invariant. Since X is

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locally connected, each component of A(M) is also open. Thecomponents of A(M) are therefore an open cover for thecompact set M. Thus only a finite number of components ofA(M) is needed to cover M. Now it is ealy to show that eachcomponent of A(M) contains only one component of M.

Theorem 5.1.13 Let M be a compact weak attractor. ThenD+ (M) is a compact asymptotically stable set with A(D*(M))Aw(M). Moreover, D+(M) is the smallest asymptotically stableset containing M.

5.2 Asymptotic stability and Liapunov's theoremIn his Memoire, Liapunov [1892] gave several theorems

dealing directly with the stability problems of dynamicalsystems. His methods were inspired by Dirichlet's proof ofLagrange's theorem on the stability of equilibrium, isreferred to by Russian authors as Liapunov's second method.Liapunov's first method for the study of stability rests onconsidering some explicit representation of the solutions,particularly by infinite series. What is known as Liapunov'ssecond (or direct) method is making essential use ofauxiliary functions, also known as Liapunov functions. Thesimplest type of such functons are C1, V: IxD - R, (t,x) -

V(t,x) where I and D as before. Note that the space X isassumed locally compact implicitly.

An interesting introduction to Liapunov's direct methodcan be found in La Salle and Lefschetz [1961]. A much morecomplete and detailed treatment of stability as well asattractivity can be found in Rouche, Habets and Laloy[1977]. Many of the material in this chapter are coming fromRouche, Habets and Laloy [1977].

If x(t) is a solution of dx/dt = f(t,x), the derivativesof the time function V'(t) = V(t,x(t)) exists and dV'(t)/dt= (aV/ax)(t,x(t))f(t,x(t)) + (aV/at)(t,x(t)). (5.2-1)

If one introduces the function dV/dt: IxD - R, (t,x) -

dV(t,x)/dt by dV(t,x)/dt = (aV/ax)(t,x)f(t,x) +(aV/at)(t,x), it then follows that dV'(t)/dt =

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dV(t,x(t))/dt. Thus, computing dV'(t)/dt at some given timet does not require a knowledge of the solution x(t), butonly the value of x at t.

Another useful type of function is the function of classK, i.e., a function a e K, a: R* -+ R+, continuous, strictlyincreasing, with a(0) = 0.

We call a function V(t,x) positive definite on D, ifV(t,x) is defined as before, and V(t,0) = 0 and for somefunction a e K, every (t,x) a IxD such that V(t,x) ? a(IIxII).

When we say that V(t,x) is positive definite withoutmentioning D, it means that for some open neighborhood D'D of the origin, V is positive definite on D'.

A well known necessary and sufficient condition for afunction V(t,x) to be positive definite on D is that thereexists a continuous function V*: D - R such that V*(0) = 0,V*(x) > 0 for x e D, x + 0, and furthermore V(t,O) = 0 andV(t,x) ? V*(x) for all (t,x) a IxD.

The following two theorems are pertinent to thedifferential equation dx/dt = f(t,x).

Theorem 5.2.1 [Liapunov 1892] If there exists a C'function V: IxD R and for some a e K, and every (t,x) e

IxD such that,

(i) V(t,x) ? a(IIxII) ; V(t,O) = 0;(ii) V(t,x) 5 0; then, the origin is stable.

Proof: Let to e I and e > 0 be given. Since V iscontinuous and V(to,0) = 0, there is a 6 = 6(to,e) > 0 suchthat V(to,xo) < a(e) for every xo e B5. Writing x(t;to,xo) byx(t), and using (ii), one obtains for every xo a Ba and

every t e J+ : a(IIxII) 5 V(t,x(t)) <_ V(to,xo) < a(c). But

since a e K, one obtains that IIx(t)II < e .

Theorem 5.2.2 In addition to the assumptions in Theorem5.2.1, if for some b e K and every (t,x) a IxD: V(t,x) <-

b(IIxII), then the origin is uniformly stable.

Let us use this theorem to derive the well-knownstability criteria for linear systems or the linearapproximation. Let x be an n-vector in R", and the systemcan be written as:

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dx/dt = Px + q(x,t),where P is a constant nonsingular matrix and the nonlinearterm q is quite small with respect to x for all t ? 0. Forsimplicity, without loss of generality, we assume that thecomponents of q have in some region D and for t >- 0

continuous first partial derivatives in xk and in t. Thus,in D and for t ? 0, the existence theorem applies. Let usassume that the characteristic roots r1, r2, ..., rn of thematrix P are all distinct and consider first the case wherethey are all real. Clearly, one can choose coordinates insuch a way that P is diagonalized and P = diag(r1,...,rn).Then, (a) if the roots are all negative. Take

V = x12 + . . . + xn2 , anddV/dt = (r,x12 + ... + rnxn2) + s(x,t),

where s is small compared with the parenthesis. Thus, in asufficiently small region D both V and -dV/dt are positivedefinite functions. Thus they satisfy the conditions ofTheorem 5.2.1, thus the origin is asymptotically stable. (b)

Some of the roots, say, rj,..., rP (p < n) are positive, andthe remaining are negative. Now take

V = x12 + ... + xP2 - xP+12 - ... - Xn2Then we have,dV/dt = (r1 x12 + ... + rPx,2 - rP+1 xP+1 - ... - rnxn2 ) + s(x,t)where s is small compared with the parenthesis. At somepoints arbitrarily near the origin V is positive. Since rp+,,..., rn < 0, thus dV/dt is positive definite, and the originis unstable. In other words, a sufficient condition for theorigin of the system to be asymptotically stable is that thecharacteristic roots all have negative real parts. If thereis a characteristic root with a positive real part, then theorigin is unstable.

As an example, there is an interesting application tothe standard closed RLC circuit with nonlinear coupling. Theequation of motion of the charge q is:

Ld2q/dt2 + Rdq/dt + q/C + g(x,dx/dt)) = 0,where dq/dt is the current, g represents nonlinear couplingwith terms of at least of second order. We can rewrite this

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second order differential equation as:dq/dt = i, di/dt = -q/LC - iR/L - g(q,i).

Clearly, the origin is a critical point and itscharacteristic roots are the roots of R2 + Rr/L + 1/LC = 0.Since R, L and C are positive, the roots have negative realparts. If R:/L2 < 4/LC or R2 < 4L/C, then the characteristicroots are both complex with negative real part -R/L. Whenthis happens, we have spirals as paths and the origin isasymptotically stable. The origin is a stable focus. If R2 >

4L/C, the origin is a stable node and is asymptoticallystable.

Next, we want to introduce the concept of partialstability and the corresponding stability theorem. Let twointegers in, and n > 0, and two continuous functions f:IXDXRm - R", g: IxDxRm - Rm, where I = (r,a) , D is a domainof R" containing the origin. Assume f(t,0,0) = 0, g(t,0,0) _0 for all t e I, and f and g are smooth enough so thatthrough every point of IxDxRm there passes one and only onesolution of the differential system

dx/dt = f(t,x,y) , dy/dt = g(t,x,y) . (5.2-2)

To shorten the notation, let z be the vector (x,y) a R"+'" and

z (t; to, zo) = (x (t; to, zo) , y (t; to, zo) ) for the solution of thesystem. Eq.5.2-2 starting from zo at to.

The solution z = 0 of Eq.(5.2-2) is stable with respectto x if given e > 0, and to a I, there exists 6 > 0 suchthat lIx(t;to,zo)II < e for all ze a B. and all t e J. Uniformstability with respect to x is defined accordingly.

Theorem 5.2.3 If there exists a C' function V: IXDXR10 -R such that for some a e K and every (t,x,y) a IXDXR°,

(i) V(t,x,y) ? a(jjxjj), V(t,0,0) = 0;(ii) dV(t,x,y)/dt 5 0; then the origin z = 0 is stable withrespect to x. Moreover, if for some b c K and every (t,x,y)e IXDx1 ,(iii) V(t,x,y) 5 b(IIxIj + Ilyll); then the origin is uniformly

stable in x.Examples: (i) Consider the linear differential equation

dx/dt = (D(t) + A(t))x, where D and A are nxn matrices,

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being continuous functions of t on I e (r,co), D diagonal, A

skew- symmetric, x e R". Choosing V(t,x) = (x,x), (where(x,x) is a scalar product), we get dV(t,x)/dt = 2(x,D(t)x).If the elements of D are <- 0 for every t e I, then V <- 0 and

from Theorem 5.2.2, the origin is uniformly stable.(ii) Next, we shall look at the stability of steady

rotation of a rigid body. Consider a rigid body with a fixedpoint 0 in some inertial frame and no external forceapplied. Let I, M, N be the moments of inertia with respectto 0, and tt is the angular velocity in the inertial frame.The Euler equations of Si, in the principal axes of inertia a

(p,q,r), areIdp/dt = (M - N)qr,Mdq/dt = (N - I)rp, (5.2-3)

Ndr/dt = (I - M)pq.The steady rotations around the first axis correspond to thecritical point p = po, q = 0, r = 0. Define new variables,x = p - p0, y = q, z = r, the critical point is shifted tothe origin, and Equations (5.2-3) becomes

dx/dt = (M - N)yz/I,dy/dt = (N - I) (po + x)z/M, (5.2-4)

dz/dt = (I - M) (po + x)y/N.If I <- M -< N, the steady rotation is around the largest axisof the ellipsoid of inertia. An auxiliary function suitablefor Liapunov's theorem isV = M(M - I)y2 + N(N - I)z2 + [My2 + Nz2 + I(x2 + 2xpo)] 2

(5.2-5)

which is a first integral, such that dV/dt = 0. It followsthat the origin is stable for Eq.(5.2-4). Furthermore, it iseven uniformly stable since the system is autonomous. If I>M ? N, one obtains another auxiliary function using thefirst integral,V = M(I - M)y2 + N(I - N)z2 + [My2 + Nz2 + I(x2 + 2xpo) ]2 .

(5.2-6)

Therefore, the steady rotations of the body around thelongest and shortest axes of its ellipsoid of inertia arestable with respect to p, q, r. Note that the auxiliary

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functionsintegralspresent ashortly.

Egs.(5.2-5) and (5.2-6) are combinations of theof energy and of moment of mementa. We shallgeneral method for constructing such combinations

(iii) The thirdglider. This may

example deals with the stability of aas well be a hovering bird. First, suppose

the plane of symmetry coincides at any moment with avertical plane in an inertia frame. Let v be the velocity ofits center of mass, and 0 the angle between v and horizontalaxis. The longitudinal "axis" of the glider is assumed tomake a constant angle a with v. Let m be the mass of theglider and g be the gravitational acceleration. Let CD(a)and CL(a) be the coefficients of drag and lift respectively.The equations of motion are:mdv/dt = - mgsinO - CD(a)v', mvd0/dt = - mgcos0 + CL (a)v'.

Letting vol = mg/CL, r = gt/vo, y = v/vo, and a = CD/CL, wethen have transformed the equations into

dy/dr = - sinO - ay'dO/dr = (-cosO + y2)/y. (5.2-7)

We have introduced a non-vanishing drag for future referenceonly.

For the moment, let a = 0, then the above equations havethe critical points yo = 1, 00 = 2rk for k an integer, allof them corresponding to one and the same horizontal flightwith constant velocity. Let us concentrate on yo = 1, 00 =0. One can easily verify that V(y,6) = (y3/3) - ycosO + 2/3is a first integral for Equations (5.2-7) with a = 0. Insome neighborhood of (1,0), V(y,A) > 0 except V(1,0) = 0.Therefore, if the critical point (1,0) is translated to theorigin, V expressed in the new variables satisfies thehypotheses of Theorem 5.2.2, thus one can prove the uniformstability. For more detail, see e.g., Etkin [1959].

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Fig. 5.2.1

We now go back to the general setting and state somesufficient conditions for instability.

Theorem 5.2.4 [Chetaev 1934] If there exist to e I, e >

0, an open set S c BE, (with BE c D), and a C' function V:(to,oo)xB, R such that on [to,oo)xS:(i) 0 < V(t,x) <_ k < oo , for some k;(ii) dV(t,x)/dt >_ a(V(t,x)), for some a e K; if further(iii) the origin of the x-space belong to aS;(iv) V(t,x) = 0 on [to ,oo)x()S n B,);then the origin is unstable.

The following two corollaries of Theorem 5.2.4 were infact established long before Theorem 5.2.4 was known.

Corollary 5.2.5 [Liapunov 1892] If there exist to e I,

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e > 0, an open set S c B,, (with B, c D), and a c' functionV: [to,-)xB, - R such that on [to,-)xS:(i) 0 < V(t,x) <_ b(jjxjj) for some b c K;(ii) dV(t,x)/dt >_ a(jjxII) for some a e K; if further(iii) the origin of the x-space belongs to aS;(iv) V(t,x) = 0 on (to,oo)x(as n B,); then the origin is

unstable.Corollary 5.2.6 [Liapunov 1892] If in Corollary 5.2.5,

(ii) is replaced by(ii') V(t,x) = c V(t,x) + W(t,x) on [to,oo)xS, where c > 0,and W: [to,-)xS R is continuous and >- 0, then the originis unstable.

Note that, if the differential equation is autonomousand if V depends on x only, then (i) and (ii) in Theorem5.2.4 can be simplified to: (i) V(x) > 0 on S; and (ii)dV(x)/dt > 0 on S.

The main use of Corollary 5.2.6 is to help proveinstability by considering the linear approximation. This isa very useful way to look at many applications. Suppose,dx/dt = f(t,x) can be specified as:

dx/dt = Ax + g(t,x) (5.2-8)

where A is an nxn real matrix and Ax + g(t,x) has all theproperties required from f(t,x). Then we have the followingtheorem.

Theorem 5.2.7 [Liapunov 1892] If at least oneeigenvalue of A has strictly positive real parts and ifIjg(t,x) II/IIxII - 0 as x -+ 0 uniformly for t e I, then theorigin is unstable for Eq.(5.2-8).

This theorem has very important applications, inparticular, when the system can be linearized. There aremany practical systems which satisfy the above criteria ofdecomposing f(t,x) into Ax + g(t,x). We shall encounter thistheorem later in Chapter 7.

As an example for immediate demonstration, we shallbriefly discuss the Watt's governor. It is a well-knowndevice and it is sufficient to present it as in Fig.5.2.2.If we disregard the friction, the equations of motion are:

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d(IdO/dt)/dt = -k(¢ - m0),d(I'dO/dt)/dt - ((d9/dt)'/2)(aI/ao) = all/am,

where I = C + 2ml2sin2O, I'= 2m1', U = 2mlcoso and C, 1, m,

k and g are positive constants.

Fig. 5.2.2

The steady motion 0 = O0, dO/dt = 0, (d6/dt)' _ (d60/dt)' _g/lcoso0 is unstable by Theorem 5.2.7. It is straightforward to show that the eigenvalue equation for the linearpart of the equation is:

(c/2m1' + sin' O0) µ3 + (d60/dt) 2 sin200(l + 3cos'.0 + c/2m1') g+ (k/2m1')00sin200 = 0.

When d90/dt < 0, then e(0) < 0, whereas e(µ) when µ --, thus there is a strictly positive eigenvalue. Likewise,when d60/dt > 0, there is a strictly negative eigenvalue.Thus at least one eigenvalue has a strictly positive realpart. Then, by Theorem 5.2.7, the steady motion beingconsidered is unstable. For more details on Watt's governorand the use of friction to stablize its steady motion, seePontryagin [1962].

The following theorem, due to Liapunov, on uniformasymptotic stability also gives an interesting estimate ofthe region of attraction. Also, uniform asymptotic stabilityhas been studied long before simple asymptotic stability.

Theorem 5.2.8 [Liapunov 1892] Suppose there exists a

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C1 function V: IxD - R such that for some functions a, b, ce K and every (t,x) a IxD: (i) a(IIxII) -< V(t,x) <_ b(IIxII);(ii) dV(t,x)/dt <_ - c(IIxII). Choosing a > 0 such that B. c D,let us set for every t e I, V"' LX = {x a D: V(t,x) <_ a(a)).

Then (a) for any to e I and any xo a V-1t a: x(t;to,xo) -' 0uniformly in to, xo when t co; (b) the origin is uniformlyasymptotically stable.

Note that, assumptions (i) and (ii) are equivalent to(i) and (ii') dV(t,x)/dt -<'- c'(V(t,x)) for some c'e K.Also, the existence of a C' function V(t,x) such that: (i)

V(t,x) >_ a(IIxII); V(t,0) = 0; (ii) dV(t,x)/dt <_ -c(jjxfl),for some a, c e K and every (t,x) a IxD does not implyuniform asymptotic stability, nor even asymptotic stability!This is demonstrated by Massera [1949] by a counter-example.

Corollary 5.2.9 The origin is uniformly globallyasymptotically stable if the assumptions of Theorem 5.2.8are satisfied for D = R" and a(r) - w as r - oo.

To illustrate Theorem 5.2.8, let us consider the scalarequation, which represents an RLC circuit with parametricexcitation, (time varying capacitance),

d2x/dt2 + a dx/dt + b(t)x = 0 (5.2-9)where a > 0, b = bo(l + ef(t)) with bo ? 0, and f(t) is abounded function from R R. Or one can view this equationrepresenting a mechanical oscillator with viscous frictionand a time-varying spring "constant". Eq.(5.2-9) isequivalent to

dx/dt = y, dy/dt = -ay - b(t)x.The auxiliary function V(x,y) _ (y + ax/2)2/2 + (a2/4 +bo)x2/2, is positive definite. Then according to Theorem5.2.8, the origin will be uniformly asymptotically stable ifthe time derivative of V, dV(t,x)/dt = - ay2/2 - (b - boxy- abx2/2 is negative definite. Following Sylvester'scriterion, this is so if for some a,

e2 bof (t) 2 - a2 (1 + of (t) ) <_ - a < 0. (5.2-10)This condition is satisfied for any sufficiently small e.One can view the differential equation as representing twoopposing forces at work: a parametric excitation

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proportional to e, and a load which is the damping forceadx/dt. Satisfying Eq.(5.2-10) tantamount to choosing theresistance, a, large enough for the load to absorb all theenergy provided by the excitation. In this case, the originis asymptotically stable. Otherwise, if the load is notlarge enough, one may expect that the balance of energy ofthe system increases and the origin becomes unstable. Thisis, of course, a heuristic view.

Another example is the damped pendulum, which wediscussed at the very beginning in Chapter 1. We have shownthat the origin is asymptotically stable by choosing thetotal energy as the auxiliary function. Nonetheless, this isnot a good choice because the time derivative of E is notnegative definite. It proves stability but not asymptoticstability. Therefore, a "natural" choice may not always fitTheorem 5.2.8. Finding a suitable auxiliary function isoften a matter of intuition, experience or an "art".

As we have noticed that it is often difficult to exhibitan auxiliary function whose time derivative is negativedefinite, so an alternative way of proving asymptoticstability will be to work out some more elaborate theoremsand propositions to allow one to use auxiliary functionswhose time derivative is non- positive (i.e., <_ 0), but ofcourse along with some more information.

Theorem 5.2.8 can be used to prove asymptotic stabilityby considering the linear approximation as illustrated inthe RLC- circuit with parametric excitation. Now supposedx/dt = f(t,x) is of the form dx/dt = Ax + g(t,x) asspecified before Theorem 5.2.7, then we have thecorresponding theorem.

Theorem 5.2.10 [Liapunov 18921 If all eigenvaleues of Ahave strictly negative real parts and if IIg(t,x)II/IIxI) - 0 asx - 0, uniformly for all t e I. then the origin is uniformlyasymptotically stable.

As an example, let us go back to the problem of theglider considering non-vanishing drag, i.e., described byEgs.(5.2-7). These equations admit a critical point,

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y0 = (1 + a2 ) -114 , 90 = - tan"1a,which corresponding to a rectilinear downward motion atconstant velocity. Without loss of generality, we assumethat 0 > 00 > - lr/2. Change the variables y = y0 + y1, A = e0+ 01, we transfer the origin to the critical point. Bycomputing the terms of first order in y1, A,, we obtain forthe linear variational equation,

dy1/dt = - (2a) y1/ (l+az ) 1'4 - 01/ (l+a' ) 1/2,d01/dt = 2y, - a91/ (l+a2 ) 1/4.

One can verify that the eigenvalues of this set of equationshave strictly negative real part. Therefore, according toTheorem 5.2.10, the critical point is asymptotically stable.

The following couple of theorems are variations ofTheorem 5.2.8.

Theorem 5.2.11 [Massera 1949] If we replace (ii) inTheorem 5.2.8 by (ii'), there exist a function U: IxD - Rand a function c e K such that: U(t,x) ? c(Iixli), U(t,0) = 0,and for any a1, a2 with 0 < al < a2, dV (t, x) /dt + U (t, x) - 0as t - ao , uniformly on at <_ 1lxII <_ a2, then the origin isequi-asymptotically stable.

Theorem 5.2.12 [Antosiewicz 1958] If the origin isuniformly stable, and if there exists a C' function V: IxD -+R such that for some functions a, c e K and every (t,x) e

IxD: (i) V(t,x) ? a(lIxjl); V(t,0) = 0; (ii) dV(t,x)/dt <5 -

c(IlxII); then the origin is equi-asymptotically stable.Next, we shall introduce a theorem which makes use of

two auxiliary functions.Theorem 5.2.13 [Salvadori 1972] Suppose there exist two

C' functions V: IxD - R and W: IxD - R such that for somefunctions a, b, c e K and every (t,x) a IxD:

(i) V(t,x) >- a(jjxjj) ; V(t,0) = 0;(ii) W(t,x) >_ b(iixll); W(t,0) = 0;(iii) dV(y,x)/dt <- -c(W(t,x));(iv) W(t,x) is bounded from below or from above. Choosing a> 0 such that B0 c D, for any t e I, we put V"1t a = (x c D:V(t,x) <- a(a)). Then (a) the region of attraction A(t0)V-1 t,Q,; (b) the origin is asymptotically stable.

217

Note that, the function W in the above theorem can be ofspecial form. For instance, if we identify W(t,x) with(xix), or W(t,x) = V(t,x), we obtain Corollary 5.2.14 or5.2.15 respectively. In fact, these two corollaries wereknown long before Theorem 5.2.13 was established.

Corollary 5.2.14 [Marachkov 1940] Suppose there existsa C1 function V: IxD -+ R such that for some functions a, c eK, and every (t,x) a IxD: (i) V(t,x) ? a(IIxII); V(t,0) = 0;(ii) dV(t,x)/dt 5 - c(xlx). If, moreover, f(t,x) is boundedon IxD, then (a) for every a > 0 such that BQ c D, theregion of attraction A(to) V"1t a; (b) the origin isasymptotically stable.

Corollary 5.2.15 [Massera 1956] Suppose there exists aC' function V: IxD - R such that for some functions a, c eK, and every (t,x) e IxD: (i) V(t,x) ? a(IIxII); V(t,0) = 0;(ii) dV(t,x)/dt <- - c(V(t,x)); then (a) for every a > 0 suchthat BQ c D, the region of attraction A(to) V"1t a; (b) the

origin is asymptotically stable.As an example for the application of Theorem 5.2.13, let

us generalize the damped pendulum by considering thependulum with a variable friction h(t), thus the equation ofmotion becomes: d29/dtz + h(t)dO/dt + sing = 0, where h is aC' function from I R, and we set the gravitationalconstant g = 1 just for simplicity. We are looking for somehypotheses concerning h(t), as mild as possible, and thesystem will entail asymptotic stability of the origin of thephase plane. Let us try the auxiliary function V(t,O,dO/dt)= (dO/dt + asinO)'/2 + b(t)(1 - cosO), which is the sum of aquadratic function. If we put b(t) = 1 + ah(t) - a2, thendV(t,O,dO/dt)/dt = - (h(t) - acosO)(dO/dt)2 - a sin20 +

ah'(t)(1 - cosO) - a'(1 - cosO)sinO d8/dt= - (h(t) - a) (dO/dt)2 - a(2 - h' (t)) (1 - cosO) + 03,

where 03 contains terms of at least third order in 0 anddO/dt, all independent of t. Now, if there exist twoconstants a > a, and P < 2 such that (i) h(t) ? a > a > 0;(ii) h1(t) <- fi < 2; then V and - dV/dt are positivedefinite. Furthermore, let us define W(9,d6/dt) = (d6/dt)2/2

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+ (1- cosO) and dW(t,O,dO/dt)/dt = - h(t)(dO/dt)= <_ 0 isbounded from above. Then from Theorem 5.2.13, the origin isasymptotically stable.

Note that, because of the requirement of f(t,x) bebounded, therefore a bound should be assumed for h(t). Thus,Corollary 5.2.14 could not be used. Corollary 5.2.15 couldnot be used either, at least not the V(t,O,dO/dt) chosenabove, because satisfying (ii) of Corollary 5.2.15 wouldalso require further hypotheses on h(t).

One can also show equi-asymptotical stability of theorigin by using an "variation" of Theorem 5.2.13.

Theorem 5.2.16 [Salvadori 1972] Suppose there exist twoC' functions V: IxD - R, and W: IxD - R such that for some ae R, some function b, c e K, and every (t,x) e IxD:

(i) V(t,x) >_ a;

(ii) W(t,x) >_ b(IIxII) ; W(t,0) = 0;(iii) dV(t,x)/dt 5 - c(W(t,x));(iv) dW(t,x)/dt _< 0;

then the origin is equi-asymptotically stable.Return to partial stability by getting back to

Eq.(5.2-2) along with the accompanying assumptions aspresented there. We suppose further that all solutions ofEq. (5.2-2) exist on [to,oo). The solution z = 0 of Eq. (5.2-2)is said to be uniformly asymptotically stable with respectto x, if it is uniformly stable with respect to x, and iffor given e > 0, to a I, there exist a > 0 and n > 0 suchthat 11x(t,to,zo) II < c for all 11zoll < n, and all t >_ to + a.

Theorem 5.2.17 Suppose there exists a C' function V:IxDxRm-+ R such that for some functions a, b, c e K andevery (t,z) a IXDXR10:

(i) a(IIxII) 5 V(t,z) <- b(IIxII)(ii) dV(t,z)/dt <_ - c(jjxDD). Then(a) for any a > 0 and any (to,zo) E Ix(BQ n D)xRm, x(t;to,zo)

- 0 uniformly in to, zo when t - oo;(b) the origin is uniformly asymptotically stable withrepsect to x.

Theorem 5.2.18 Assume u e R"'k, 0 _< k <_ m, as a vector

219

containing all components of x and k components of y.Suppose there exists a C' function V: IxDxRm R such thatfor some functions a, b, c e K and every (t,z) e IxDxRm:

(i) a(iixii) <- V(t,z) <_ b(iixii);(ii) dV(t,z)/dt -< - c(11xii) ;then the origin is uniformly asymptotically stable withrespect to x.

When the whole space is the region of asymptoticstability, we say that we have complete stability. Then wehave the following:

Theorem 5.2.19 For an autonomous system, let thereexists a C' function V: IxD -+ R such that, V(x) > 0 for allx + 0 and dV(x)/dt <- 0. Let E be the locus dV/dt = 0 and letS be the largest invariant set contained in E. Then allsolutions bounded for t > 0 tend to S as t - oo.

In addition, if we know that V(x) oo as iixii co, theneach solution is bounded for t >- 0, and one can concludethat all solutions approach S as t -+ co. Furthermore, if S isthe origin, we then have complete stability. Thus, in orderto establish complete stability we need to show that everysolution is bounded for t >- 0 and that S is the origin.

Suppose that V(x) - co as Iixii - co and that dV(x)/dt < 0for x + 0. Then S is certainly the origin, and it is easy toshow that every solution is bounded for t ? 0. Let x(t) bethe solution through xo, then for some sufficiently large r,V(x) > V(xo) for all lixil >_ r. But since V(x(t)) decreaseswith t, we note that iix(t)iI < r for all t >_ 0. Hence everysoulution is bounded. Thus,

Theorem 5.2.20 Let V be C'. Suppose (i) V(x) > 0 for x0; (ii) dV/dt < 0 for x + 0; and (iii) V - co as jlxii - oo.

Then the autonomous system is completely stable.In many applications, it occurs that one can construct a

Liapunov function V satisfying Theorem 5.2.20. Examples willbe given in section 7.7.

The question arises whether it is true that stability,asymptotic stability, etc., imply the existence of Liapunovfunctions such as described in various theorems in this

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section. We shall answer this in the next section. It shouldbe pointed out that even though this is a good mathematicalproblem, it is not of great practical importance as we shallsee in the next section.

5.3 Converse theoremsSo far, in most theorems given, the existance of a

Liapunov- like function is assumed. The question arisesnaturally whether such a function actually exists, i.e.,given some stability or attractivity properties of theorigin, can one build up an auxiliary function suitable forthe corresponding theorem? It is answered in the so-calledconverse theorems.

Before we get to the converse theorems, there areseveral comments are in order. First of all, most conversetheorems are proved by actually constructing the suitableauxiliary function. Such construction almost always assumesthe knowledge of the solutions of the differential equation.This is why converse theorems give no clue to the practicalsearch for Liapunov's functions. Second, when the existenceof such and such function is necessary and sufficient forcertain stability property, then any other sufficientcondition will imply the originial one. Nonetheless, thisobservation should not prevent anyone from looking for othersufficient conditions which might be more practical, easierto apply. Third, sometimes a stability property of a systemcan be studied by considering first a simplified system. Letus suppose that the stability of a simplified system can beestablished easily. Then we can deduce from a conversetheorem the existence of a suitable auxiliary function. Thenunder appropriate conditions, it might also be a goodauxiliary function for the original system and prove thestability property of the original system.

Hence we shall give the converses of three importanttheorem given earlier. The general setting is the same asbefore, i.e., considering a continuous function f: IxD - R",

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I = (r,-), D c R" containing the origin, and thedifferential equation dx/dt = f(t,x), where the origin is anequilibrium or critical point.

Theorem 5.3.1 (converse of Theorem 5.2.1) [Persidski1933] If the function f is C1 and if the origin is stable,then there exist a neighborhood U c D of the origin and a C'function V: IxU - R such that for some a e K and every (t,x)c IxU: (i) V(t,x) ? a(IIxII); V(t,O) = 0; (ii) dV(t,x)/dt <_ 0.

Theorem 5.3.2 (converse of Theorem 5.2.2) [Kurzweil1955] If the function f is C1 and if the origin isuniformly stable, then there exist a neighborhood U c D ofthe origin and a C' function V: IxU -+ R such that for somea, b e K, and every (t,x) c IxU:

(i) b(IIxII) ? V(t,x) >- a(IIxII); V(t,0) = 0;(ii) V(t,x) _< 0.

Theorem 5.3.3 (converse of Theorem 5.2.8) [Massera 1949,1956] If the function f on IxD is locally Lipschitzian in xuniformly with respect to t, and if the origin is uniformlyasymptotically stable, then there exist a neighborhood U c Dof the origin and a function V: IxU - R possessing partialderivatives in t and x of arbitrary order, such that forsome functions a, b, c e K and every (t,x) e IxU:

(i) a(IIxII) -< V(t,x) -< b(IIxII);(ii) dV(t,x)/dt <_ - c(IIxII).

Corollary 5.3.4 If f on IxD is Lipschitzian in xuniformly with respect to x, V can be chosen such that allpartial derivatives of any order of V are bounded on IxU,with the same bound for all of them. If on IxD f isindependent of t, or periodic in t, V can be chosenindependent of t or periodic in t respectively.

5.4 Comparison methodsIn calculus or even in algebra, we have learned that

there are many methods of testing the convergence of aseries. Among them is the comparison test. Similarly, indetermining the stability of a equation at a point, one can

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find the relationship between the said equation and a givenequation with stability properties known at the origin. Moreexplicitly, consider the equation

dx/dt = f(t,x), (5.4-1)

and suppose that there exists a scalar differential equationdu/dt = g(t,u) (5.4-2)

with a critical point at the origin u = 0 and some knownstability properties. The comparison method studies therelationship which should exist between Egs.(5.4-1) and(5.4-2) in order that the stability properties of Eq.(5.4-2)entail the corresponding properties for Eq.(5.4-1).

In order to avoid needless intricacies in computation,we shall define g(t,u) of Eq. (5.4-2) on IXR*, where R' =(0,oo), in this section. Of course, the uniqueness of thesolutions of the comparison equation is assumed here forconvenience. Furthermore, when it is supposed thatEq.(5.4-2) admits of u = 0 as a critical point and thatpoint is stable, or asymptotically stable, only positiveperturbations will have to be considered.

Lemma 5.4.1 [Wazewski 1950] Let g: IxR' - R, (t,u) -+

g(t,u) be continuous and such that Eq.(5.4-2) has a uniquesolution u(t;to,uo) through any (to,uo) a IXR`. Let [to,b) bethe maximal future interval where u(t;to,uo) is defined. Letv: [t,b) R be (i) v(to) <_ uo; (ii) dv(t)/dt <- g(t,v(t)) on[to,b); then v(t) 5 u(t) on [to,b).

Theorem 5.4.2 (Corduneanu 1960] Suppose that thereexists a function g as in Lemma 5.4.1, with g(t,0) = 0, anda C' function V: IxD - R, such that for some function a e Kand every (t,x) a IxD:

(i) V(t,x) ? a(IIxII), V(t,0) = 0;(ii) dV(t,x)/dt <_ g(t,V(t,x)); then(a) stability of u = 0 implies stability of x = 0;(b) asymptotic stability of u = 0 implies equi-asymptoticstability of x = 0.Furthermore, if for some function b e K and every (t,x) c

IxD: (iii) V(t,x) 5 b(IlxIl) ; then(c) uniform stability of u = 0 implies uniform stability of

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X = 0;(d) uniform asymptotic stability of u = 0 implies uniformasymptotic stability of x = 0.

Let us apply this theorem to some particular cases. Forg(t,u) = 0, then (a) is reduced to Liapunov's theorem 5.2.1,(c) yields Persidski's theorem 5.2.2 on uniform stability.By choosing g(t,u) = - c(u) for some c e K, we notice thatfor the equation du/dt = - c(u), the origin u = 0 isuniformly asymptotically stable, i.e., (b) reduces toMassera's theorem [1956], and (d) is reduced to Liapunov'stheorem 5.2.8. This is because dV(t,x)/dt <_ - c(aIjxII) and

that is a function of class K.Various ways of choosing the function g(t,u) can be

illustrated in the following:(1) If c e K and if 0: I - R' is continuous, then the originu = 0 is uniformly stable for the equation du/dt =-0(t)c(u). If, moreover, f0 0(t)dt = co, the origin isequi-asymptotically stable.(2) If /3: I - R is continuous, the equation du/dt = p(t)uhas a critical point at the origin, which is stable,uniformly stable, or equi-asymptotically stable accordingto:

(a) for all to e I there exist A > 0 and t >_ to such thatf tot /3 (s) ds 5 A, or(b) therer exist A > 0, for all to e I and t >- to such thatfto p(s)ds <- A, or(c) for all to e I such that f tot /3 (s) ds -+ -oo, as trespectively.(3) Suppose there exist two C' functions V: IxD - R, k: I -R and a continuous function g: IXR' - R such that g(t,O) _

0, that the solutions of du/dt = g(t,u) are unique, and forsome function a e K and every (t,x) c IxD:

(i) V(t,x) >_ a(lIxIl), V(t,0) = 0;(ii) k(t) dV(t,x)/dt + dk(t)/dt V(t,x) _< g(t,k(t)V(t,x));(iii) k(t) > 0; and moreover,(iv) k(t) - oD as t oo;

then stability of u = 0 implies equi-asymptotic stability of

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x = 0 [Bhatia and Lakshmikantham 1965].(4) The following result pertains to the system of equationsEq. (5.2-2) concerning partial stability. Suppose thereexist a function g as in Lemma 5.4.1, with g(t,O) = 0, and aC' function V: IxDxRm - R, such that for some function a eK and every (t,x,y) a IXDXR"':

(i) V(t,x,y) >_ a(IIxjj), V(t,0,0) = 0;(ii) dV(t,x,y)/dt <- g(t,V(t,x,y)); then(a) stability of u = 0 implies stability with respect to xof x = y = 0;(b) asymptotic stability of u = 0 implies equi-asymptoticstability with respect to x of x = y = 0, provided that thesolutions of Eq.(5.2-2) do not approach w in a finite time.If moreover, for some function b e K and every (t,x,y) e

IxDx Rm: (iii) V(t,x,y) < b(pxll + IIYII); then(c) uniform stability of u = 0 implies uniform stabilitywith respect to x of x = y = 0;(d) uniform asymptotic stability of u = 0, along with theexistence of solutions of Eq.(5.2-2) which do not approach coin a finite time, implies uniform asymptotic stability withrespect to x of x = y = 0.

For more details on the comparison method, see, forinstance, Rouche, Habets and Laloy [1977], Ch. 9.

5.5 Total stabilityUp to now, all the considerations on stability pertain

to variations of the initial conditions. Here we shallconsider another type of stability which takes into accountthe variations of the second member of the equation. Formost practical problems, significant perturbations do occurnot only at the initial time, but also during the evolution.

We shall still assume that the differential equationdx/dt = f(t,x) (5.5-1)

is such that f(t,0) = 0 for all t e I. Also, we shallconsider

dy/dt = f(t,y) + g(t,y) (5.5-2)

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along with Eq.(5.5-1), where g: IxD -+ R" satisfies the sameregularity conditions as f. This ensures global existenceand uniqueness for all solutions of Eq.(5.5-2). Thisfunction g will play the role of a perturbation term addedto the second member of Eq.(5.5-1). Particularly, it willnot be assumed that g(t,O) = 0, and therefore the originwill not in general be a solution of Eq.(5.5-2). Thesolution x = 0 of Eq.(5.5-1) is called totally stablewhenever for any e > 0 there exist 61, 62 > 0 for all to e I

and any yo a B. and for any g such that for all t to andfor any x e B, where II g (t, x) II < 62, then for any t to, wehave y(t;to,yo) a B,.

Theorem 5.5.1 [Malkin 1944] If there exist a C'function V: IxD - R, three functions a, b, c e K and aconstant M such that, for every (t,x) c IxD:

(i) a(IIxII) <- V(t,x) < b(IIxII);(ii) dV(t,x)/dt <_ - c(IIxII), dV/dt computed along thesolution of Eq.(5.5-1);(iii) II V(t,x)/ xli <- M;then the origin is totally stable for Eq.(5.5-1).

Indeed, Malkin [1952] showed that asymptotic stabilityimplies total stability.

Theorem 5.5.2 [Malkin 1944] If f is Lipschitzian in xuniformly with respect to t on IxD and if the origin isuniformly asymptotically stable, then the origin is totallystable.

Clearly, the hypotheses of Theorem 5.5.1 do not implythat any solution y(t;to,yo) tends to 0 as t - oo; in fact,g(t,y) does not vanish, nor does it fade down as t - oo.Nonetheless, some kind of asymptotic property can be found.

Theorem 5.5.3 [Malkin 1952] In the hypotheses ofTheorem 5.5.1, for any e > 0 there exist 6, > 0 and for anyn > 0, there exists 62' > 0 and for all to a I, if yo a B6and if for any t >_ to and x e B, such that IIg(t,y)II < 62',then there is a T > 0 such that for any t >- T, we have

y(t;to,yo) a B,1.

Note that in the definition of total stability, one can

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replace 6, and d2 by a single 61 = 62 = S. Also, in Theorem5.5.1 the condition V(t,x) -< b(IIxII) can be replaced by

V(t,O) = 0 [Hahn 1967]. Massera [1956] has shown that if theorigin is totally stable for a linear differential equationdx/dt = A(t)x, where A is a continuous nxn matrix, then itis uniformly asymptotically stable. Nonetheless, in the samepaper, it is demonstrated that for an equation dx/dt = f(x),f e C', f(0) = 0, total stability does not imply uniformasymptotic stability.

Auslander and Seibert [1963] relates hitherto unrelatedpoints of view of stability. The first is a generalizedversion of Liapunov's second method. The second is theconcept of prolongation, which is a method of continuingorbits beyond their !1-limit sets. And the third is theconcept of total stability. We shall briefly relate them viasome of the theorems.

A generalized Liapunov function for a compact invariantset C is a nonnegative function V. defined in a positivelyinvariant neighborhood W of M. and satisfying the following:(a) if e > 0, then there exists µ > 0 such that V(x) > µ,for x not in S,(C) a (y c XI d(y,C) < e); (b) if p > 0,there exists n > 0 such that V(x) < µ, for x e S7) (C); (c) if

x e W. and t >_ 0, V(xt) <_ V(x). By a gereralized Liapunovfunction at infinity we mean a nonnegative function Vdefined on all of X satisfying the following: (a) V is

bounded on every compact set; (b) the set (xl V(x) _< p) is

compact, for all µ ? 0; (c) if x e X, and t ? 0, then V(xt)< V(x).

Theorem 5.5.4 [Lefschetz 1958] The compact set M isstable iff there exists a generalized Liapunov function forM.

Theorem 5.5.5 Let M be compact. Then the followingstatements are equivalent:(a) M is absolutely stable.(b) M possesses a fundamental sequence of absolutely stablecompact neighborhoods.(c) There exists a continuous generalized Liapunov function

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for M.

We have not formally defined the term absolutely stable.The definition has to be based on some mathematicalmachinary which we will not use later on. Thus, one canconsider the above theorem also as a definition. One canconsider the following two theorems in the same light.

Theorem 5.5.6 A dynamical system is 1-bounded iff thereexists a generalized Liapunov function at infinity.

Theorem 5.5.7 The following statements are equivalent:(a) A dynamical system is absolutely bounded.(b) Every compact set is contained in an absolutely stablecompact set.(c) There exists a continuous Liapunov function at infinity.

A dynamical system is said to be ultimately bounded ifthere exists a compact set A such that n(X) is a subset ofA. The following two theorems relate asymptotic stability,ultimate boundedness, absolute stability, and absoluteboundedness.

Theorem 5.5.8 If the compact set M is asymptoticallystable, it is absolutely stable. In fact, M isasymptotically stable iff there exists a continuous Liapunovfunction V for M such that, if x does not belong to M, and t> 0, then V(xt) < V(x).

Theorem 5.5.9 If a dynamical system is ultimatelybounded, it is absolutely bounded. Furthermore, there existsa compact set M which is globally asymptotically stable.

The following theorem is another definition of totalboundedness and also relates it to prolongation.

Theorem 5.5.10 The following statements are equivalent:(a) The dynamical system is totally bounded.(b) If A is a compact subset of X, then the prolongation ofA is also compact.

The result of Milkin [1952] together with the followingtheorem gives an interesting relationship between asymptoticstability and absolute stability.

Theorem 5.5.11 (a) Total stability implies absolutestability. (b) Boundedness under perturbations implies

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absolute boundedness.Let M be a positively invariant set of the flow 0. M is

strongly stable under perturbations if (a) it is weaklystable under perturbations and (b) there exists a constant awith the following property: given any e, there exist r and6 such that the flow is ultimately bounded whenever p*satisfies O(r) = S.

Theorem 5.5.12 Uniform asymptotic stability and strongstability under perturbations are equivalent.

For a detailed discussion of the connections betweenasymptotic stability and stability under perturbations, seeSeibert [1963]. For higher prolongations and absolutestability, see Ch. 7 of Bhatia and Szego [1970].

As an example, let us consider the systemd2x/dt' + f(x' + (dx/dt)')dx/dt + x = 0. To every zero ofthe function f(r2) = f(x' + (dx/dt)2) corresponds a limitcycle x' + (dx/dt)' = r'. The orbits between two neighboringlimit cycles are spirals with decreasing or increasingdistance from the origin, depending on the sign of f. (a) If

f(r2) = sin(,r/r'), the origin is totally stable, thereforeabsolutely stable, but not asymptotically stable. (b) Iff(r') = sin(,rr'), the system is totally bounded, thereforeabsolutely bounded, but not ultimately bounded.

Examples for the non-autonomous system under persistentperturbations will be discussed and illustrated in Ch.7, inparticular, in section 7.2.

5.6 Popov's frequency method to construct a Liapunovfunction

In this section we shall use a nuclear reactor as anexample to illustrate one of the very practical methods,namely the frequency method, to construct a Liapunovfunction. This method is due to Popov [1962]. For moredetails, see for instance, Lefschetz [1965] and Popov[1973].

Basic variables to describe the state of a nuclearreactor are the fast neutron's density D, D > 0, and the

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reactor are the fast neutron's density D, D > 0, and thetemperatures T e R" of its various constituents. Theneutron's density satisfies

dD/dt = kD (5.6-1)

where the reactivity k is a linear function of the state k =k0 - rtT - t7 D, where r e R", rt the transpose of r, t e R. ByNewton's law of heat transfer, one gets the temperatureequations

dT/dt = AT - bD (5.6-2)

where A is a non-singular real nxn matrix and b e R. Thesystem of Equations (5.6-1) and (5.6-2) has the equilibriumvalues

To = k0A-'b/ (f1 + rtA-'b) ,Do = ko/ (n + rtA-1b) .

Recall that D > 0, and changing of the variablesx = T - To, µ = - D0(ln(D/DO) + rtA-tx)/k0,after some tedious computations, one obtains

dx/dt = Ax - bo(a), dµ/dt = O(a),a = - rtA'ix - k0µ/Do, (5.6-3)

where 0(a) = D0 (e° - 1). Note that aO(a) > 0 when a + 0. Asystem like Eq.(5.6-3) is also known as an indirect controlsystem [see Lefschetz 1965]. We shall not get into thetechnical details about the reactor, but rather prove somesufficient conditions for the global asymptotic stability ofthe critical point at the origin for Eq.(5.6-3).

Lemma 5.6.1 Let A be a non-singular nxn matrix, whoseeigenvalues have strictly negative real parts, D asymmetric, positive definite non-singular matrix of order n.Let b e R", b + 0, k e R", and let r and a be real, where r

0, e > 0. Then a necessary and sufficient condition forthe existence of a symmetric positively definite,non-singular matrix B of order n and a g e R" such that (a)AtB + BA = - qqt - cD, (b) Bb - k = qr, is that a be smallenough and that the inequality

r + 2 Re(kt(iflI - A)-'b) > 0be satisfied for all real fl [Yacuborich 1962; Kalman 1963].

Theorem 5.6.2 Pertain to Eq.(5.6-3), suppose that all

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and ctb + r > 0. If furthermore, there exist real constants

a and Q such that (i) a ? 0, Q ? 0, a + p > 0; (ii) forall real n,

Re[(2ar + inI)(ct(inI - A) -'b + r/in)] > 0;

then the origin x = 0, a = 0 is globally asymptoticallystable for the system Eq.(5.6-3).

Remark: The results obtained by this frequency methodare stronger and more effective than those consideredearlier. And in fact they apply to a whole class of systemscorresponding to any function 0 with O(a)a > 0 when a + 0,and they yield the auxiliary function explicitly. Moreover,the frequency criterion is the best possible result in thesense that the proposed Liapunov function proves asymptoticstability iff this criterion is satisfied. Nonetheless, thescope of the method is undoubtedly narrower than theLiapunov's direct method.

Before ending this chapter, we would like to point outthat all auxiliary functions introduced up to now are C'functions. Nonetheless, it may happen, such as the exampleof a transistorized circuit may show, that the "natural"Liapunov function is not that regular. Thus, it is nature togeneralize the theorems of Liapunov's second method toemcompass the case of a less smooth function V. Using someresults of Dini derivatives, one can prove most theorems inthis chapter, while imposing on the auxiliary functions onlya local Lipschitz condition with respect to x and continuityin V(t,x). For Dini derivatives see, e.g., McShane [1944],Rouche, Habets and Laloy [1977]. For non- continuousLiapunov functions, see for instance, Bhatia and Szego[1970].

5.7 Some topological properties of regions of attractionIn this section, we will present some additional

properties of weak attractorsand their regions ofattraction. This section can be skipped over for the firstreading, or if the reader is not particularly interested in

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the topological properties of attraction.As before, the space X is locally compact. From the

definition of stability, it is clear that if a singleton (x)is stable, then x is a critical or fixed point. Inparticular, if (x) is asymptotically stable, then x is afixed point. The next theorem concerns an importanttopological property of the region of attraction of a fixedpoint in R".

Theorem 5.7.1 If a fixed point p e R' is asymptoticallystable, then A(p) is homeomorphic to R".

Corollary 5.7.2 If p is an asymptotically stable fixedpoint in R", then A(p) - (p) is homeomorphic to R" - (0),where 0 is the origin in R.

Theorem 5.7.3 Let a subset M of R" be a compactinvariant globally asymptotically stable set in R". Then R"- M = C(M) is homeomorphic to R" - (0).

Theorem 5.7.4 Let a subset M of R" be a compactpositively invariant set, which is homeomorphic to theclosed unit ball in R". Then M contains a fixed point.

Theorem 5.7.5 Let a subset M of R" be a compact setwhich is a weak attractor. Let the region of weak attractionAw(M) of M be homeomorphic to R. Then M contains a fixedpoint. In particular, when Aw(M) = R" (i.e., M is a globalweak attractor), then M contains fixed point.

Corollary 5.7.6 If the dynamical system defined in R"is admitting a compact globally asymptotically stable set(equivalent to the system being ultimately bounded), then itcontains a fixed point.

Corollary 5.7.7 The region of attraction of a compactminimal weak attractor M cannot be homeomorphic to R",unless M is a fixed point.

Corollary 5.7.8 If M is compact minimal and a globalweak attractor, then M is a singleton. Consequently, Mconsists of a fixed point.

It should be interesting to note that if M is a compactinvariant asymptotically stable set in X, then therestriction of the dynamical system to the set A(M) - M is

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parallelizable, thus dispersive. Now we shall discussasymptotic stability of closed sets and its relation withthe Liapunov functions.

Let S(M,6) = (y: d(y,M) < 6, where M is a subset of X,and 6 > 0). A closed subset M of X will be called: (i) a

semi-weak attractor, if for each x e M, there is a 6x > 0,and for each y e S(x,6x) there is a sequence (tn) in R, to -ao, such that d(ytn,M) -+ 0, (ii) a semi-attractor, if foreach x e M, there is a 6x > 0, such that for each y eS(x,6x), d(yt,M) - 0 as t - co, (iii) a weak attractor, ifthere is a 6 > 0 and for each y e S(M,6), there is asequence (tn) in R, to ao, such that d(ytn,M) -. 0, (iv) anattractor, if there is a 6 > 0 such that for each y eS(M,6), d(yt,M) - 0 as t - oo, (v) a uniform attractor, ifthere is an a > 0, and for each e > 0 there is a T = T(e) >0, such that x[T,-) is a subset of S(M,e) for each x eS[M,a], (vi) an eaui-attractor, if it is an attractor, andif there is an µ > 0 such that for each e, 0 < e < A. and T> 0, there exists a 6 > 0 with the property that x[O,T] nS(M,6) = 0 whenever a <_ d(x,M) <_ µ, (vii)

semi-asymptotically stable, if it is stable and asemi-attractor, (viii) asymptotically stable, if it isuniformly stable and is an attractor, (ix) uniformlyasymptotically stable, if it is uniformly stable and auniform attractor. The following figures are orbits ofcertain dynamical systems in R2. See the following figuresFig.5.7.1.

y

x

Stable but not equistable

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Equistable but not stableyA

Semi-attractor

y

Weak attractor

234

*y

Attractor

-> X

x

Uniformly stable semi-attractor

x

Stable attractorTheorem 5.7.9 Let M be a closed set. Then M is

asymptotically stable iff there is a function 0(x) definedin X with the following properties:(i) 0(x) is continuous in some neighborhood of M whichcontains the set S(M,6) for some 6 > 0,(ii) 0(x) = 0 for x e M, O(x') > 0 for x' not belong to M,(iii) there exist strictly increasing functions a(µ), Q(µ),where a(0) = Q(0) = 0, defined for µ >_ 0, such that

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c(d(x,M)) <- O(x) <_ Q(d(x,M)),

(iv) O(xt) <- O(x) for all x e X, t > 0, and there is a d > 0such that if x e S(M,d) - M then O(xt) < O(x) for t > 0, andO(xt) 0 as t - -.

Theorem 5.7.10 Let a closed invariant set M beasymptotically stable. Let A(M) - M (or in particular thespace X) be locally compact and contain a countable densesubset. Then the invariant set A(M) - M is parallelizable.

Proposition 5.7.11 Let M be a closed invariantuniformly asymptotically stable subset of X with A(M) as itsregion of attraction. Then A(M) - M is parallelizable.

In the following we shall discuss the concepts andproperties of relative stability and attraction of a compactset. X is assumed to be locally compact. For a given compactset M, a subset of X, and a subset U of X, the set M is saidto be: (i) stable relative to U, if given an e > 0 thereexists d > 0 , such that r (S(M,6)nU) is a subset of S(M,e),(ii) a weak attractor relative to U, if f(x)nM + 0, for eachx e U, (iii) an attractor relative to U, if 12(x) + 0, 12(x)is a subset of M, for each x e U, (iv) a uniform attractorrelative to U, if J'(x,U) + 0, J'(x,U) is a subset of M,for each x e U, (v) asymptotically stable relative to U if Mis a uniform attractor relative to U and it is positivelyinvariant.

Note that, if in the above definitions U is aneighborhood of M, then the stability, weak attraction,attraction, uniform attraction and asymptotic stability of Mrelative to U reduces to the stability, weak attraction,attraction, uniform attraction and asymptotic stability of Mrespectively.

Theorem 5.7.12 A compact subset M of X is stablerelative to a subset U of X iff M contains D'(M,U).

Theorem 5.7.13 Let a subset M of X be compact and suchthat A.(M) - M + 0. Let U be a subset of AM(M) be a set withthe following properties: (i) U is closed and positivelyinvariant; (ii) U Is M + 0. Then the set D`(M,U) is compactand asymptotically stable relative to U.

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Theorem 5.7.14 Let a subset M of X be compact andpositively invariant and let M' be the largest invariantsubset contained in M. Then M' is a stable attractor,relative to M.

For example, consider a limit cycle r in R2 with theproperty that all orbits outside the unit disk bounded bythe limit cycle r, has r as their sole positive limit set,and all orbits in the interior of the disk tend to theequilibrium point 0. See Fig.5.7.2. We shall meet thisauto-oscillatory system in Section 7.2. Suffice to note thatif U is the complement of the disk bounded by r, then r isrelatively stable with respect to U. Note also that if r isan asymptotically stable limit cycle, then r is stable withrespect to every component of R2 - r.

Fig.5.7.2These considerations lead to the following definition andtheorem. Let M be a compact subset of X. M is said to becomponent-wise stable if M is relative stable with respectto every component of X - M.

Theorem 5.7.15 Let a compact subset M of X bepositively stable. Then M is component-wise stable.

The converse is not true in general. The followingexample illustrates this conclusion. Let X be a subset of R2given by X = ( (x,y) c R2: y = 1/n, n is any integer, or y =0). Clearly the space is a metric space with the distancebetween any two points being the Euclidean distance betweenthe points in R2. We can define a dynamical system on X bydx/dt = lyl, dy/dt = 0. Then the set ((0,0)) in X is

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component-wise stable, but is not stable.It is then nature to ask under what conditions the

converse of Theorem 5.7.15 is true. We first introduce thefollowing definition, then followed by a couple of theorems.

Let M be a compact subset of X. We say the pair (M,X) isstability-additive if the converse of Theorem 5.7.15 holdsfor every dynamical system defined on X which admits M as aninvariant set.

Theorem 5.7.16 The pair (M,X) is stability-additive ifX - M has a finite number of components.

Theorem 5.7.17 The pair (M,X) is stability-additive ifX - M is locally connected.

5.8 Almost periodic motionsWe have discussed periodicity and recurrence. In the

following we shall briefly discuss the concept intermediatebetween them, namely that of almost periodicity. Forconvenience, we assume that the metric space X is complete.

A motion Ox is said to be almost periodic if for every e> 0 there exists a relatively dense subset of numbers (Tn)called displacements, such that d(xt, x(t+T,)) < e for all te R and each Tn. It is clear that periodic motion and fixedpoints are special cases of almost periodic motions. And itis also easy to show that every almost periodic motion isrecurrent. We shall show that not every recurrent motion isalmost periodic, and an almost periodic motion need not beperiodic. The following theorems show how stability isdeeply connected with almost periodic motions.

Theorem 5.8.1 Let the motion Ox be almost periodic andlet the closure of r(x) be compact. Then:(i) every motion 0Y with y e closure of r(x), is almostperiodic with the same set of displacements (Tn) for anygiven e > 0, but with the strict inequality < replaced by <_;(ii) the motion ¢x is stable in both directions in theclosure of r(x).

Corollary 5.8.2 If M is a compact minimal set, and if

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one motin in M is almost periodic, then every motion in M isalmost periodic.

Corollary 5.8.3 If M is a compact minimal set of almostperiodic motions, then the motions through M are uniformlystable in both directions in M.

Theorem 5.8.4 If a motion 0X is recurrent and stable inboth directions in r(x), then it is almost periodic.

Theorem 5.8.5 If a motion 0, is recurrent andpositively stable in r(x), then it is almost periodic.

Theorem 5.8.6 If the motions in r(x) are uniformlypositively stable in r(x) and are negatively Lagrangestable, then they are almost periodic.

A semi-orbit r+(x) is said to uniformly approximate itslimit set, n(x), if given any e > 0, there is a T = T(e) > 0such that n(x) is a subset of S(x[t,t+T],e) for each t e R+.We want to find out under what conditions a limit set iscompact and minimal. We have the following:

Theorem 5.8.7 Let the motion 0X be positively Lagrangestable. Then the limit set n(x) is minimal iff thesemi-orbit r+(x) uniformly approximates n(x).

Theorem 5.8.8 Let the motion 0X be positively Lagrangestable, and let the motions in r+(x) be uniformly positivelystable in r+(x). If furthermore r+(x) uniformly approximatesn(x), then n(x) is a minimal set of almost periodic motions.

It should be pointed out that no necessary andsufficient conditions are known yet. In closing, let usdiscuss an example of an almost periodic motion which isneither a fixed point nor a periodic motion.

Consider a dynamical system defined on a torus by thefollowing set of simple differential equations:

dt/dt = 1, dB/dt = a,where a is irrational. For any point p on the torus, theclosure of r(p) = the torus, and since a is irrational, noorbit is periodic. The torus is thus a minimal set ofrecurrent motions. To show that the motions are almostperiodic, we note that if p, = (01,01), and p2 = (02,82), then

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d(P1t,2t) = (01 - 02). + (0 - 62)= = d(P1,P2),where the values of 01 - 02 and A, - 62 ae taken as thesmallest in absolute value of the differences, and also notethat any motion on the torus is given by 0 = 0o + t, 0 = 00+ at. Thus the motions are uniformly stable in bothdirections in the torus. Thus, from Theorem 5.8.4, the torusis a minimal set of almost periodic motions.

In Section 4.4 we have derived the van der Pol'sequation of the nonlinear oscillator. Cartwright [1948] andher later series of papers dealt with a generalized van derPolls equation for forced oscillations. The periodic andalmost periodic orbits are obtained. For detail, see alsoGuckenheimer and Holmes [1983]. Krasnosel'skii, Burd andKolesov [1973] discusses broader classes of nonlinear almostperiodic oscillations. The book by Nayfeh and Mook [1979]gives many detailed discussions on nonlinear oscillations.The Annual of Mathematics series on nonlinear oscillationsare highly recommended for further reading (Lefschetz 1950,1956, 1958, 1960]. Indeed, there are many current problems,such as in phase locked laser arrays, where some of theresults are very applicable to the problems. We shallbriefly point this out in the next chapter.

Most of this chapter is based on several chapters ofRouche, Habets and Laloy (1977]. This is still one of thebest sources for Liapunov's direct method.

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Chapter 6 Introduction to the General Theoryof Structural Stability

6.1 IntroductionHowever complex it may seem, our universe is not random,

otherwise it would be futile to study it. Instead, it is anendless creation, evolution, and annihilation of forms andpatterns in space which last for certain periods of time.One of the central goals of science is to explain, and ifpossible, to predict such changes of form. Since theformation of such structures or patterns and theirevolutional behavior are "geometric" phenomena, uncoveringtheir common bases is a topological problem. But theexistence of topological principles may be inferred fromvarious analogies found in the critical behavior of systems.It should be emphasized that recognizing analogies is animportant source of knowledge as well as an importantmethodology of acquiring knowledge.

There is a striking similarity among the instabilitiesof convection patterns in fluids, cellular solidificationfronts in crystal growth, vortex formation insuperconductors, phase transitions in condensed matter,particle physics, laser physics, nonlinear optics,geophysical formations, biological and chemical patterns anddiffusion fronts, economical and sociological rhythms, andso forth. Their common characteristic is that one or morebehavior variables or order parameters undergo spontaneousand discontinuous changes or cascades, if competing, butslowly and continuously control parameters or forces cross abifurcation set and enter conflicting regimes. Consequently,an initially quiescent system becomes unstable at criticalvalues of some control variables and then restabilize into amore complex space or time-dependent configuration. If othercontrol parameters cause the disjoint bifurcation branchesto interact, then multiple degenerate bifurcation pointsproduce higher order instabilities. Then, the systemundergoes additional transitions into more complex states,

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giving rise to hysteresis, resonance and entramment effects.These ultimately lead to states which are intrinsicallychaotic.

In the vicinity of those degenerate bifurcation points,a system is extremely sensitive to small changes, such asimperfection, or external fluctuations which lead tosymmetry breaking. Consequently, the system enhances itsability to perceive and to adapt its external environment byforming preferred patterns or modes of behavior. Prigoqine'sconcept of dissipative structures [1984], Haken'ssynergetics [1983], and Thom's catastrophe theory [1973] aremost prominent among the theoretical study of these generalprinciples.

As we have discussed earlier, the guiding idea of astable system is to find a family of dynamical systems whichcontains "almost all" of them, yet can be classified in somequalitative fashion. It was conjectured that structurallystable systems would fit the bill. Although it turns out notto be true except for low dimensional cases, structuralstability is such a natural property, both mathematicallyand physically, that it still holds a central place in thetheory of dynamical systems.

As we have pointed out in Section 1.2, there is thedoctrine of stability in which structurally unstable systemsare regarded as suspicious. This doctrine states that, dueto measurement uncertainties, etc., a model of a physicalsystem is valuable only if its qualitative properties do notchange under small perturbations. Thus, structural stabilityis imposed as a prior restriction on "good" models ofphysical phenomena. Nonetheless, strict adherence to suchdoctrines is arguable to say the least. It is very true thatsome model dynamical systems, such as an undamped harmonicoscillator, the Lotka-Volterra equations of thepredator-prey model, etc., are not good models for thephenomena they are supposed to represent becauseperturbations give rise to different qualitative features.Nonetheless, these systems are indeed realistic models for

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the chaotic behavior of the corresponding deterministicsystems since the presumed strange attractors of thesesystems are not structurally stable.

If we turn to the other side of the coin, as we havealso pointed out in Section 1.2, since the systems are notstructurally stable, details of their dynamical evolutionswhich do not persist under perturbations may not correspondto any verifiable physical properties of the systems.Consequently, one may want to reformulate the stabilitydoctrine as the only Properties of a (or a family of)dynamical system(s) which are physically (or quantitatively)relevant are those preserved under perturbations of thesystem(s). Clearly, the definition of (physical) relevancedepends on the specific problem under study. Therefore, wewill take the spirit that the discussions of structuralstability requires that one specify the allowableperturbations to a given system.

The two main ingredients of structural stability are thetopology given to the set of all dynamical systems and theequivalence relation placed on the resulting topologicalspace. The former is the Cr topology (1 <- r <- c). This

topology has been discussed in Chapter 3, and the idea inour context is clear. For instance, two diffeomorphisms areC''-close when their values and values of correspondingderivatives up to order r are close at every point. Once wehave defined the Cr topology, we may be able to be morespecific about what we mean by "almost all" of the systems.The latter attribute is topological equivalence for flowsand topological conjugacy for diffeomorphisms.

Before we get into a more general discussion ofstructural stability for manifolds, diffeormorphisms,function spaces of maps, and so forth, let us discuss theconcept for R".

Let F e Cr (R"), we want to specify what we mean by aperturbation G of F. Let F be as above, r, k are positiveintegers, k 5 r, and e > 0, then G is a Ck perturbation ofsize a if there is a compact subset K of R" such that F = G

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_on the set R" - K and for all with iI + ... + ini 5 k we have I (a'/ax1' l ... ax"'") (F - G) I < e. F and G canalso be vector fields. Note also that this definition can bestated in terms of k-jets as introduced in Chapter 3.

Let us recall that two Cr maps F and G are Ck (k <_ r)

equivalent or Ck conjugate if there exists a Ckhomeomorphism h such that C° equivalence iscalled topological equivalence. Two Cr vector fields f and gare Ck (k <_ r) equivalent if there exists a Ckdiffeormorphism h which takes orbits of f to orbits of g,preserving senses but not necessarily parameterization bytime. If h does preserve parameterization by time, then itis called a conjugacy. Intuitively, it is not difficult tosee that parameterization by time cannot be preserved ingeneral. This is because the periods of closed orbits inflows can be different.

We say a map F e Cr (Rn) (or a Cr vector field frespectively) is structurally stable if there is an e > 0such that all C' e- perturbations of F (resp. of f) aretopologically equivalent to F (resp. f).

Let us consider a small perturbation of a linear systemdx/dt = Ax, x e R2: dx/dt = Ax + ef(x), where f has supportin some compact set. Since A is invertible, then by theimplicit function theorem, the equation Ax + ef(x) = 0continues to have a unique solution x = 0 + O(e) near x = 0,for sufficiently small E. Since the matrix of the linearizedsystem dt/dt = (A + eDf(x))$ has eigenvalues which dependcontinuously on e, no eigenvalues can cross the imaginaryaxis if a remains small with respect to the magnitude of thereal parts of the eigenvalues of A. Thus the perturbedsystem has a unique fixed point and invariant manifolds ofthe same dimensions as those of the unperturbed system, andwhich are a-close locally in position and slope to theunperturbed manifolds. The problem is finding ahomeomorphism which takes orbits of the linear system tothose of the perturbed nonlinear system.

Structurally stable systems have "nice" properties,

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namely, any sufficiently close system has the samequalitative behavior. Nonetheless, structurally stablebehavior can be very complex for flows of dimension >_ 3 ordiffeomorphisms of dimension ? 2. It should be stressed thatthe definition of structural stability is relative to theclass of systems we will be dealing with. In fact,structural stability is not even a generic property, weshall come to this in Section 6.3, because we can findstructurally unstable systems which remain unstable undersmall perturbations, and some, in fact, continually changetheir topological equivalence class as we perturb them. Weshall meet such systems shortly.

The idea is the following. Let Diffr(M) (1 _< r 5 -) be

the set of all Cr-diffeomorphisms of the C°-manifold M,provided with the Cr-topology. We say that f e diffr(M) isstructurally stable if it is in the interior of itstopological conjugacy class. In other words, f isstructurally stable iff any Cr- "small" perturbation takesit into a diffeomorphism that is topologically conjugate tof. Similarly, let Xr(M) be the set of all Cr vector fieldson M topologized with the Cr topology. Then the vector fieldX C Xr(M) (or the corresponding integral flow 0) isstructurally stable if it is in the interior of itstopological equivalence class. This type of definition ofstructural stability is due to Andronov and Pontryagin[1937]. This concept of structural stability has played adominant and extremely important role in the development ofdynamical systems during the past thirty years. It not onlyhas helped to emphasize the global point of view, but alsohas been very fruitful in stimulating conjecturesculminating in the powerful theorems we shall discussshortly.

Before we get into any details of the notion ofstructural stability, we would like to discuss the stablemanifolds of the dynamical system at a fixed point.

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6.2 Stable manifolds of diffeomorphisms and flowsIf we look at a phase diagram of a saddle point

(Fig.6.2.1) and try to analyze which qualitative featuresgive the saddle point its characteristic appearance, we arebound to single out the four special orbits that begin orend at the fixed point. These four orbits together with thefixed point form the stable manifolds and unstable manifoldsof the dynamical system at the fixed point. In Chapter 4 wehave noted the importance of hyperbolicity in the theory ofdynamical systems, and a hyperbolic structure always impliesthe presence of such manifolds. Thus, if we know where the"singular elements" of a system are (including periodicpoints for diffeomorphisms, fixed points and closed orbitsof flows), and if we also know the way in which their stableand unstable manifolds fit together, we then have a verygood idea on the orbit structure of the system.

Fig. 6.2.1Let f: M - M be a diffeomorphism, p be a fixed point of

f. The stable set of f at p is the set {x e M : f"(x) - p asn - co ). Note that this set is always non-empty because itcontains p. The unstable set of f at p is the stable set off-1 at p. For any open neighborhood U of p, the local stableset of flU at p is the set of all x e U such that {f"(x): n

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> 0) is a sequence in U converging to p. One can show thatif p is hyperbolic then the global stable set is an immersedsubmanifold of M and is at least as smooth as thediffeomorphism f. This submanifold is called the stablemanifold of f at p, and denoted by WS(p). Clearly, it neednot be an embedded submanifold, for instance, as in Fig.6.2.2.

Now we shall deduce the global version of the stablemanifold theorem from the local one. Let T be a hyperboliclinear automorphism of a Banach space B. From Chapter 4, Bsplits into stable and unstable summands B = BS(T) a Bu(T) _BS(T) x Bu(T). The norm we use is the max (I-IS, I u) on B.

As in Chapter 4, suppose T has skewness a and let k be anynumber where 0 < k < 1 - a. Let Bb = (Bb)S X (Bb)u be the

closed ball with center at 0 and radius b (possibly b = oo)in B. Let the graph for T be

IBU

--+--- Bs

f

After a Lipschitz perturbation n with constant k satisfyingInfo <- b(1-a), the local stable set of (T + n)IBshown in the picture below, the graph of a map,

is,

h:

as

(Bb)S -'

(Bb)u. Furthermore, h is as smooth as n.

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(B,)S

(B,)

Let T subject to a Lipschitz perturbation n with constant ksuch that 1771, <_ b(1-a). One can show that T+n has a uniquefixed point in Bb, and by transferring the origin to thispoint one may assume, for simplicity, that n(O) = 0. Then wehave:

Theorem 6.2.1 (local stable manifold theorem) Let n: BbB be Lipschitz with constant k, where k < 1 - a. Suppose(O) = 0. Then there is a unique map h: (Bb)S - (Bb) u such

that graph h is the stable set of (T + n)IBb at 0, and h isLipschitz, and is Cr when n is Cr.

Note that the stable manifold theorem works under muchweaker assumptions than Hartman's theorem (Theorem 4.6.6),i.e., it is not necessary to assume n to be bounded as b =oo, nor a condition to ensure T + n to be homeomorphism isimposed.

There are two additional features of the local stablemanifold we would like to establish. First, the tangentspace at 0 to the local stable manifold is the stablesummand of the tangent map To(T + n). Second, iterationsunder (T + n) of points of the local stable manifold do notdrift in towards 0; they approach it exponentially.

Theorem 6.2.2 (i) If n is C' in Theorem 6.2.1, then thetangent to graph h at 0 is parallel to the stable manifoldof the hyperbolic linear map T + Dn(O). (ii) The map h andfi(graph h) are Lipschitz with constant µ where f = T + n, g

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= a + k < 1. For any norm 11 11 on B equivalent to norm I I onB, there exists A > 0 such that for all n ? 0 and for allx,y a graph h,

IIf"(x) - f^(y) II <- Aµ"IIx - ylITheorem 6.2.3 Let p be a hyperbolic fixed point of a Cr

diffeomorphism (r ? 1) f of M. Then, for some openneighborhood U of p, the local stable set of flu at p is aCr imbedded submanifold of M, tangent at p to the stablesummand of TPf.

Theorem 6.2.4 (global stable manifold theorem) Let p bea hyperbolic fixed point of Cr diffeomorphism f (r >- 1) ofM. Then the global stable set S of f at p is a Cr immersedsubmanifold of M, tangent at p to the stable summand of TPf.

If one looks at the proof of the above theorem, onenotices that the nature of the charts 0; strongly suggeststhat the global stable manifold of f at p is an immersedcopy of the stable manifold of the linear approximation.Indeed, by extending the map n to the whole of B, we canconstruct a locally Lipschitz bijection of the Banach spaceonto the stable manifold of f. With the help of Theorem6.2.3, one can show that the bijection gives a Cr immersionof (Bb), in X.

In fact, one may extend the above theorem from fixedpoints to periodic points of a diffeomorphism with littleextra effort. Indeed, if p is a periodic point of adiffeomorphism f: M - M, then p is a fixed point of thediffeomorphism fk, where k is the period of p. Thus:

Theorem 6.2.5 Let p be a hyperbolic periodic point ofperiod k of a Cr diffeomorphism f of M (r >_ 1). Then thestabe set of f at p is a Cr immersed submanifold of Mtangent at p to the stable summand of TPfk.

Now we turn to the stable manifold theory for flows.Note that the stable manifold theorem for a hyperbolic fixedpoint of a flow is a simple corollary of the correspondingtheorem for diffeomorphisms. Let 0 be a flow on M, and let dbe an admissible distance function on M. The stable set of 0at p e M is {x e M : d(O(t,x),O(t,p)) - 0 as t - oo) and the

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unstable set of 0 at p is {x e M : d(O(t,x),O(t,p)) - 0 as t

- -00). Recall that if p is a fixed point of 0 then 0(t,x) -p as t - - iff p(n,x) - p as n oo, where t c R and n e Z.Thus the stable set of the flow 0 at p is precisely thestable set of the diffeomorphism 01 at p, or of Ot for any

other t > 0. Thus we obtain the following theorem.Theorem 6.2.6 (global stable manifold theorem) Let p be

a hyperbolic fixed point of a Cr flow (r ? 1) ¢ on M. Thenthe stable set of 0 at p is a Cr immersed submanifold of M,tangent at p to the stable summand of TP01.

Theorem 6.2.7 (stable manifold theorem for closedorbits) Let r be a hyperbolic closed orbit of a Cr (r ? 1)flow 0 on a manifold M with distance function d. The stableset W,(p) of 0 at p is a Cr immersed submanifold tangent atp to the stable summand of M with respect to TP0', where ris the period of F. If q = O (p), then Ws(q) = Ot(Ws(p)). Thestable set Ws(r) of 0 at t is a Cr immersed submanifoldwhich is Cr foliated by (W5(q): q e r).

Here the submanifold Ws(p) is called the stable manifoldof g5 at p. Since Ws(p) is also the stable manifold of thediffeomorphism 0' at p, it is independent of the distancefunction d, as also is WS(r) = Uger W, (q)

We have seen that when a point p e X is periodic and Xhas a hyperbolic structure with respect to f, its stable setis a manifold. We want to know whether we can extend thisnotion of hyperbolicity to a non-periodic point p and get astable manifold theorem for such p. The problem is that TPf"does not map T

PX to itself for n > 0, and thus we cannot

define hyperbolicity in terms of eigenvalues of this map.Let A be any invariant subset of a Riemannian manifold

X. and let TAX be the tangent bundle of X over A. We saythat A has a hyperbolic structure with respect to f if thereis a continuous splitting of TAX into the direct sum ofTf-invariant sub-bundles ES and E. such that, for someconstants K and a and for all v e Es, w e E, and n >_ 0,

ITf"(v)I -< Kanivl, ITf-"(w)1 <- KanIwI, where 0 < a < 1. Thatis, Tf will eventually contract on E. and expand on Eu. A

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hyperbolic subset of X (with respect to f) is a closedinvariant subset of X that has a hyperbolic structure. For ageneral manifold, the existence of a hyperbolic structure onthe manifold implies there are some topological restrictions(or requirements) on the manifold. For instance, thefrequently used hyperbolic structure on a manifold is theLorentz structure, which requires the existence of a nowherezero vector field on the manifold. For more complexhyperbolic structures (locally, indefinite metric), thetopological requirements are given by a series of theoremsin Steenrod [1951]. We shall not consider these here, onlythat such a hyperbolic structure exists.

Before we state the generalized stable manifold theorem,let us make several definitions so that the theorem can bestated. Let B(x,a) be an open ball in X with center x andradius a, with respect to the Riemannian distance functiond, and let D(x,b) be the set (y a X: d(f"(x),f"(y)) < b forall n ? 0). For b >_ a >_ 0 we define the stable set of size(b.a) of f at x to be B(x,a)nD(x,b). A map of an open subsetof the total space of a vector bundle into a manifold isdenoted by Fr (r times continuously fiber differentiable)if, with respect to admissible atlases, all partialderivatives in the fiber direction up to order r exist andare continuous as functions on the total space.

Theorem 6.2.8 (Generalized stable manifold theorem)Let f be a Cr diffeomorphism of X, and let A be a compacthyperbolic subset of X, with associated decomposition TAX =ES a Eu. Then there exists an open neighborhood W of thezero section in Es and an Fr map h: W - X such that for someb ? a >_ 0 and for all x e A, g restricted to the fiber Wxover x is a Cr imbedding with image W." (x), the stable setof size (b,a) at x. The tangent space to WsLo`(x) at x is(E,) x.

The above theorem is a corollary of an even more generaltheorem formulated by Smale [1967], proved by Hirsch andPugh (1970], with addendum by Hirsch, Pugh and Shub [1970].

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6.3 Low dimensional stable systemsFor the simplest example of a low dimensional stable

system, consider a vector field v on S' with no zeros. Thenthere is precisely one orbit and it is periodic. In fact, wemay identify T(S1) = S' x R. Let v: S' - R be the principalpart of v. Since v is continuous and S' is compact, Iv(x)I

is bounded below by some constant a > 0. Then anyperturbation of v less than a does not introduce any zero.Thus the perturbed vector field still has only one orbit,and is topologically equivalent to v. So v is structurallystable (in fact, C°-structurally stable).

Now suppose that v has finitely many zeros, all of whichare hyperbolic. We shall call such a vector field on S1 aMorse-Smale vector field on S1. (We shall give a definitionof a Morse-Smale vector field on a higher dimensional spacelater. Here we merely want to demonstrate the idea for thesimplest case first). Then there must be an even number (2n)of zeros, with sources and sinks alternating around S1. See,for instance:

n=l n=2 n=3Note that the hyperbolic zeros are each individuallystructurally stable, so that a sufficiently smallC1-perturbation of v leaves the orbit configurationunchanged on some neighborhood of the zeros. Furthermore, asufficiently small C°-perturbation introduces no furtherzeros outside this neighborhood. Thus the perturbed vectorfield is topologically equivalent to the original one, and vis C' structurally stable.

It should be noted that there are vector fields which

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are not Morse-Smale but nonetheless have the above orbitstructures. One can show, through the construction ofexamples, that structurally stable vector fields on S' withfinitely many zeros are Morse- Smale. In fact the followingstronger result is true.

Theorem 6.3.1 A vector field on S' is C' structurallystable iff it is Morse-Smale. Moreover, Morse-Smale vectorfields form an open, dense subset of rrS' (1 5 r 5 co).

In attempting to prove that certain properties are openand dense (i.e., it holds for systems in an open, densesubset of Diff(M) or rr(M)) or generic, one usestransversality theory (Chapter 2). Recall that this theoryinvestigates the way submanifolds of a manifold cross eachother, and how a map of one manifold to another throws thefirst across a submanifold of the second. Sard's theorem(Theorem 2.4.14) usually provides the opening shots. Alsonote that if a property is dense, then clearly a stablesystem is equivalent to a system has the property. Forinstance, the property of having only hyperbolic zeros isopen and dense for flows on a compact manifold of anydimension, and any structurally stable flow possesses thisproperty. The proof is essentially the one for S'.Similarly,

Theorem 6.3.2 [Kupka 1963, Smale 1963] Fordiffeomorphisms of a compact manifold M the followingproperties are generic, and satisfied by structurally stablediffeomorphisms: (i) all periodic points are hyperbolic,(ii) for any two periodic points x and y, the stablemanifold of x and the unstable manifold of y intersecttransversally. (That is, the tangent spaces of the twosubmanifolds at any point of intersection p generate (span)the whole tangent space of the manifold M at p. In otherwords, (Ws(x))P + (W"(y))P = MP and it is denoted by W.(x),

We say that a dynamical system is Morse-Smale if (i) itsnon- wandering set is the union of a finite set of fixedpoints and periodic orbits, all of which are hyperbolic, and

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(ii) if and are any two orbits of the non-wanderingset then Wu Thus the non-wandering set of a2-dim. flow consists of at most five types of orbits, nmely,hyperbolic source, sinks, saddle points, and hyperbolicexpanding and contracting closed orbits.

Peixoto shows that Theorem 6.3.1 also holds for vectorfields on compact orientable 2-manifolds and fordiffeomorphisms of S1. That is:

Theorem 6.3.3 [Peixoto 1962] A vector field on acompact orientable 2-dim. manifold M (resp. a diffeomorphismof S) is C' structurally stable iff it is Morse-Smale.Morse-Smale systems are open and dense in r'(M) (resp.

DiffrSl) for 1 - r -< oo.

The theorem for non-orientable 2-manifolds is still anopen question.

Although the results above failed to generalize tohigher dimensions, it is fortunate to find that for alldimensions:

Theorem 6.3.4 A Morse-Smale system on a compactmanifold is structurally stable.

In fact, it gets even better. Since every compactmanifold admits a Morse-Smale system [Smale 1961], thusevery compact manifold admits a structurally stable system(Palis and Smale 1970].

Theorem 6.3.4 and the above statement are corollaries ofthe following theorem due to Palis and Smale [1970]:

Theorem 6.3.5 Let M be a compact e manifold withoutboundary, and for r >_ 1 let Diff(M) be the set of Crdiffeomorphisms of M with the uniform Cr topology. For f eDiff(M) we denote by fl(f) the set of nonwandering points off. If f e Diff(M) satisfies: (a) fl(f) is finite, (b) fl(f) ishyperbolic, and (c) transversality condition, then f isstructurally stable.

One can extend the result to flows or vector fields.Using the above theorem and a converse known for some time,we have:

Theorem 6.3.6 Let f e Diff(M), with fl(f) being finite.

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Then f is structurally stable iff f satisfies (a)-(c) inTheorem 6.3.5.

This result gives a good characterization of structuralstability in the n-finite case and leads to the question offinding a similar characterization for the general case.Palis and Smale also proposed the following conjecture. f eDiff(M) is structurally stable iff f satisfies: (a) axiom A,i.e., n(f) is hyperbolic and the set of periodic points of fis dense in n(f); (b) strong transversality condition, i.e.,for all x, y e n(f), WS and W, intersect transversally.

Armed with these results, Afraimovich and Sil'nikov[1974] considered the system:

dx/dt = f(x), (6.3.1)

with f e Cr, r> 1, in some region D c R. Let G be abounded region which is homeomorphic to a ball and has asmooth boundary. Then they considered the system,

dx/dt = f(x), dO/dt = 1, (6.3.2)

defined in G x S1. They have the following theorem.Theorem 6.3.7 If system (6.3.1) is a Morse-Smale system

in G, then for sufficiently small d in C', the6-neighborhoods of system (6.3.2) will be everywhere denseMorse-Smale systems.

From Palis and Smale [1970] and Robbin [1971] oneobtains the following useful result.

Corollary 6.3.8 If system (6.3.1) is structurally stablein G and does not have periodic motions, then system (6.3.2)is also structurally stable.

In the following, we shall briefly discuss perturbationproblems for the two dimensional system dx/dt = f(x). By aperturbation of the system, we mean a system

dx/dt = f(x) + µg(x,t), (6.3.3)

where x = (u,v) a R2, g << 1. Equivalently, we have thesuspended system:

dx/dt = f(x) + gg(x,9),dO/dt = 1, (x,9) a R' x S1. (6.3.4)

Here f(x) is a Hamiltonian vector field and gg(x,t) is asmall perturbation which need not even be Hamiltonian

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itself. We want to discuss the bahavior of the orbits of(6.3.3) for (x,µ) in a neighborhood of rx(0) where r is theperiodic orbit of the unperturbed system.

The basic idea is due to Melnikov [1963]. It is to makeuse of the globally computable solutions of the unperturbedintegrable system in the computation of perturbed solutions.In order to do this, we must first ensure that theperturbation calculations are uniformly applicable onarbitrarily long time intervals. Let us make the assumptionsprecise. Consider the system (6.3.4) which is sufficientlysmooth, Cr, r >_ 2, and bounded on the bounded region G, andg is periodic in t with period T, and f = (ft(x),f2(x))t, andg = (g1(x,t),g2(x,t))t. For simplicity, we assume that theunperturbed system is Hamiltonian with fl = aH/av, f2 =-aH/au. Furthermore, we assume the unperturbed flow is: (a)

For µ = 0, the system (6.3.4) possesses a homoclinic orbitq°(t), to a hyperbolic saddle point pO; (b) Let t° = {q°(t)I

t e R)U(pO), the interior of t° is filled with a continuousfamily of periodic orbits q°(t), a c (-1,0). Letting d(x,rO)= infg,r°Ix - qj we have lima,o supt,R d(ga(t),ro) = 0. (c) Letha = H(ga(t)) and T. be the period of qa(t). Then Ta is adifferentiable function of ha and dTa/dha > 0 inside t°.

Before we state any results, we would like to remarkthat many of them can be proved under less restrictiveassumptions. Note that assumption (a) implies that theunperturbed Poincare map P° has a hyperbolic saddle point pOand that the closed curve t° = Wu(pO) n Ws(p0) is filled withnontransverse homoclinic points for P0. Finally, theMelnikov function is defined as:M(tO) = J_m f (q°(t - to) ) A g(q°(t - to) , t) dt (6.3.4)Then we have the following important theorem which allows usto test for the existence of transverse homoclinic orbitsfor specific systems:

Theorem 6.3.9 If M(to) has simple zeros and isindependent of M. then for sufficiently small µ > 0, W.(pand Ws(pµ) intersect transversely. If M(to) remains away fromzero then Wu(pA) n Ws(pA) = 0.

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Remarks: (i) M(to) is periodic with period T.(ii) If the perturbation g is derived from a time-dependentHamiltonian function G(u,v) such that g, = aG/av, g2 =-aG/au; then:M(t°) = J_°° {H(q°(t - t°)) , G(q°(t - t°) ,t) ) dt, (6.3.5)where (H,G) denotes the Poisson bracket (Section 2.8).(iii) If g = g(x), not explicitly time dependent, then usingGreen's theorem we obtain: M(to) = fintr° trace Dg(x) dx.(iv) By changing variables t - t + to, we have a moreconvenient Melnikov integral:

M(to) = f_m f (q°(t) ) A g(q°(t) ,t+t°)dt.Now, let us return to the more general case, where g =

g(x,t;6).Theorem 6.3.10 Let the system dx/dt = f(x) + µg(x,t;6),

9 e R, satisfies the assumptions (a)-(c) before we definedthe Melnikov function. Suppose that the Melnikov functionM(to,0) has a quadratic zero M(r,Ob) = (aM/ato)(r,8b) = 0 but(a2M/at02) (r,() b) + 0 and (aM/aO) (r,9b) + 0. Then ee = eb +O(µ) is a bifurcation value for which quadratic homoclinictangencies occur in the family of systems.

If we rewrite system (6.3.3) as: dx/dt = f(x,µ), where x_ (x1,x2), and for µ = 0, the system becomes dx/dt = f(x,0)has a periodic orbit r with period T. Let us define

co = tr[af(0,0)/ax) + 0.Theorem 6.3.11 The homoclinic orbit r is asymptotically

stable if ao < 0 and unstable if ao > 0. There can be atmost one periodic orbit bifurcating from r and it isasymptotically orbitally stable if ao < 0 and unstable if ao> 0.

Thus, if ao = 0, one can have either periodic orbits inany neighborhood of r, or t can be asymptotically stable orunstable. To illustrate this, consider: dx/dt = 2y, dy/dt =12x -3x2. This system has a potential V(x,y) = x3 - 6x2 +y2, and the solution curves are given as

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Clearly, the origin 0 is a hyperbolic saddle and point A =(4,0) is a center. There is a periodic orbit in anyneighborhood of the homoclinic orbit I' and ao = 0. Take notethat r c ((x,y): V(x,y) = 0). Consider the perturbed systemto be:

dx/dt = 2y - AV(x,y)(12x - 3x') f1(x,y,9),

dy/dt = 12x - 3x' + MV(x,y)2y = f2(x,y,µ).

It is easy to show thatao(p) = tr a(f,(0,0,µ),fz(0,0,µ))/a(x,y) = 0 for all A. Theperturbed system has only the equilibrium points 0 and Awith 0 a hyperbolic saddle and A a hyperbolic focus for µ0 sufficiently small. A is stable for µ > o and unstable forµ < 0. The solution curves in (x,y)-space defined by V(x,y)= 0 is invariant under the perturbation. Thus, I' is again a

homoclinic orbit for the perturbed system for every A.Noting that the perturbed system is obtained from theoriginal system by a rotation through an anglearctan(MV(x,y)), it follows that no curve V(x,y) = constantinside the curve r can be tangent to the vector field of theperturbed system. Thus, there can be no periodic orbits ofthe perturbed system inside r for µ + 0. Since µ > 0 (µ < 0)implies the focus A is stable (unstable), it follows that ris unstable (asymptotically stable) for µ > 0 (µ < 0).

Chan [1987) considered the equationd'x/dt' + g(x) = - adx/dt + gf(t),

where x is real, g is smooth, f is periodic, i.e., f(t+l) _f(t), and a and µ are small parameters. By using periodicMelnikov functions and the method of Liapunov, all the

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Floquet multipliers of the bifurcating subhamonic solutionsare determined. Recently, Ling and Bao (1987) have deviced anumerical implementation of Melnikov's method. The procedureis based on the convergence of the integral and theuniqueness of the boundary of the horseshoe region in theparameter space under the conditions that it be inversesymmetric, i.e., f(x) = -f(-x), and there exists more thanone homo(hetero)clinic orbits. Meanwhile, Salam [1987]presented explicit calculations which extended theapplicability of the Melnikov's method to include a generalclass of highly dissipative systems. The only requiredcondition is that each system of this class possesses ahomo(hetero)clinic orbit. Furthermore, it was shown thatsufficiently small time-sinusoidal perturbation of thesesystems resulted in transversal intersection of stable andunstable manifolds for all but at most discretely manyfrequencies.

For further details for bifurcations of autonomous orperiodic planar equations, and their applications, see,e.g., Chapter 4 of Guckenheimer and Holmes [1983]; Chapter9-11 of Chow and Hale [1982]; Chapter 7 of Lichtenberg andLieberman (1983); and for applications, see, e.g., Smoller[1983].

Recently, Wiggins [1988] has developed a globalperturbation technique similar to that of Melnikov [1963]for detecting the presence of orbits homoclinic tohyperbolic periodic orbits and normally hyperbolic invarianttori in a class of ordinary differential equations. Thistechnique is more general then Melnikov's because it appliesto systems undergoing large amplitude excitation at lowfrequencies and to systems undergoing quasiperiodicexcitation.

When the paper by Palis and Smale appeared, it was knownthat there were other structurally stable systems besideMorse-Smale systems. One of them was the toral automorphisms(see Section 4.1). These are structurally stable, but theirnon-wandering sets are the whole of the tori, so they

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certainly fail condition (i) of Morse-Smale. Next, we shallvery briefly discuss the Anosov systems.

6.4 Anosov SystemsA diffeomorphism f: M - M of a manifold M is Anosov if M

has a hyperbolic structure with respect to f. Recall thatthis means that the tangent bundle TM splits continuouslyinto a Tf- invariant direct-sum decomposition TM = ES + Eusuch that Tf contracts ES and expands Eu with respect tosome Riemannian matric on M. Trivially, hyperbolic linearmaps f possess this property, since one has theidentification TR" = R" x R" and Tf(x,v) = (f (x) , f (v)) . Inthe case of toral automorphisms, this splitting is carriedover to the tours when the identification was made. So,toral automorphisms are Anosov.

Similarly, a vector field on M is Anosov if M has ahyperbolic structure with respect to it. As examples ofsuch, we have all suspensions of Anosov diffeomorphisms.

Theorem 6.4.1 (Anosov 1962) Anosov systems on compactmanifolds are C1 structurally stable.

There are some unsolved problems about Anosovdiffeomorphisms. For instance, is their non-wandering setalways the whole manifold? Do they always have a fixedpoint? Since not all manifolds admit Anosov diffeomorphism,do all n-dim. manifolds which do admit them have R" asuniversal covering space?

6.5 Characterizing structural stabilityRealizing the diagonal differences of systems such as

Morse-Smale and Anosov systems, it is a challenging problemto characterize structural stability. The essential linkcomes about when Smale recognizes the fact that by replacingthe term "closed orbits" in Morse-Smale definition by "basicsets" (to be defined shortly), then Anosov and other systemsare also encompassed.

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A dynamical system has an n-decomposition if its non-wandering set n is the disjoint union of closed invariantsets n1, 0Z,..., nk. If the system is topologicallytransitive on n, (i.e., n, is the closure of the orbit ofone of its points) for all i, we say that n1U.... Unk is aspectral decomposition, and that the n1 are basic sets. Onecan also define a basic set individually by saying that aclosed invariant set, A c n is basic if the system istopologically transitive on A but A does not meet theclosure of the orbit of nlA. Note that a basic set isindecomposable since it is not the disjoint union of twonon-empty closed invariant sets.

A dynamical system satisfies Axiom A if itsnon-wandering set (a) has a hyperbolic structure, and (b) isthe closure of the set of closed orbits of the system. Itwas conjectured that (a) implies (b). Newhouse and Palis[1970] have shown that it is true for diffeomorphisms of2-dim manifolds, but false for higher dimension manifolds[Danberer 1977].

Theorem 6.5.1 (Spectral Decomposition Theorem) The non-wandering set of an Axiom A dynamical system on a compactmanifold is the union of finitely many basic sets.

In order to visualize an Axiom A system, one thinks of asystem of 2-dim manifold with finitely many fixed points andperiodic orbits, all hyperbolic, such as the gradient of theheight function on the torus, or a Morse-Smale system for Sz(see Fig. 6.5.1). But in higher dimensions, one replace thefixed points and periodic orbits by more general basic sets.

It should be pointed out that there is no need for abasic set to be a submanifold. Such basic sets are termedstrange or exotic.

Following Smale, we say that a system is AS if itsatisfies both Axiom A and the strong transversalitycondition, which is, for all x and y in the non-wanderingset of the system, the stable manifold of the orbit of x andthe stable manifold of the orbit of y intersecttransversally. The strong transversality c 3ition is the

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general version of the second condition in the definition ofa Morse-Smale system. For C2 diffeomorphisms, the best setof criteria of structural stability is due to Robbin [1971]and for C' diffeomorphisms and flows Robinson [1975a; 1975b;1976; 1977; 1980].

Fig. 6.5.1

Theorem 6.5.2 Any AS system is C' structurally stable.It is known that structural stability is equivalent to

AS system when fl(f) is finite [Palis and Smale 1970], andthat structural stability and Axiom A imply strongtransversality [Smale 1967]. It seems that the converse ofTheorem 6.5.2 is also true, which would be verysatisfactory. Nonetheless, the closest to this is the resultdue to Franks [1973]. A diffeomorphism f: M - M isabsolutely C1-structurally stable if for some C1-neighborhood N c Diff(M) of f, there is a map a associatingwith g e N a homeomorphism a(g) of Diff(M) such that (i)a(f) = id,, (ii) for all g e N, g = hfh"' where h = a(g),(iii) a is Lipschitz at f with respect to the C° matric d(i.e., for some k > 0 and all g e N, d(a(g),id,) <- kd(g,f).

Theorem 6.5.3 Any diffeomorphism is absolutely C'structurally stable iff it is AS.

As an example, in order to classify the transitionsbetween interacting time-periodic and steady-state solutions

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of nonlinear evolution equations by using the normal formsof the imperfect bifurcation theory (Section 6.6), two typesof bifurcation of solutions of general evolution equationsare fundamental: the bifurcation of a steady state withamplitude x and the Hopf bifurcation of a time-periodicsolution with amplitude y from a stationary one.Interactions between them lead to secondary bifurcations ofperiodic solutions and to tertiary bifurcations ofdouble-periodic motions lying on tori and eventually tochaotic motions. Such interactions occur if a controlparameter g in the evolution equation crosses some criticalvalues. By Liapunov-Schmidt reduction we can show that x andy satisfy two algebraic normal form equations a(x,g,y') = 0and yb(x,g,y2) = 0. The solution of the first is a twodimensional multisheeted surface y = O(x,&) in(x,y,µ)-space, and that of the second equation is anothersurface y = Q(x,µ). The lines along which both surfacesintersect are the bifurcation diagrams of the evolutionequations from which the behavior of the system can beinferred as µ varies. The intersection of the two surfacesmay be transversal so that any perturbation of a and b,i.e., a slight deformation or shifting of the surfaces,causes no new type of intersections. In this case structuralstability of the bifurcation diagram is ensured at theoutset. Nonetheless, if the two surfaces intersect withtangential contact, or just touching, then a slightdeformation or shifting of them produces new intersectionsand gives rise to new bifurcation diagrams which then arestable against any further perturbations. Theseperturbations can be thought to be induced by variations ofsystem-imminent imperfection parameters (such as impurity ofmaterial parameters etc.) in the original evolutionequations. Since the forms of the perturbed polynomials aand b can be classified into a finite set by imperfectbifurcation theory, the problem of interacting spatial andtemporal patterns has thus been reduced to linking togetherthe possible basic bifurcation diagrams. Then a variety of

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new phenomena, such as gaps in Hopf branches, periodicmotions not stably connected to steady states, and thediscovery of formation of islands, which one can expect tofind in general systems of evolution equations. Many of thenew phenomena, predicted on topological grounds alone, stillawait experimental confirmation [Dangelmayr and Armbruster1983; Armbruster 1983].

As we have mentioned in Sections 1.2 and 6.1,structurally stable systems may not be dense. Indeed,Peixoto and Pugh [1968] have shown that structurally stablesystems are not dense on any noncompact manifold ofdimension >- 2. Finally, Williams [1970] showed thatstructurally stable diffeomorphisms are not dense on thetwo-dimensional torus. Thus, we are left with two courses ofapproach. We can either alter the equivalence relation onthe space of all dynamical systems hoping that stabilitywith respect to the new equivalence relation may be dense,or we can ask for some structures which are less than densein the given topology. One of the new equivalence relationwhich aroused most interest is n-stability. This is based onn-equivalence discussed in Chapter 4. n-stability isstability with respect to n-quivalence. Unfortunatelyn-stability is not any more successful than structuralstability as far as the dense of the structure is concerned.For examples, see Abraham and Smale [1970] and Newhouse[1970a,b].

For the second approach, it is natural to ask thefollowing question: Given an arbitrary dynamical system, canwe deform it into a structurally stable system? If we can,how small a deformation is necessary? Clearly, we cannotmake it arbitrarily C1-small, otherwise it would implyC1-density of structural stability. Thus we may be able todeform it by an arbitrarily Co- small deformation. Here weonly talk about the size of the deformation needed toproduce structural stability, and we leave the smoothness ofthe maps and the definition of structural stability asbefore. The following theorem by Smale [1973] and Shub

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[1972] answers these questions.Theorem 6.5.4 Any C' diffeomorphism (1 5 r <_ m) of a

compact manifold is Cr isotopic to a C' structurally stablesystem by an arbitrarily C°-small isotopy.

Thus the structural stability is dense in DiffrX withrespect to the C°-topology. It should be noted that thestructural stability is no longer open in this topology. DeOliviera [1976] showed an analogous theorem for flows.

Recall that the motivation of studying the structuralstability of a dynamical system is because one is requiredto make measurements, but since the measurements are limitedby their measurement uncertainties (the measured systems areonly approximations of the true systems. It is important toknow whether the qualitative behavior of the approximatesystem and the true system are the same. Structuralstability of a system guarantees this if the approximationis sufficiently good. To make things more complicated (alsomore interesting), in most of the situations the measuredquantities would not be completely time independent, butonly be approximately constant during the measuringinterval. In other words, the true dynamical system is notreally autonomous but to a certain extent, time dependent.Thus, we are asking under what conditions an autonomoussystem is structurally stable when it is perturbed to a timedependent system. Franks [1974] gave a solution for C2diffeomorphisms on compact manifolds:

Theorem 6.5.5 If f: M - M is a C2 diffeomorphism of acompact manifold, then f is time dependent stable iff fsatisfies Axiom A and the strong transversality condition.

In closing, we would like to point out that severalother notions of stability have been proposed in hope thatthey might be generic, nonetheless, none as yet has beencompletely sucessful. After all, maybe it is too optimisticto expect to find a single natural equivalence relation withrespect to which stability is dense. More recently,attention has been focused on the interesting and importantquestion of bifurcation of systems due mainly to R. Thom

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[1975]. Which we shall discuss very briefly in the nextsection.

In a series of papers, Hirsch [1982, 1985, 1988, 1989a,1989b, 1989c] has studied a vector field in n-spacedetermines a competitive (or cooperative) system ofdifferential equations provided all the off-diagonal termsof its Jacobian matrix are nonpositive (or nonnegative). Hehas found that orthogonal projection along any positivedirection maps a limit set homeomorphically andequivariantly onto an invariant set of a Lipschitz vectorfield in a hyperplane. And limit sets are nowhere dense,unknotted and unlinked. In other words, most trajectoriesare stable and approach stationary points, and limit setsare invariant sets of systems in one dimension lower. Indimension 2 every trajectory is eventually monotone, and indimension 3 a compact limit set which does not contain anequilibrium is a closed orbit or a cylinder of closedorbits. Furthermore, Hirsch [1985] has found that acooperative system cannot have nonconstant attractingperiodic solutions. The persistent trajectories of then-dimensional system are studied under the assumptions thatthe system is competitive and dissipative with irreducibleJacobian matrices. Then it is shown that there is acanonically defined countable family of disjoint invariantopen (n-l)-cells which attract all nonconvergent persistenttrajectories. These cells are Lipschitz submanifolds and aretransverse to positive rays. Furthermore, if the Jacobianmatrices are strictly negative then there is a closedinvariant (n-1)-cell which attracts every persistenttrajectory. In 3 dimensional system, the existence of apersistent trajectory implies the existence of a positiveequilibrium. It is then shown that among 3-dimensionalsystems which are competitive or cooperative, thosesatisfying the generic conditions of Kupka- Smale alsosatisfy the conditions of Morse-Smale and are thereforestructurally stable. This provides a new and easilyrecognizable class of systems which can be approximated by

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structurally stable systems. For three-dimensional systems,a certain type of positive feedback loop is shown to bestructurally stable.

6.6 BifurcationAs we have discussed earlier at the beginning of Chapter

5, the most important systems are the ones which can be usedto model the dynamics of real life situations. But rarelycan real life situations ever be exactly described, and weshould expect to lead to slight variations in the modelsystem. Consequently, a theory making use of qualitativefeatures of a dynamical system is not convincing nor has itsutility unless the features are shared by "nearby" systems.That is to say that good models should possess some form ofqualitative stability. Hence our contempt for extremelyunstable systems. Furthermore, in a given physicalsituation, there may be factors present which rule outcertain dynamical systems as models. For instance,conservation laws or symmetry have this effect. In thiscase, the subset of these dynamical systems that areadmissible as models may be nowhere dense in the space ofall systems, and thus the stable systems that we areconsidering are really irrelevant. Thus one has to considerafresh which properties are generic in the space ofadmissible systems! On the other hand, even if the usualspace of systems is the relevant one, the way in which asystem loses its stability due to perturbation may be ofimportance, since the model for an event consists of a wholefamily of systems. In his theory of morphogenesis, Thomenvisions a situation where the development of a biologicalorganism, say, is governed by a collection of dynamicalsystems, one for each point of space time.

Bifurcation is a term which has been used in severalareas of mathematics. In general, it refers to a qualitativechange of the object under study due to change of parameterson which the object depends. For the kinds of applications

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we have in mind, the following more precise definitionsuffices. Let X and Y be Banach spaces, U c X, and F: U - Y.Suppose there is a one-to-one curve r = (x(t): t e (O,1)) cU such that for z e r, F(z) = 0. A point p e r is abifurcation point for F with respect to r (more simply abifurcation point) if every neighborhood of p contains zerosof F not in r. In most applications, possibly after making achange of variables, one usually has X = RxB where B is areal Banach space, F = F(a,u), and r = {(a,O): a e (a,b) cR). Here, the members of r will be called trivial solutionsof F(a,u) = 0. Thus, we are interested in nontrivial zerosof F.

We would like to mention several models of phenomena toillustrate the motivation for studying bifurcation.

First, an infinite horizontal layer of a viscousincompressible fluid lies between a pair of perfectlyconducting plates. A temperature gradient T is maintainedbetween the plates, the lower plate being warmer. If T isappropriately small, the fluid remains at rest, the heat istransported through the fluid solely by conduction, and thetemperature is a linear function of the vertical height.When T exceeds a certain value, the fluid undergoestime-independent motions called convection current and heatis transpoted through the fluid by convection andconduction. In actual experiments, the fluid breaks up intocells whose shape depends in part on the shape of thecontainer. This is called Benard instability.Mathematically, the equilibrium configuration of the fluidis described by a system of nonlinear partial differentialequations. Formulated in the general Banach space framework,the pure conduction solutions correspond to the trivialsolutions, while the value of T at which convection beginscorresponds to a bifurcation point. For reference, see, forinstance the classic, Chandrasekar [1961]. See alsoKirchgassner and Kielhofer [1973].

Another interesting problem in fluid motion is theTaylor problem of rotating fluid. A viscous incompressible

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fluid lies between a pair of concentric cylinders whose axisof rotation is the symmetry axis which is vertical. Theinner cylinder rotates at a constant angular velocity nwhile the outer one remains at rest. If n is sufficientlysmall, the fluid particles move in circular orbit withvelocity depending on their distance from the axis ofrotation. Equilibrium states of the fluid are solutions ofthe time-independent Navier-Stokes equations, and they arecalled Couette flow. When n exceeds a critical value, thefluid breaks up into horizontal bands called Taylor vorticesand a new periodic motion in the vertical direction issuperimposed on the Couette flow. Here Couette flowcorresponds to the trivial solutions in the generalframework, and the values n at which the onset of Taylorvortices taking place corresponds to a bifurcation point.

Buckling phenomena of a flat plate is another example ofbifurcation. A thin, planar, clamped elastic plate issubjected to a compressive force along its edges. If themagnitude of this compressive force f is small enough, theplate remains motionless and in equilibrium. But if fexceeds a certain value, the plate deflects out of the planeand assumes a nonplanar equilibrium position called abuckled state. Equilibrium configurations of the platesatisfy a system of nonlinear partial differential equationcalled the von Karman equations. The unbuckled states aretrivial solutions of these equations, while the value of fat which buckling taking place corresponds to a bifurcationpoint. See, for example, Friedrichs and Stoker [1941];Berger and Fife [1968]; Keller and Autman [1969]; Berger[1977]. Thompson [1979] has shown that elastic structureunder dead and rigid loadings can assess the stable regionsthrough a succession of folds, and the examples of bucklingof elastic arches, shallow domes and the incipientgravitational collapse of a massive cold star aredemonstrated. A closely related mechanical phenomenon hasbeen described by Duffing's equation. The book byGuckenheimer and Holmes [1983] gives a very detailed study

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of the four systems, namely Van der Pol's equation fornonlinear electronic oscillator, Duffing's equation forstiffed spring with cubic stiffness, Lorenz equation fortwo-dimensional fluid layer heated from below, and abouncing ball on a vibrating table. This book is highlyrecommended. The applications of these systems are farbeyond their original problems. We shall see this in Chapter7. For an elementary yet detailed discussion of lowdimensional bifurcation, the book by Iooss and Joseph [1980]is recommended. A very recent book by Ruelle [1989] is alsohighly recommended. A much more advanced treatment ofbifurcation theory, Chow and Hale [1982] is indispensible.

There are also many interesting applications ofbifurcation theory in other physical and nonphysicalsciences, such as chemical reactions, geophysics,atmospherical science, biology, and social science.

Let us return to the general theory of bifurcation,where there are three main questions of interest: (a) Whatare the necessary and sufficient conditions for (a,O) e r tobe a bifurcation point? (b) What is the structure of the setof zeros of F(a,u) near (a,O)? (c) In problems such asdescribed above, where there is an underlying evolutionequation of which the solutions described are equilibriumsolutions, determine which solutions are stable or unstable.For the detailed "mechanism" of the bifurcation, see Holmesand Rand [1978]. For the forced van der Pol-Duffingoscillator applies to the trubulence flow, the routes toturbulence are discussed in Coullet, Tresser and Arneodo[1980].

A catastrophe is a point where the form of the organismchanges discontinuously, this corresponds to topologicalchange in the orbit structure of the dynamical systems. Wesay that the family of dynamical systems bifurcates at thepoint where the change is discontinuous.

Let us give some simple examples of bifurcations offlows. There are local changes which can happen on anymanifold. For convenience and simplicity, let us take a

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suitable chart in R". First, a vector field v on R by v(x) _a + xz and a e R. We are interested in how the orbitstructure varies with a. We find that: (i) for a > 0, thereis no zero, and the whole of R is an orbit orientedpositively; (ii) for a = 0, we have a zero at x = 0 which isone way zero in the positive direction; (iii) for a < 0, wehave two zeros, a sink at -,/-a and a source at ,/-a. Thus thebifurcation occurs at a = 0.

a > 0 a = 0

U < 0

If we take the product of vQ with a fixed (i.e.,independent of a) vector field on

R""1

having a hyperbolicfixed point at 0, we obtain a bifurcation of the resultingvector field on R". All such bifurcations are known assaddle-node bifurcations such as the following pictures forn = 2 depicted. A saddle point and node come together,amalgamate, and cancel each other out!

(a<O) -- ---- -i /

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(a>0)

Saddle-node bifurcations are important because they arestable as bifurcations of one-parameter families. Roughlyspeaking, one-parameter families near a family with asaddle- node bifurcation also exhibit something that istopologically like a saddle-node bifurcation near theoriginal one. One speaks of them as codimension onebifurcations; one can visualize the set of systemsexhibiting zeros of the a = 0 type in the above example (insome sense) as a submanifold of codimension one inI'r(M), and the one-parameter families are being given by anarc in rr(M) crossing the submanifold transversally.

Note that, the bifurcation illustrated below (a node,i.e., a -< 0, bifurcating into two nodes and a saddle point,a > 0) is not stable for the one-parameter families. It canbe perturbed slightly so that there is a saddle-nodebifurcation pair near to, but not at, the original node.

sink (node) node saddle node

a -< 0 a > 0

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The saddle-node bifurcation is the typical bifurcationresulting when the sign of a real eigenvalue of thedifferential at a zero is changed by varying a singleparameter governing the system. There is a typicalcodimension one bifurcation which comes about when the signof the real part of a complex conjugate pair of eigenvaluesis changed by varying a single parameter. This is known asthe Hopf bifurcation [Hopf 1943].

For instance, consider the vector field vQ on R2 givenby va(x,y) = (-y - x(a + x2+ y2 ), x - y(a + x2+ y2)). Here ais a single real parameter. For all a e R, vQ has a zero atthe origin, and the linear terms make this a spiral sourcefor a < 0 and a spiral sink for a > 0. For a = 0, the linearterms would give a center, but the cubic terms make theorbits spiral weakly inwards. The interesting feature is theunique closed orbit one obtains at x2 + y2 = - a for each a< 0. The bifurcation is illustrated in the following:

a < 0 a <_ 0

That is, the periodic attractor (or orbit) decreases in sizeuntil it amalgamates with the spiral source to form a spiralsink. It is intriguing to note that the reverse bifurcationcan happen where a spiral sink splits into a spiral sourceand a periodic attractor. This is because when one hassomething from an inert (or dead) source, one creates apulsating (and alive) periodic orbit.

Even for a simple recurrence equation,x(t+l) = Ax(t)[l - x(t-1)],

which is analogous to the logistic model, is found to showHopf bifurcation [Morimoto 1988]. It has a fixed point atzero for A e [0,1), and at 1- 1/A for A e [1,2), and thefixed point is destabilized at A = 2. For A > 2 the

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oscillatory behavior appears, which is Hopf bifurcation.Recently, Hale and Scheurle [1985] investigated the

smoothness of bounded solutions of nonlinear evolutionequations and they have found that in many cases globallydefined bounded soutions of evolution equations are assmooth in time as the corresponding operator, even if ageneral solution of the initial value problem is much lesssmooth. In other words, initial values for bounded solutionsare selected in such a way that optimal smoothness isattained. In particular, solutions which bifurcate fromcertain steady states, such as periodic orbits, almostperiodic orbits, homo- and heteroclinic orbits, have thisproperty.

Recently, Baer and Erneux [1986] have studied thesingular Hopf bifurcation from a basic steady state torelaxation oscillation characterized by two quite differenttime scales of the form dx/dt = f(x,y,a,e) and dy/dt =eg(x,y,a,e) where e << 1 and is the control parameter. Theirbifurcation analysis shows how the harmonic oscillationsnear the bifurcation point progressively change to becomepulsed, triangular oscillations. They further presented anumerical study of the FitzHugh-Nagumo equations for nerveconduction. They also considered the switching from a stablesteady state to a stable periodic solution, or the reversetransition. Baer et al [1989] further expanded their studyof the FitzHugh-Nagumo model of nerve membrane excitabilityas a delay or memory effect. It can occur when a parameterpasses slowly through a Hopf bifurcation point and thesystem's response changes from a slowly varying steady stateto slowly varying oscillations.

Next, let us briefly discuss and state the centermanifold theorem, which provides a mean for systematicallyreducing the dimension of the state spaces needed to beconsidered when analyzing bifurcations. Later in thischapter, we shall use the Lorenz system and its bifurcationsas an example to illustrate the role of center manifoldtheorem in bifurcation calculations.

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Suppose we have an autonomous dynamical system dx/dt =f(x) such that f(O) = 0. If the linearization of f at theorigin has no pure imaginary eigenvalues, then Hartman'slinearization theorem (Theorem 4.6.6) states that thenumbers of eigenvalues with positive and negative real partsdetermine the topological equivalence of the flow near theorigin. If there are eigenvalues with zero real parts, thenthe flow can be quite complicated near the origin. We haveseen such situations before. Let us consider the followingsystem: dx/dt = xy + x3, dy/dt = -y -x'y.We will not go into any detail to analyze the above system.It suffices to say that one of the eigenvalues is -1 andhence a one-dimensional stable manifold exists (indeed, they-axis). Direct calculation shows that the x-axis is asecond invariant set tangent to the center eigenspace E`.This is an example of a center manifold, an invariantmanifold tangent to the center eigenspace. Let us give asimple example before stating the main theorem. Thefollowing example is due to Kelley [1967] which also gavethe first full proof of the main theorem we shall stateshortly.

Let us consider the very simple system:dx/dt = x', dy/dt = -y.

The parametric solutions to this system have the followingform:

x(t) = a/(1 - at), y(t) = be-t.By eliminating t, we have the solution curves which aregraphs of the functions y(x) = (be"1/e)e1/". Clearly, for x <0, all of these solution curves approach the origin in sucha way that all of their derivatives vanish at x = 0. Whilefor x >- 0, the only solution curve approaches the origin isthe x-axis. Thus, the center manifold is not unique. Indeed,we can obtain a C° center manifold by piecing together anysolution curve in the left half plane with the positive halfof the x-axis. The center manifolds (heavy curves) are shownin the following figure. Nonetheless, the only analyticcenter manifold is the x-axis itself.

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X

Theorem 6.6.1 (Center manifold theorem) Let f be a Crvector field on R" vanishing at the origin and let A =Df(O). Let us divide the spectrum of A into three parts,namely, as, ac, au with Re µ < 0 if µ e as, Re µ = 0 if µ ea., and Re µ > 0 if µ e cu. Let us denote the eigenspaces ofas, ac, and au by Es, E`, and Eu respectively. Then thereexist Cr stable and unstable invariant manifolds WS and Wutangent to Es and Eu at 0 and aCr-1 center manifold W` tangent to E` at 0. The manifolds Wsare all invariant to the flow of f, and both the stable andunstable manifolds are unique, but the center manifold neednot be.

In general, the center manifold method isolates thecomplicated asymptotic behavior by locating an invariantmanifold tangent to the subspace spanned by the eigenspaceof eigenvalues on the imaginary axis. As we have noted inthe example and in the theorem, there are technicaldifficulties involving the nonuniqueness and the loss ofsmoothness of the invariant center manifold which are notpresent in the invariant stable manifold.

For further examples and detailed discussion includingthe existence, uniqueness, and smoothness of centermanifolds and the proof of the above theorem, see, e.g.,Marsden and McCracken [1976], Carr [1981], Chow and Hale[1982], Guckenheimer and Holmes [1983].

After Guckenheimer and Holmes [1983], Guckenheimerpublished a long paper about multiple bifurcations with

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multiple degeneracy in some features of the system and amulti-parameter in its definition. Multiple bifurcationsoccur in the mathematical descriptions of many naturalphenomena, and more importantly provide a means oforganizing the understanding of simple bifurcations and alsoprovide a powerful analytic tool for locating complicateddynamical behavior in some models. This paper is highlyrecommended, and one may consider this paper as an appendixto the book mentioned.

Although the phenomena of Hopf bifurcations depending onsome autonomous external parameters are well understood,nonetheless, the parametrically perturbed Hopf bifurcationshave not received enough attention. The effects of periodicperturbation of a bifurcating system have been considered byRosenblat and Cohen [1980, 1981] and Kath [1981],nonetheless, they neglected to examine any possiblesecondary bifurcations which may exist in these systems. SriNamachchivaya and Ariaratnam [1987] studied small periodicperturbations on two-dimensional systems exhibiting Hopfbifurcations in detail, and obtained explicit results forvarious primary and secondary bifurcations, and theirstabilities. Here the center manifold theorem and othertechniques are utilized.

In addition to the above suggested reading list, Ioossand Joseph [1980] and Ruelle [1989] are recommended.

Before we end this section, let us briefly discuss theunfolding of singularities and describe the elementarycatastrophes. Intuitively, unfoldings means that we embed asingularity of a map in a higher dimensional domain, so thatthe "bigger" map offers some insights and advantages.

Let f be a finite sum of products of two elements, eachof which from those germs f: R" - R for which f(O) = 0. So,f has a singularity at O. An unfolding of f is a germ f':R r - R, with f'(O) = 0, such that if x c R", f'(x,O) =f(x). Of course, here 0 = (0,...,0) with r entries. Theunfolding f' have r-parameters. Note that the constantunfolding f', defined by f(x,y) = f(x). We say that f' (with

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r parameters) is versal if any other unfolding of the germ fis induced from f'. A versal unfolding of a germ f isuniversal if the number of parameters is minimal.

Theorem 6.6.2 (a) A germ f of sum of products of twoelements has a versal (thus a universal) unfolding iff f hasfinite codimension; (b) If f' of r-parameter is a universalunfolding of f, then r = codim f. All universal unfoldingsare isomerphic; (c) The universal unfolding of a germ isstable (even if the germ is not).

The elementary theorem of Thom classifies singular germsof codim <- 4, and these are the elementary catastrophes.This result is stated in germs, that is, as a local theorem.

Theorem 6.6.3 (Thom's elementary catastrophes) Let f bea smooth germ (f(O) = 0 with 0 a singularity). Let 1 <- c 5 4

be the codim of f. Then f is 6-degree-determined. Up to signchange, and the addition of a non-degenerate quadratic form,f is (right) equivalent to one of the germs in the table.

Germ Codim Universal manifold Popular namex3 1 x3 + ux foldX4 2 X4 + ux2 + vx cuspx5 3 x5 + ux3 + vx2 + wx swallow-tailx3+y3 3 x3 + y3 + wxy - ux - vy hyperbolic umbilicx3-xy2 3 x3 - xy2 + w (x2 +y2) - ux - vy

elliptic umbilicx6 4 x6 + tx4 + ux3 + vx2 + wx butterflyx2 y+y4 4 x2 y + y4 + wx2 + ty2 - ux - vy

parabolic umbilic

There are several places where the details of thisclassification theorem are carried out. Brocker's lecturenotes [1975] are excellent. For more details includingcomments on higher dimensions, see for instance: Wasserman[1974]; Zeeman [1976]. Zeeman's article points out thatwhile germs such as x3 are not locally stable, theiruniversal unfoldings are. Furthermore, the global point ofview, as well as genericity for maps R"- R are also

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discussed.For those readers who may want to pursue the details and

specifics of this theory, one needs at least the followingbackground:(i) Some ring theory to get to the basic results of Matheret al on finite determination and codimension.(ii) The Malgrange preparation theorem which generalizes tothe smooth case a famous theorem of Weierstrass from severalcomplex variables. See for instance: Brocker [1975],Malgrange [1964]; a chapter in Golubitsky and Guillemin(1973).

(iii) Some basic algebraic geometry.Catastrophe theory has brought with it a wealth of

applications, however the risk of oversimplification inapplications are enormous. Prudent caution is required!There are several concrete examples of applications ofcatastrophe theory, such as: in optics [Berry and Upstill1980], relativity [Barrow 1981, 1982a,b], geophysics(Gilmore 1981], particle scattering from surface [Berry1975], rainbow effect in ion channeling in very thin crystal(Neskovic and Perovic 1987], elastic structure under deadand rigid loadings assess the stable regions of anequilibrium path which exhibits a succession of folds[Thompson 1979], just to name a few. Gilmore [1981] alsoprovides some other applications of catastrophe theory.

swallow tail

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Elliptic umbilic

Hyperbolic umbilicAs we have given some references for further reading on

the subject of bifurcation theory, we will not discuss themore well- known applications which can be found in mosttexts or references. Instead, in the next chapter we shalldiscuss some nonlinear dynamical systems in variousdisciplines. In the discussion, we shall utilize theconcepts and techniques we have discussed here and earlier,and we would also like to point out the common mathematicalstructures which transcend the boundaries of diversedisciplines.

6.7 ChaosRecently, chaos is a very fashionable word in the

natural sciences. Roughly speaking, chaos is defined as anirregular motion stemming from deterministic equations.Nonetheless, there are somewhat different definitions of the

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term chaos in the literature. The difference is mainly dueto different ways of defining "irregular motion". Irregularmotion of stochastic processes, such as Brownian motion,occurs due to random, i.e., unpredictable, causes orsources. Thus these random irregular motions areuncorrelated. We do not consider this type of irregularmotion as chaotic. Instead, one can discuss the atypicalbehavior of the correlation function of chaotic processes.

There are a number of introductory articles on chaos.The following are a partial list. Chernikov, Sagdeev andZaslavsky [1988], Chua and Madan [1988], Bak [1986]Crutchfield et al (1986], May [1976] and Gleick [1987]. Thefollowing review articles on chaos and routes to chaos arehighly recommended: Eckmann [1981], Ott [1981], and Tomita(1982] for nonlinear oscillators. There are also manyintroductory or popular books on chaos. Holden [1986] ishighly recommended. Prigogine and Stengers [1984] gives anontechnical account of order and chaos and raises somephilosophical issues. It stimulates and challenges numerousquestions and thoughts, nonetheless, due to its lack ofspecifics, it does not help the "advanced beginners" to findways or approaches for the solutions.

The idea of chaos has been applied to almost anysubject. In addition to the classical applications inhydrodynamics (for instance, even in the periodical laminarflows through curved pipes, under certain conditions theflows exhibit period- tripling, which is reminiscent of oneof the routes to chaos [Hamakiotes and Berger 1989]), plasmaphysics, classical mechanics, nonlinear feedback control,there are chaotic phenomena observed or predicted in optics,chemical and biochemical reactions, semiconductor physics,interacting population and delayed feedback, competitiveeconomy, just to name a few. We shall discuss theseapplications in next chapter. Here we just want to point outa few examples which we will not go into any further detail,but give references for interested readers to pursue thesubjects. For instance, Papantonopoulos, Uematsu and

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Yanagida [1987] presented a chaotic inflationary model ofthe universe, in which nonlinear interaction of dilaton andaxion fields, in the context of the super-conformal theory,can dynamically give rise to the initial conditions for theinflation of the universe. Buchler and Eichhorn [1987]discussed various chaotic phenomena in astrophysics. Also,Fesser, Bishop and Kumar [1983] have shown numerically thatthere are parameter ranges of radio frequency (rf)superconducting quantum interference device (SQUID) forchaotic behavior. They have shown that the strange attractorcharacterizing the chaotic regimes can be described by aone-dimensional return map. Recently, Herath and Fesser[1987], motivated by the rf-SQUID device and using differentmode expansions, investigated nonlinear single welloscillators driven by a periodic force with damping. Veryrecently, Miles [1988] investigated the symmetricoscillations of an inverted, lightly damped pendulum underdirect sinusoidal force, and analytically predictedsymmetry-breaking bifurcations and numerically confirmed.Similar results were also obtained for Josephson junctioncircuit by Yeh and Kao [1982], Kautz and Macfarlane (1986),Yao [1986] and Hadley and Beasley [1987].

Damped, driven pendulum systems have been used to modelcomplicated behavior in nonlinear systems successfully.Varghese and Thorp [1988] chose to study the transientlyforced pendulums and they shifted their emphasis to thecomposition of the boundaries separating the domains ofattraction of the various asymptotically stable fixedpoints. A simple proof of the existence of diffeomorphismsfrom connected basins to striated basins is also presented.

Since we are on the subject of effects ofsuperconducting material, it is interesting to note that aseries of papers on the nonlinear hysteretic forces due tosuperconducting materials have been published by the Cornellgroup [Moon, Yanoviak and Ware 1989, Moon, Weng and Chang1989]. Moon [1988] has demonstrated the period doubling andchaos for the forced vibration due to the nonlinear

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hysteretic force of a small permanent magnet near thesurface of a high temperature superconducting disk. Theforces are believed to be related to flux pinning and fluxdragging effects in the superconductor of type II state.Based on the displacement of the magnet, a return map underiteration exhibits a bifurcation structure similar to theexperimental results obtained.

In fact, much earlier, Huberman and Crutchfield [1979],Huberman, Crutchfield and Packard [1980] have shown that thenonlinear dynamics of anharmonically interacting particlesunder periodic fields resulted in a set of cascadingbifurcations and into chaos. Turschner [1982] has presentedan analytic calculation of the Poincare section of thedriven anharmonic oscillator based on proper canonicaltransformations. Linsay [1981] has demonstrated periodicdoubling and chaotic behavior of a driven anharmonicoscillator and the experimental results are in quantitativeagreement with the theory by Feigenbaum [1978, 1979]. Testa,Perez and Jeffries [1982] have also experimentally observedsuccessive subharmonic bifurcations, onset of chaos, andnoise band merging from a driven nonlinear semiconductoroscillator. See also, [Wiesenfeld, Knobloch, Miracky andClarke 1984]. Rollins and Hunt [1982] have shown that both afinite forward bias and a finite reverse recovery time arerequired if the diode resonator is to exhibit chaos.Furthermore, this anharmonic oscillator also exhibits periodtripling and quintupling. Nozaki and Bekki [1983] have shownthat a nonlinear Schrodinger soliton behaves stochasticallywith random phases in both time and space in the presence ofsmall external oscillating fields and emits small-amplitudeplane waves with random phases. They also have found thatthe statistical properties of random phases give the energyspectra of the soliton and plane waves. Bryant and Jeffries[1984] studied a forced symmetric oscillator containing asaturable inductor with magnetic hysteresis, approximated bya noninvertible map of the plane. The system displays a Hopfbifurcation to quasiperiodicity, entrainment horns, and

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chaos. Within an entrainment horn, they observed symmetrybreaking, period doubling, and complementary band merging.Very recently, Gunaratne et al [1989] have studied thechaotic dynamics from a nonlinear electronic cirsuit whichhas shown to exhibit the universal topological structure ofmaps on an annulus. They further suggested thatlow-dimensional strange attractors fall into a few classes,each characterized by distinct universal topologicalfeatures.

Recently, Holmes [1986] (using the averaging method) andMelnikov [1963] (using the perturbation technique) haveshown that an N-degree of freedom model of weakly nonlinearsurface waves due to Miles [1976] has transverse homoclinicorbits. And this implies that sets of chaotic orbits existin the phase space. Relevance of their results toexperimental work on parametrically excited surface waves[Ciliberto and Gollub 1985] are also briefly discussed.Funakoshi and Inoue [1987] have experimentally demonstratedthe chaotic behavior of the surface water wave, modeled byMiles [1976] with the assumption of weak nonlinearity andlinear damping, when a container is oscillated resonantly ina horizontal direction with appropriate amplitude andfrequency. The experimental work of Funakoshi and Inoue[1987] are directly relevant to Holmes' [1986] results, yetthey are not aware of Holmes' work.

Melnikov's method detects transverse homoclinic pointsin differential equations which are small perturbations ofintegrable systems. This together with the Smale-Birkhoffhomoclinic theorem [Smale 1967] implies the existence ofchaotic motions among the solutions of the equation withqualitative information. Recently, Brunsden and Holmes[1987] proposed a method which provides quantitativestatistical measures of solutions. They compute powerspectra of chaotic motions which are perturbations ofhomoclinic orbits and their approach relies on the existenceof global homoclinic structures, verified by Melnikovtheory, and derived from the notion of coherent structures

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in turbulence theory.Another example of the route to chaos via

period-doubling is a linear, viscously damped oscillatorwhich rebounds elastically whenever the displacement dropsto zero. It exhibits a family of subharmonic resonant peaksbetween which there are cascades of period-doublingbifurcations leading to chaotic regimes [Thompson andGhaffari 1982].

Yet another example of routes to chaos is the chaoticscattering. Bleher et al [1989] have shown that the onset ofchaotic behavior in a class of classical scattering problemscould occur in two possible ways. One is abrupt and isrelated to a change in the topology of the energy surface,while the other is a result of a complex sequence ofsaddle-node and period- doubling bifurcations. The former,the abrupt bifurcation represents a new generic route tochaos and yields a characteristic scaling of the fractaldimension associated with the scattering function as [ln(Ec- E)-']-', for particle energies E near the critical value Ecat which the scattering becomes chaotic.

It is known that steady planar propagation of acombustion front is unstable to disturbances correspondingto pulsating and spinning waves. Recently, Margolis andMatckowsky [1988] considered the nonlinear evolutionequations for the amplitudes of the pulsating and spinningwaves in a neighborhood of a double eigenvalue of thesystem, in particular, near a degenerate Hopf bifurcationpoint, and new quasi-periodic modes of combustion were alsodescribed.

1/f noise is found ubiquitously in various scientificand engineering disciplines. It has been speculated whether1/f noise can be explained as a chaotic phenomenon.Recently, Geisel, Zacherl and Radons [1987] proposed a newmechanism for 1/f noise as a generic phenomenon in thevelocity fluctuations of a particle in a 2-dim periodicpotential, and is closely related to the generic structureof phase space of nonintegrable Hamiltonian systems. On the

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other hand, Bak, Tang and Wiesenfeld [1987] have shown thatdynamical systems with spatial degrees of freedom naturallyevolve into a self-organized critical state and 1/f (orflicker) noise can be identified with the dynamics of thecritical state. They also related this to fractal objects.

Phase transitions of trapped particles and ions,initially observed in 1959 by collisional cooling and morerecently by laser cooling, can also be explained asorder-chaos transitions [Hoffnagle et al 1988].

In the following we shall briefly discuss one of thewell-known examples of deterministic chaos, the Lorenz modelof turbulence. For details, see Guckenheimer and Holmes[1983]. C. Sparrow [1982] gives an extensive treatment ofLorenz equations. There are several other well-knownexamples of deterministic chaos, such as Van der Pollsequations of damped nonlinear oscillator [see, e.g., Holmes1979, Holmes and Rand 1978], Duffing's equations ofnonlinear mechanical oscillator with a cubic stiffness[Novak and Frehlich 1982, Liu and Young 1986], we eitherhave touched upon earlier or we shall meet in next chapter.For a nice review, see, e.g., Holmes and Moon [1983]. Wewill not be able to get into any detail to discuss theirroutes to chaos. For details, once again the reader isreferred to Guckenheimer and Holmes [1983] and Moon [1987].Recently, Byatt-Smith [1987] studied the 2v period solutionof the forced-damp Duffing's equation with negativestiffness, with linear damping proportional to the velocity.It has very rich structures.

Many problems in physical systems involve the nonlinearinteraction of two oscillators with different frequencies.When these frequencies are incommensurate, the interactioninvolves the amplitude rather than the phases of eachoscillator. On the other hand, when the frequencies are in aratio closely corresponding to a rational fraction of asmall denominator, phase locking occurs and the dynamics ismuch richer, see, e.g., Perez and Glass [1982] andCoppersmith [1987]. Wiesenfeld and Satija [1987] studied the

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effect of noise on systems having two competing frequencies.When the system is mode-locked, random perturbations areeffectively suppressed, while outside the locking intervalrelatively high levels of broadband noise are presented.Such effects have also been observed in mode-locked lasers.We shall discuss this in the next chapter. Van Buskirk andJeffries [1985] have observed chaotic dynamics from thenonlinear charge storage of driven Si p-n junction passiveresonators. And the behavior is in good agreement withtheoretical models. Yazaki, Takashima and Mizutani [1987]investigated the Taconis oscillations, which are spontaneousoscillations of gas columns thermally induced in a tube withsteep temperature gradients. Near the overlapping region,the intersection of the stability curves for two differentmodes with incommensurate frequencies has been found thatboth modes can be excited simultaneously and competitionbetween them can lead to complex quasiperiodic and chaoticstates. These problems are of particular interest for sometechnologies, such as phase locking of several independentlasers of the same kind, called the phased array. We shallcome to this in the next chapter. Knobloch and Proctor[1988] have investigated fully the special case where thefrequencies are in the ratio of 2:1. On the subject ofresonance, Parlitz and Lauterborn [1985] have shownnumerically a periodic recurrence of a specific finestructure in the bifurcation set of the Duffing equationwhich is closely related with the nonlinear resonance of thesystem. Gray and Roberts [1988a,b,c,d], in a series ofpapers, re-examined chemical kinetic models described by twocoupled ordinary differential equations containing at mostthree control parameters, originally studied by Sal'nikov[1949]. They have found some interesting effects and detailswhich have missed earlier. Recently, Wiesenfeld and Hadley[1989] have described a novel feature of certain arrays of Ncoupled nonlinear oscillators. They have found that thenumber of stable limit cycles scales as (N-1)!. In order toaccommodate this very large multiplicity of attractors, the

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basins of attraction crowd more tightly in phase space withincreasing N. Their simulations have shown that for largeenough N, even minute levels of noise can cause the systemto hop freely among the many coexisting stable attractors.

Lorenz [1963] presented an analysis of a coupled set ofthree quadratic ordinary differential equations, one influid velocity and two in temperature, for fluid convectionin a two-dimensional layer heated from below to modelingatmospherical dynamics. The Lorenz equations are:

dx/dt = a(y - x),dy/dt = ax - y - xz, (6.7-1)

dz/dt = -pz + xy,where (x,y,z) c R3, a (the Prandtl number), a (the Rayleighnumber), and $ (an aspect ratio) are real positiveparameters. For any further detail of Lorenz equations, see,e.g., Sparrow [1982], Guckenheimer and Holmes [1983], Moon[1987]. Amazingly, equations completely equivalent to theset of Lorenz equations, Eq.(6.7-1), occur in laser physicsexplaining the phenomenon of irregularly spiking of lasers.For the discussion of Lorenz equations in laser physics, seefor instance, Haken [1983], Haken [1975a], Sparrow [1986].For a more comprehensive review of cooperative phenomena insystems far from thermal equilibrium including lasers,nonlinear wave interactions, tunnel diodes, chemicalreactions, and fluid dynamics, see Haken [1975b] as well ashis book on synergetics [1983].

It suffices to say that for a < 1, the origin (0,0,0) isglobally attracting, i.e., the fluid is at rest and withlinear temperature gradient (corresponding to no laseraction in lasre physics). This is because the trace of theJacobian (the divergence of the vector field) is equal to-(a + 1 + Q) < 0. In fact, for a < 1, the origin is ahyperbolic sink and is the only attractor. For a = 1, one ofthe eigenvalues of the linearized system is zero, and theothers, µ, = -R, µ2 = -(a + 1), thus a pitchfork bifurcationoccurs, that is, the pure conductive solution becomesunstable and the convective motion starts (corresponding to

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laser at threshold and lasing starts). As for a > 1, theorigin is a saddle point with a one-dimensional unstablemanifold, and there is a pair of nontrivial steady solutions(or fixed points S. and S_) at (x, y, z) = (±f3 (a-1) , ±13 (a-1) ,a-1), and these are sinks for 1 < a < ah = a(a + /3 + 3)/(a -$ - 1). At a = ah a Hopf bifurcation occurs at the fixedpoints, since the eigenvalues of the matrix of the linearsystem are:

g, = -(a + (3 + 1), and µz = ±i2a(a+l)/(a-$-1).To allow imaginary roots, here we assume that a > 1 + /3.Otherwise, µ2 are real. For a > ah, the nontrivial fixedpoints are saddle points with two-dimensional unstablemanifolds. Thus, for a > ah, all three fixed points aresaddles, i.e., unstable. Nonetheless, an attracting set doesexist and may contain complicated bounded solutions. One maythink that the Hopf bifurcation occurring as a passesthrough ah will give rise to stable periodic orbits, butsubsequently it has been found that the bifurcation issubcritical [Marsden and McCracken 1976], so that unstableperiodic orbits shrink down to the sinks as a increasestowards ah and no closed orbits exist near these fixedpoints for a > ah.

Thus, qualitatively we can anticipate that all thesolutions will have the following behavior in the phasespace. For all positive values of the parameters, allsolutions of the equations eventually lie in some boundedregion and they all tend towards some set in three-dimensional phase space with zero volume. This follows fromthe dissipative nature of the flow, and implies thatsolutions do not wander about the whole three-dimensionalspace but eventually come close to point-like, line-like, orsheet-like objects in the phase space.

Note also that the fixed points S. and S_ will be stablefor a < /3 + 1, but lose stability at some a if a > E3 + 1. Byusing an averaging procedure, one can show that there is astable symmetric periodic orbit or limit cycle for all largeenough a if 3a > 2/3 + 1. Note that the Lorenz equations are

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symmetric under the mapping (x,y,z) - (-x,-y,z). Also noticethat the region 3a > 2Q + 1 is larger than and includes theregion of a > /3 + 1. That is, whenever S. or S_ losestability at a finite a, there is a stable periodic orbit atlarge enough a. In the case when 3a > 20 + 1 but a < p + 1,it can be shown that there are two unstable periodic orbits,i.e., a symmetric pair of non-symmetric orbits which existfor all large enough a, in addition to the stable symmetricorbit. Various other regions and other limits of theequations can be analysed in different ways. For instance,the limit a ~ a - - and / z 1 has been analysed by Shimadaand Nagashima [1978], Fowler and McGuinness [1982, 1984].Again, the references cited earlier are urged to consultwith.

If it was not for the complicated but beautiful resultsobtained by numerically integrating the Lorenz equations ona computer, the Lorenz equations would not have receivedsuch interest and attention. Indeed, a computer is necessaryto proceed much beyond the qualitative description earlier.In the following we will give a few well-known figures justto entice the reader for further reading.

We will fix two parameters a and p, and let a vary. Itshould be noted that the values used by Lorenz and mostother researchers are a = 10, /3 = 8/3, and similar behavioroccurs for other values. With these parameters fixed, onefind ah z 24.74. Lorenz then fixed a = 28 and integratedEgs.(6.7-1) numerically. Fig.6.7.1 shows a two-dimensionalprojection of a typical orbit calculated with the above setof parameters. The transients have been allowed to die awaybefore plotting begins. The orbit appears to oscillate backand forth, rotating first on one side and then on the otherand never closing up! Fig.6.7.2 show numerically calculatedorbits at other parameter values. For the same a and Q, (a)

a = 60, resulted in a chaotic behavior; (b) a = 126.515resulted in a stable orbit; (c) a = 198 resulted in anasymmetric chaotic attractor; and (d) a = 350 resulted in atypical large-a stable symmetric periodic orbit.

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60

0

401

20 -

-20 0 20

X

Fig.6.7.1 [Sparrow 1986] This is a plot of x vs z for a =10, fi = 8/3, and a = 28.0.

(a)

(C)

(b)

(d)

Fig.6.7.2 [Sparrow 1986] Same as Fig.6.7.1 for a = 10, /3 =

8/3. (a) a = 60.0, chaotic behavior; (b) a = 126.515, astable orbit; (c) a = 198.0, an asymmetric chaoticattractor; (d) a = 350.0, a large a stable symmetricperiodic orbit.

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For any further details on the chaotic behavior of theLorenz equations as well as other well-known nonlinearsystems, see the references cited earlier. A prototypeequation to the Lorenz model of turbulence contains just onenonlinearity in one variable has been proposed by Rossler[1976]:

dx/dt = -(y + z), dy/dt = x + 0.2y,dz/dt = 0.2 + z(x - 5.7). (6.7-2)

There is only a single nonlinear term, and of course thereis no longer any immediate physical interpretation.Nonetheless, the flow in phase space allows for "folded"diffeomorphisms, called horseshoe maps [Smale 1967], arewell-known in the geometric theory of dynamical systems. Infact, each of them can give rise to a three-dimensionalsuspension, and the limit set is the strange attractor andwhose cross-section is a two-dimensional Cantor set. Theflow is nonperiodic and structurally stable [Ruelle andTakens 1971] even though all orbits are unstable. Thus, mostof the results of the Lorenz model turn out to be true forEgs.(6.7-2) also. The simplicity of Egs.(6.7-2) has theadded attribute that some other results one would like toobtain about the strange attractors, such as basinstructure, behavior through bifurcations, etc., can beobtained easier. Indeed, Rossler's results stimulated andhelped a better understanding of the Lorenz equations.Recently, Holmes and Williams [1985] constructed asuspension of Smale's horseshoe diffeomorphism of thetwo-dimensional disc as a flow in an orientable threedimensional manifold. Such a suspension is natural in thatit occurs frequently in periodically forced nonlinearoscillators such as the Duffing equation. From thesuspension, they constructed a know-holder or template insuch a way that the periodic orbits are isotopic to those inthe full three-dimentional flow. Theorems of existence,uniqueness and nonexistence for families of torus knots, andthese families to resonant Hamiltonian bifurcations whichoccur as horseshoes are created in a one-parameter family of

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area preserving maps.Saravanan et al [1985] have studied the effect of

modulating the control parameter a with a frequency n, i.e.,in the second equation of Egs.(6.7-1), replace a by a[l +ecos(nt)]. They have found that this modulation is a verysensitive probe of the limit cycle of frequency no near theHopf bifurcation point. For n = 2no, and low amplitudes,such modulation leads to a hastening of chaos. But forhigher amplitudes, the destabilization ceases and beyond acritical amplitude the limit cycle of frequency no isstabilized. The existence of such a critical amplitudefollows from a variant of the perturbation theory. Theynoted that the possibility of stabilizing the limit cyclesis particularly interesting as it restores thecorrespondence with the real hydrodynamics up to the onsetof chaos. The routes to chaos in this modulated systems areof interest for further study.

Shimizu and Morioka [1978] have shown that by changingvariables, Lorenz equations can be written as a differentialsystem describing a particle in some fourth order potentialdepending on a varying parameter. Coullet, Tresser andArneodo [1979] have shown that such simple differentialsystems lead to the transition to stochasticity. Andrade[1982] used the Carleman embedding to obtain some resultsfor the Liapunov exponents of the Lorenz model. Birman andWilliams [1983a,b] in a series of papers have asked thequestions such as can the periodic orbits of a dynamicalsystem on S3 or R3 be knotted, and if so, what kinds ofknots can occur, and what are the implications? Franks andWilliams [1984] have further shown that with positivetopological entropy, there are infinitely many distinct knottypes. Results are obtained for the Lorenz model. Williams[1983] proved that Lorenz knots are prime. Holmes andWilliams [1985] have constructed an suspension of Smale'shorseshoe diffeomorphism (we have not discussed this topicdue to its complicated construction, we refer the reader toSmale [1967], Irwin [1980], Guckenheimer and Holmes [1983])

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of a two-dimensional disk as a flow in an orientable threedimensional manifold. Such a suspension occurs frequently inperiodically forced nonlinear oscillators, such as theDuffing equation. From this suspension they constructed atemplate, which is a branched two-manifold with a semiflow,in such a way that the periodic orbits are isotopic to thosein the full three-dimensional flow. Some existence,uniqueness, and nonexistence theorems for families of torusknots are obtained. They then were able to connect acountable subsequence of one-dimensional bifurcations with asubsequence of area-preserving bifurcations in a twoparameter family of suspensions by using knot theory,kneading theory and Hamiltonian bifurcation theory. Themoral of the results is that there are no universal routesto chaos! For more recent work on the classification ofknotted periodic orbits in periodically forced nonlinearoscillators, as well as a review of earlier work, see Holmes[1988].

Recently, Agarwal, Banerjee and Bhattacharjee [1986]have shown that at the threshold of period-doubling chaos ina dynamical system such as the Lorenz system, the fractaldimension of the associated strange attractor assumes auniversal value. Dekker [1986] using computer simulationdemonstrated that the tunnelling orbits under a symmetricdouble-well potential can be assigned a fractal dimension.

The three-dimensional Lorenz flow is approximated by atwo-dimensional flow with a branch curve with the use of theapproximation of the Lorenz attractor by invariant two-dimensional manifolds [Dorfle and Graham 1983], and theprobability density generated by the flow on the invariantmanifolds in the steady state is obtained. It is also shownthat the probability density arises in the Lorenz modelsubject to stochastic forces as a self-consistentapproximation for very small but finite noise.

Recently, Lahiri and Nag [1989] have consideredintermittency in inverted-pitchfork bifurcation of a 1Ddissipative map and of a 2D conservative map. They have also

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considered the effect of noise and the scaling laws in thepresence of noise. Furthermore, they have obtained theresults for the saddle-node bifurcation in 2D area-preserving maps.

For a broader view of chaos, with its implications, see,for instance, Holden [1986], and Schuster (1984]. For a morecurrent review of chaotic phenomena in nonlinear systems,see Hao (1988]. For chaotic phenomana in astrophysics, seefor instance Buchler and Eichhorn [1987], Contopoulos[1985]. For nonlinear dynamical phenomena in chemicalsystems, see for instance, Vidal and Pacault [1984]. Forfluid or plasma phenomena, see for instance, Marsden (1984],Sagdeev [1984], Rand and Young [1981]. For thoes readers whoare interested in computer graphic presentations andnumerical integration of nonlinear dynamical systems, thereare several PC compatible software available for suchpurpose.

Although, we have mainly concentrated on nonlineardynamical systems represented by ordinary differentialequations, similar techniques can also be applied tononlinear dynamical systems represented by partialdifferential equations. Birnir and Morrison [1987] havediscussed structural stability and chaotic solutions ofperturbed Benjamin-Ono equations for characterizing theturbulent motion. Bishop et al [1983, 1988] have discussedchaotic solutions to other perturbed nonlinear partialdifferential systems such as Sine-Gordon, and nonlinearSchrodinger equations. Olsen and Samuelsen [1987] haveexamined the one-dimensional sine-Gordon soliton equation inthe presence of driving and damping numerically. They havefound that by increasing the driving strength, the systemexhibits an infinite sequence of period doubling leading tochaos. They noticed that the first bifurcation occured aftera narrow regime characterized by intermittency-type of chaosand quasi-periodic oscillations. They further attributed theorigin of the intermittency-type of chaos by the competitionbetween two spatial patterns described by the presence of

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one or two breather-like modes respectively. Brandstater etal [1983] have presented evidence for low dimensionalstrange attractors in Couette-Taylor flow data. Doering etal [1987] have obtained an exact analytic computation of theLiapunov dimension of the universal attractor of the complexGinzburg-Landau equation for a finite range of parametervalues. Meanwhile, Sirovich and Rodriguez [1987] have usedthe Ginzburg-Landau equation to extract a complete set ofuncorrelated coherent structures, which are then used as abasis for the dynamical description of coherent structuresin the attractor set. Chate and Manneville [1987] discussthe transition to turbulence via spatiotemporalintermittency observed in a partial differential equationdisplays statistical features. The transition to chaosthrough intermittency has recently been observed in asteady-state plasma [Cheung, donovan and Wong 1988].

Even the traditional problems, such as the Navier-Stokesequations for an incompressible fluid on a two-dimensionaltorus, are rich in phenomenology. For instance, Franceschini[1983a,b] has numerically investigated two truncations ofthe above problem.

Over a decade of active research, a number ofexperiments have concluded that the production of complex,unpredictable turbulence for low-dimensional systems is dueto few-dimensional attractors. Indeed, based on these andother observations and analysis, the "conventional wisdom"has concluded that the Newhouse-Ruelle-Takens route offew-dimensional chaos [Ruelle and Takens 1971, Newhouse,Ruelle and Takens 1978] is the proper explanation of thenature of turbulence. Nonetheless, there are severalalternative explanations of turbulence which do not employthe attractive hypothesis, namely, the spin-glass relaxation[Walden and Ahlers 1980, Crutchfield 1984], spatial noiseamplification [Deissler and Kaneko 1987], and transients[Crutchfield and Kaneko 1988]. Let us briefly describe threemost common scenarios for turbulence. By the way, here chaosand weak turbulence are interchangeable.

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The Newhouse-Ruelle-Takens (NRT) scenario is the oldestone in describing the route to turbulence. This can bestated in the following theorem [Newhouse, Ruelle and Takens1978]:

Theorem 6.7.1 Let x be a constant vector field on then-dim torus T" = R"/Z". If n >_ 3, every C2 neighborhood of xcontains a vector field x' with a strange Axiom A attractor.If n >_ 4, we may take C° for C2 .

Under all these assumptions, the NRT scenario assertsthat a strange attractor is likely to occur in the followingsense. In the space of all differential equations, someequations have strange attractors, and others have none.Those which do have strange attractors form a set containinga subset which is open in the C2 topology. The closure ofthis open set contains the constant vector fields on thetorus T3. The measurable consequences of the presence ofstrange attractor following the NRT scenario is thefollowing: If a system starting from a stationary statesolution undergoes three Hopf bifurcations when a parameteris varied, then it is very likely that the system possessesa strange attractor with sensitivity to initial conditionsafter the third bifurcation. And the power spectrum of sucha system will exhibit first one, then two, and possiblythree independent basic frequencies. If there is a strangeattractor, when the third frequency is about to appear somebroad-band noise will simultaneously also appear. When thesehappens, we consider the system chaotic or turbulent. Thenit is natural to ask whether or not three-frequencyquasiperiodic orbits are to be expected in typical nonlineardynamical systems? This is the question raised by Grebogi,Ott and Yorke (1983). Nonetheless, at that time the answerwas positive but incomplete.

Another scenario leading to turbulence is the Feigenbaumscenario. These type of systems usually can be reduced toPoincare return maps or area-preserving maps, and thebifurcations of the orbit structure are pitchforkbifurcations, i.e., a stable fixed point loses its

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stability and gives rise to a stable periodic orbit as aparameter is changed. This corresponds to a crossing of oneeigenvalue of the tangent map through -1. In an experiment,if one observes subharmonic bifurcations at µl, µ2, thenaccording to the scenario, it is very probable for a further

bifurcation to occur near µ3 = µ2 - (Al - µ2)/61 where 6 =4.66920... is universal [Feigenbaum 1978, 1979, 1980; Colletand Eckmann 1980, Greene, MacKay, Vivaldi and Feigenbaum1981]. Some other universal fine structures in periodicdoubling systems have also been discussed by Geisel andNierwetberg [1981, 1982]. Furthermore, if one has seen threebifurcations, a fourth one becomes more probable then thethird one after the first two, etc. And at the accumulationpoint, one will observe aperiodic behavior, but withoutbroad-band spectrum. This scenario is extremely well testedboth numerically and experimentally. The periodic doublingshave been observed in most low dimensional dynamicalsystems. Giglio, Musazzi and Perini [1981] presentedexperimental results of a Rayleigh-Benard cell. Byappropriate preparation of the initial state, the system canbe brought into a single frequency oscillatory regime. Whena further increase of the temperature gradient makes thesystem undergo a reproducible sequence of period-doublingbifurcations, the Feigenbaum universal numbers are alsodetermined. Gonzalez and Piro [1983] considered a nonlinearoscillator externally driven by an impulsive periodic force,and an exact analytical expression for the Poincare map forall values of parameters is obtained. This model alsodisplays period-doubling sequences of chaotic behavior andthe convergence rate of these cascades is in good agreementwith the Feigenbaum theory.

The third scenario is the Pomeau-Manneville scenario[Pomeau and Manneville 1980], which is also termedtransition to turbulence through intermittency. Themathematical status of this scenario is less thansatisfactory. Nonetheless, this scenario is associated witha saddle-node bifurcation, i.e., the amalgamation of a

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stable and an unstable fixed point. One can state thisscenario for general dynamical systems as follows. Assume aone- parameter family of dynamical systems has Poincare mapsclose to a one-parameter family of maps of the interval, andthat these maps have a stable and unstable fixed point whichamalgamate as the parameter is varied. As the parameter isvaried further to µ = 1.75 from the critical parameter valueµ,, one will see intermittently turbulent behavior of randomduration. The difficulty with this scenario is that it doesnot have any clear- cut or visible precursors like the othertwo scenarios. Recently, Keeler and Farmer [1987]investigated a one dimensional lattice of coupled quadraticmaps. They found that the motion of the spatial domain wallscauses spatially localized changes from chaotic to almostperiodic behavior and the almost periodic phases haveeigenvalues very close to one and with a 1/f low frequencyspectrum. This behavior has some aspects of Pomeau-Manneville intermittency, but is quite robust under changesof parameters.

We would like to emphasize that since a given dynamicalsystem may have many attractors, several scenarios mayevolve concurrently in different regions of phase space.Therefore, it is natural if several scenarios occur in agiven system depending on how the initial state of thesystem is prepared. Furthermore, the relevant parameterranges may overlap, thus although the basins of attractionfor different scenarios must be disjoint, they maybeinterlaced. Recently, Grebogi, Kostelich, Ott and Yorke[1986] using two examples to show that basin boundarydimensions can be different in different regions of phasespace. They can be fractal or nonfractal depending on theregion. In addition, they have shown that these regions ofdifferent dimension can be intertwined on an arbitrarilyfine scale. They further conjectured that a basin boundarytypically can have at most a finite number of possibledimensions. Indeed, a series of papers by Gollub andcoworkers using laser-Doppler methods have identified four

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distinct sequences of instabilities leading to turbulence atlow Prandtl number (between 2.5 to 5.0) in fluid layers ofsmall horizontal extent [Gollub and Benson 1980, Gollub andMcCarriar 1982, Gollub, McCarriar and Steinman 1982].McLaughlin and Martin [1975] proposed a mathematical themeof the transition to turbulence in statically stressed fluidsystems. Systems are classified according to Hopfbifurcation theorem. They have found certain kind of flowsobey the Boussinesq conditions exhibit hysteresis,finite-amplitude instabilities, and immediate transition toturbulence. They have also found another kind of flow, suchas a model of fluid convection with a low-Prandtl number, inwhich as the stress increases, a time-periodic regimeprecedes turbulence. Nonetheless, the transition tononperiodic behavior in this model is found to proceed inaccordance with the NRT scenario.

Although we have not discussed stochastic influence onthe nonlinear dynamical systems, there are many interestingresults in this area. For instance, Horsthemke andMalek-Mansour [1976], using the method of Ito [1944, 1951]stochastic differential equations, have shown that in thevicinity of the bifurcation point the external noise, eventhey are characterized by a small variance, can influenceprofoundly the macroscopic behavior of the system and givesrise to new phenomena not predicted by the deterministicanalysis. In Ruelle [1985], the ergodic theory ofdifferentiable dynamical systems is reviewed, and it isapplied to the Navier-Stokes equation. He also obtains upperbounds on characteristic exponents, entropy, and Haussdorffdimension of attracting sets. Recently, Machacek [1986]presented a general method for calculating the moments ofinvariant measure of multidimensional dissipative dynamicalsystems with noise. In particular, moments of the Lorenzmodel are calculated. Nicolis and Nicolis [1986] have castedthe Lorenz equations in the form of a single stochasticdifferential equation where a deterministic partrepresenting a bistable dynamical system is forced by a

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noise process. An analytically derived fluctuation-dissipation-like relationship linking the variance of thenoise to the system's parameters provides a satisfactoryexplanation of the numerical results. This leads the authorsto suggest that the emergence of chaos in the Lorenz modelis associated with the breakdown of the time scaleseparation between different variables. This gives somecredence to the frequent assertion in the literature thatthe elimination of fast variables generates noise in thesubset of the slow variables.

Horsthemke and Lefever (1977] have demonstrated on achemical dynamics model that even though the system is abovethe critical point, phase transitions can still be inducedsolely by the effect of external noise. Jeffries andWiesenfeld [1985] have measured the power spectra of aperiodically driven p-n junction in the vicinity of adynamical instability. They have found that the addition ofexternal noise introduces new lines in the spectra, whichbecome more prominent as a bifurcation (either perioddoubling or Hopf bifurcation) is approached. Furthermore,they have found that the scaling of the peak, width, areaand lineshape of these lines are in excellent agreement withthe predictions [Wiesenfeld 1985]. Experimentally, the onsetof chaos in dynamical systems can often be analyzed in termsof a discrete-time map with a quadratic extremum [Collet andEckmann 1980]. Kapral and Mandel (1985] investigated thebifurcation structure of a nonautonomous quadratic map. Theyfound that alghough nontrivial fixed points do not exist forsuch system, a bifurcation diagram can be constructedprovided that the sweep rate is not too large. Morris andMoss [1986] constructed an electronic circuit model of anonautonomous quadratic map, different from the one byKapral and Mandel [1985]. They observed that bifurcationpoints are postponed by amounts which depend on both thesweep velocity and the order of the bifurcation. Themeasured results obey two scaling laws predicted by Kapraland Mandel [1985] and provide evidence for the universality

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of these scaling laws.Newell, Rand and Russell [1988] recently suggested that

the transport properties and dissipation rates of a wideclass of turbulent flows are determined by the randomoccurrence of coherent events that correspond to certainorbits in the non- compact phase space which are attractedto special orbits which connect saddle points in the finiteregion of phase space to infinity and represent coherentstructures in the flow field.

Recently, Chernikov et al [1987] have proposed thatchaotic web may be able to explain the origin of ultra highenergy cosmic rays. Eckmann and Ruelle [1985] gave aninteresting review of the mathematical ideas and theirconcrete implementation in analyzing experiments in ergodictheory of dynamical systems.

Recently, Ruelle [1987] suggested the study of timeevolutions with adiabatically fluctuating parameters, i.e.,evolutions of the form xt+1 = f(x,g(t)) for discrete time.Similar to the nonlinearly coupled oscillators, Yuan, Tung,Feng and Narducci [1983] analyzed some general features ofcoupled logistic equations. They have found that forselected values of the three control parameters, chaoticbehavior may emerge out of an infinite sequence ofperiod-doubling bifurcations. There also exist large regionsof control parameter space, where two characteristicfrequencies exist, and the approach to chaos follows the NRTscenario. Like the coupled oscillators, as the ratio ofthese frequencies is varied, phase locking and quasiperiodicorbits are observed enroute to chaos. Recently,Klosek-Dygas, Matkowsky and Schuss [1988] have consideredthe stochastic stability of a nonlinear oscillatorparametrically excited by a stationary Markov process.Harikrishnan and Nandakumaran [1988] recently analyzednumerically the bifurcation structure of a two-dimensionalnoninvertible map of the form suggested by Ruelle and theyshowed that different periodic cycles are arranged in it inthe same order as in the logistic map. Indeed, this map

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satisfies the general criteria for the existence ofSarkovskii [1964] ordering, which is simply the order inwhich different period cycles are arranged along theparameter axis. On the other hand, the Henon [1976] map canbe considered as the two-dimensional analogue of thequadratic map, yet the Sarkovskii theorem does not hold[Devancy 1986].

One may also expect that external noise can influencethe nature of chaotic systems. Surprisingly, the theorem dueto Kifer [1974] states that for a dynamical system with anAxiom A attractor, the system is insensitive to smallexternal noise. This is experimentally demonstrated forRayleigh-Benard system by Gollub and Steinman [1980]. Ruelle[1986a,b] expanded to consider the analytic properties ofthe power spectrum for Axiom A systems near resonance.Recently, Ciliberto and Rubio [1987] studied experimentallythe spatial patterns in temporal chaotic regimes ofRayleigh-Benard convection.

Recently, Cumming and Linsay [1987], using a simpleoperational-amplifier relaxation oscillator driven by a sinewave which can be varied in frequency and amplitude, havepresented experimental evidence for deviations fromuniversality in the transition to chaos fromquasiperiodicity in a nonlinear dynamical system. In fact,they have shown that the power spectrum, tongue convergencerate, and spectrum of critical exponents all differ from thetheory. Meanwhile, in a different domain, Cvitanovic [1985]argued that period doubling and mode lockings for circlemaps are characterized by universal scaling and can bemeasured in a variety of nonlinear systems. Indeed, suchphenomena not only have been observed in physical phenomena,but also have been observed in biological and physiologicalsituations. For instance, the spontaneous rhythmic activityof aggregates of embryonic chick heart cells was perturbedby the injection of single current pulses and periodictrains of current pulses. The regular and irregulardynamical behavior produced by periodic stimulation, as

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period doubling bifurcations, were predicted theoretically

and observed experimentally by Guevara, Glass and Shrier[1981].

The above paragraphs show that even though we have someunderstanding of low dimensional (one- or two-dimensional)chaos, but there is still much more we do not understand andalso with lots of confusion, to say the least!

In the past several years, a great deal of interest hasbeen focused on the temporal evolution and behavior ofchaos, which arises in spatially constrained macroscopicsystems while some control parameters are varied. Recently,it has become increasingly more popular to study the spatialpattern formation and the transition to turbulence inextended systems [Thyagaraja 1979, Wesfreid and Zaleski1984, Bishop, Campbell and Channell 1984], and bifurcationsin systems with symmetries [Stewart 1988 and referencescited therein, Sattinger 1983], and nonlinear modecompetition between instability waves in the forced freeshear layer [Treiber and Kitney 1988]. Recently, Pismen[1987] has investigated the strong influence of periodicspatial forces on the selection of stationary patterns neara symmetry-breaking bifurcation. Using simple modelequations of long-scale thermal convection, he demonstrated:(i) the transition between alternative patterns, (ii)emergence of spatially quasiperiodic patterns, and (iii)evolution of patterns due to rotating phases. The presenceof symmetry usually can simplify the analysis, yet symmetrycan also lead to more complicated dynamical behavior. Swiftand Wiesenfeld [1984] have examined the role of symmetry insystems displaying period-doubling bifurcations. They havefound that symmetric orbits usually will not undergoperiod-doubling, and those exceptional cases cannot occur ina large class of systems, including the sinusoidally drivendamped oscillators, and the Lorenz model. Experimentally,conservation laws are manifested through the observation ofrestrictions on transport of chaotic orbits in phase space.Recently, Skiff et al [1988] have presented experiments

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which demonstrated the conservation of certain integrals ofmotion during Hamiltonian chaos. Recently, Coullet, Elphickand Repaux [1987] presented two basic mechanisms leading tospatial complexity in one-dimensional patterns, which arerelated to Melnikov's theory of periodically driven one-degree-of-freedom Hamiltonian systems, and Shilnikov'stheory of two-degrees-of-freedom conservative systems [see,for instance, Guckenheimer and Holmes 1984]. It isinteresting to note that Coullet and Elphick (1987) haveused Melnikov's analysis to construct recurrence time mapsnear homoclinic and heteroclinic bifurcations, and they haveshown that an elegent method developed by Kawasaki and Ohta[1983] to study defect dynamics is equivalent to Melnikov'stheory. Recently, Newton [1988] considered the spatiallychaotic behavior of separable solutions of the perturbedcubic Schrodinger equation in certain limits. Unfortunately,the limits under consideration were not applicable tononlinear optics, otherwise, its applications would be muchbroader.

On the subject of Hamiltonian systems, recently Moser[1986] gave a very interesting and broad review of recentdevelopments in the theory of Hamiltonian systems. Althoughmany real world problems are dissipative, nonetheless, as wehave pointed out in Ch. 2, Hamiltonian systems can give ussome global information on the structure of the solutions.Hopefully, we can treat the dissipative or dispersivesystems as perturbations of Hamiltonian systems and use KAMtheory to determine the characteristics of the solutions.

On the subject of KAM theory, it is known that the KAMtori survive small but finite perturbation and are expectedto break at a critical value of the parameter which dependsupon the frequency. Recently, Farmer and Satija [1985] andUmberger, Farmer and Satija [1986] studied the breakup of atwo-dimensional torus described by a critical circle map forarbitrary winding numbers. They have demonstrated that sucha breakdown can be described by a chaotic renormalizationgroup where the chaotic renormalization orbits converge on a

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strange attractor of low dimension. And they have found thatthe universal exponents characterizing this transition areglobal quantities which quantify the strange attractor.Recently, MacKay and van Zeijts [1988] have found universalscaling behavior for the period- doubling tree intwo-parameter families of bimodal maps of the interval. Arenormalization group explanation is given in terms of ahorseshoe with a Cantor set of two-dimensional unstablemanifolds instead of the usual fixed point with one unstabledirection. Satija [1987] recently has found that similarresults are also valid in Hamiltonian systems, i.e., thebreakup of an arbitrary KAM orbit can be described by auniversal strange attractor, and the global criticalexponents can characterize the breakup of almost all KAMtori.

Numerically, Sanz-Serna and Vadillo [1987] consideredthe leap-frog (explicit mid-point) discretization ofHamiltonian systems, and it is proved that the discreteevolution preserves the symplectic structure of the phasespace. Under suitable restrictions of the time step, thetechnique is applied to KAM theory to guarantee theboundedness of the computed points.

Dimension is one of the basic properties of anattractor. Farmer, Ott and Yorke [1983] discussed andreviewed different definitions of dimension, and computedtheir values for typical examples. They have found thatthere are two general types of definitions of dimension,those that depend only on metric properties and those thatdepend on the frequency with which a typical orbit visitsdifferent regions of the attractor. They have also foundthat the dimension depending on frequency is typically equalto the Liapunov dimension, which is defined by Liapunovnumbers and easier to calculate. An algorithm whichcalculates attractor dimensions by the correlation integralof the experimental time series was proposed by Grassbergerand Procaccia [1983]. Nonetheless, due to nonstationarity ofthe system, consistent results may require using a small

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data set and covering short time intervals during which thesystem can be approximately considered stationary. Undersuch circumstances, several sources of error in calculatingdimensions may result. Recently, Smith [1988] has computedintrinsic limits on the minimum size of a data set requiredfor calculation of dimensions. A lower bound on the numberof points required for a reliable estimation of thecorrelation exponent is given in terms of the dimension ofthe object and the desired accuracy. A method of estimatingthe correlation integral computed from a finite sample of awhite noise is also obtained. Havstad and Ehlers [1989] haveproposed a method to resolve such problems. Eckmann, Ruelleand Ciliberto [1986] discussed in detail an algorithm forcomputing Liapunov exponents from an experimental timeseries, and a hydrodynamic experiment is investigated as anexample. Recently, Ellner [1988] has come up with amaximum-likelihood method for estimating an attractor'sgeneralized dimensions from time-series data. Meanwhile,Braun et al [1987], Ohe and Tanaka [1988] have investigatedthe ionization instability of weakly ionized positivecolumns of helium glow discharges, and the transition to theturbulent state is experimentally observed. By calculatingthe correlation integral, they have determined that theperiodic instability has low dimensionality while turbulentinstability has a higher, but finite, dimensionality.Nonetheless, they have not determined the dimensions ofstrange attractors in their system by computing the Liapunovexponents for the experimental time series. This can be aninteresting problem to relate the dimensions obtained viatwo approaches. Earlier, Farmer [1982] studied the chaoticattractors of a delay differential equation and he has foundthat the dimension of several attractors computed directlyfrom the definition of dimensions agrees to withinexperimental resolution of the dimension computed from thespectrum of Liapunov exponents according to a conjecture ofKaplan and Yorke [1979]. Farmer and Sidorowich [1987] havepresented a technique which allows us to make short-term

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predictions of the future behavior of a chaotic time seriesusing information from the past. Recently, Tsang [1986]proposed an analytical method to determine thedimensionality of strange attractors in two-dimensionaldissipative maps. In this method, the geometric structuresof an attractor are obtained from a procedure developedpreviously. From the geometric structures, the Hausdorffdimension for the Cantor set is determined first, then forthe attractor. The results have compared well with numericalresults. Recently Auerbach et al (1987) were able toapproximate the fractal invariant measure of chaotic strangeattractors by the set of unstable n-periodic orbits ofincreasing n, and they also presented algorithms forextracting the periodic orbits from a chaotic time seriesand for calculating their stabilities. The topologicalentropy and the Hausdorff dimension can also be calculated.For nonlinear and nonstationary time series analysis, seePriestley [1988].

On the other hand, it is important to realize that thereare other types of invariant sets in dynamical systems thatare not attracting, and these non-attracting invariant setsalso play a fundamental role in the understanding ofdynamics. For instance, these non-attracting invariant setsoccur as chaotic transient sets [Grebogi, Ott and Yorke1983, Kantz and Grassberger 1985, Szepfalusy and Tel 1986],fractal basin boundaries (McDonald, Grebogi, Ott and Yorke1985), and adiabatic invariants (Brown, Ott and Grebogi1987]. In many cases, these invariant sets have complicated,Cantor set-like geometric structures and almost all thepoints in the invariant sets are saddle points. Hsu, Ott andYorke [1988] called such non-attracting sets strangesaddles. They discussed and numerically tested formularelating the dimensions of strange saddles and their stableand unstable manifolds to the Liapunov exponents and foundto be consistent with the conjecture formulated by Kaplanand Yorke [1979].

Recently, Lloyd and Lynch [1988] have considered the

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maximum number of limit cycles for a dynamical system ofLienard type: dx/dt = y - F(x), dy/dt = - g(x), with F andg are polynomials. They have found that for several classesof such systems, the maximum number of limit cycles that canbifurcate out of a critical point under perturbation of thecoefficients in F and g can be represented in terms of thedegree of F and g.

In their classic paper Li and Yorke [1975] posed aquestion that for some nice class of functions whether theexistence of an asymptotically stable periodic point impliesthat almost every point is asymptotically periodic.Recently, Nusse [1987] established the following twotheorems which affirm the question posed by Li and Yorke inpart.

Theorem 6.7.2 Let f be a mapping of C'`Q for somepositive real number a, from a nontrivial interval X intoitself. Assume that f satisfies the following conditions:(i) The set of asymptotically stable periodic points for fis compact (if this set is empty, then there exists at leastone absorbing boundary point of X for f). (ii) The set ofpoints, whose orbits do not converge to an asymptoticallystable periodic orbit of f or converge to an absorbingboundary point of X for f, is a nonempty compact set, and fis an expanding map on this set. Then we have: (a) The setof points whose orbits do not converge to an asymptoticallystable periodic orbit of f or to an absorbing boundary pointof X for f, has Lebesgue measure zero. (b) There exists apositive integer p such that almost every point in X isasymptotically stable periodic with period p, provided thatf(X) is bounded.

As a consequence, the set of aperiodic points for f, orequivalently, the set on which the dynamical behavior of fis chaotic, has Lebesgue measure zero. Furthermore, one canshow that the conditions in Theorem 6.7.2 are invariantunder the conjugation with a diffeomorphism. Nonetheless,since condition (i) and part of (ii) can not be ascertaineda priori, they are not very useful for practical purposes.

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Theorem 6.7.3 Assume that f is a chaotic C3-mappingfrom a nontrivial interval X into itself satisfying thefollowing conditions:(i) f has a nonpositive Schwarzian derivative, i.e.,(d3 f(x)/dx3)/(df(x)/dx) - (3/2)[(d 2 f(x)/dx2)/(df(x)/dx) ] 2 <_

0 for all x e X with df(x)/dx + 0;(ii) The set of points, whose orbits do not converge to anabsorbing boundary point(s) of X for f, is a nonemptycompact set;(iii) The orbit of each critical point for f converges to anasymptotically stable periodic orbit of f or to an absorbingboundary point(s) of X for f;(iv) The fixed points of f2 are isolated. Then we have:(a) The set of points whose orbits do not converge to anasymptotically stable periodic orbit of f or to an absorbingboundary point(s) of X for f, has Lebesgue measure zero;(b) There exists a positive integer p such that almost everypoint in X is asymptotically periodic with period p,provided that f(X) is bounded.

In the following, we will give some simple examples:(A) X = [-1,1], f: X - X is defined by f(x) = 3.701x3 -2.701x. It can be shown that f has two asymptotically stableperiodic orbits with period three. Since f has a negativeSchwarzian derivative and df/dx(x=l) = df/dx(x=-l) > 1, byTheorem 6.7.3 we have that almost every point in X isasymptotically periodic with period three.(B) Let f be a chaotic map of C3 from a compact interval[a,b] into itself with the following properties [Collet andEckmann 1980]: (i) f has one critical point c which isnondegenerate, f is strictly increasing on [a,c] andstrictly decreasing on [c,b]; (ii) f has negative Schwarzianderivative; (iii) the orbit of c converges to anasymptotically stable periodic orbit of f with smallestperiod p, for some positive integer p. But since theexistence of an asymptotically stable fixed point in[a,f2(c)) has not been excluded, thus the following casescan occur: (1) f has an asymptotically stable fixed point in

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[a,f2(c)) and f has an asymptotically stable periodic orbitwhich contains the critical point in its direct domain ofattraction. (2) f has an asymptotically stable fixed pointin [a,f2(c)) and the orbit of the critical point convergesto this stable fixed point. Moreover f has 2" unstableperiodic points with period n for each positive integer n.(3) f has no asymptotically stable fixed point in (a,f2(c));consequently, the critical point is in the direct domain ofattraction of the asymptotically stable periodic orbit.Since the map f satisfies the conditions of Theorem 6.7.3and we have that almost each point in the interval [a,b] isasymptotically periodic with period p.

For one-dimensional maps x,1 = mF(xn), there are fourkinds of bifurcation likely to take place in general,namely: (a) regular period doubling, (b) regular periodhalving, (c) reversed period doubling, and (d) reversedperiod halving. It is known that if the Schwarzianderivative is negative at the bifurcation point, then either(a) or (b) can take place, (if the Schewarzian derivative ispositive, then either (c) or (d) will take place) [Singer1978, Guckenheimer and Holmes 1983, Whitley 1983]. Thus,having a negative Schwarzian derivative is a condition whichimplies that at the parameter value where a period doublingbifurcation occurs, the periodic orbit is stable due tononlinear terms and prevent (c) and (d) to occur. There werequestions raised as to whether or not having a negativeSchwarzian derivative ruled out regular period halving.Recently, Nusse and Yorke [1988] presented an example of aone-dimensional map where F(x) is unimodal and has anegative Schwarzian derivative. They showed that for theirexample, some regular period halving bifurcations do occur,and the topological entropy can decrease as the parameter mis increased.

So far we have been discussing bifurcations of nonlinearcontinuous or discrete systems. George [1986] hasdemonstrated that bifurcations do occur in a piecewiselinear system.

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Before we end this section on chaos, we would like topoint out that it is fairly simple to demonstratedeterministic chaos experimentally. Briggs [1987] gives fivesimple nonlinear physical systems to demonstrate the ideasof period doubling, subharmonics, noisy periodicity,intermittency, and chaos in a teaching laboratory, such asin senior or graduate laboratory courses.

So far we have only scratched the surface of chaos,nonetheless, we hope that we have provided some sense of therichness and diversity of the phenomenology of chaoticbehavior of dynamical systems. It will remain to be anexciting field for sometime to come, because not only therewill be new phenomena to be discovered, but moreimportantly, there will be many surprises along the way.

For instance, Grebogi, Ott and Yorke [1982] haveinvestigated the chaotic behavior of a dynamical system atparameter values where an attractor collides with anunstable periodic orbit, specifically, the attractor iscompletely annihilated as well as its basin of attraction.They call such events crises, and they found suddenqualitative changes taken place and the chaotic region cansuddenly widen or disappear. It is well-known that inperiodically driven systems, the intersection of stable andunstable manifolds of saddle orbits forms two topologicallydistinct horseshoes, and the first one is associated withthe destruction of a chaotic attractor, while the other onecreates a new chaotic attractor. Abrupt annihilation ofchaotic attractors has been observed experimentally in adriven CO2 laser and p-n junction circuits. Recently,Schwartz [1988] used a driven CO2 laser model to illustratehow sequential horseshoe formation controls the birth anddeath of chaotic attractors. Hilborn [1985] reported thequantitative measurements of the parameter dependence of theonset of an interior crisis for a priodically drivennonlinear diode-inductor circuit. The measurements are inreasonable agreement with the predictions of Grebogi, Ottand Yorke [1982]. Grebogi, Ott and Yorke [1986] recently

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have also found a theory of the average lifetime of a crisisfor two- dimensional maps. Recently, Yamaguchi and Sakai[1988] studied the structure of basin boundaries for atwo-dimensional map analytically, and they have shown thatbasin boundaries do not play the role of a barrier when thecrisis occurs as the two parameters are changed.

The studies of nonlinear dynamical systems or mapsmodeling these systems have been concerned with how chaosarises through successive series of instabilities. Theconverse questions are: is there a nonlinear system or mapwhereby at certain control parameters the dynamics of thesystem can lead from chaotic states to ordered states? orhow can the interaction of chaotic attractors lead toordered states? Recently, Phillipson [1988] has demonstratedthe emergence of ordered states out of a chaotic backgroundof states by suitably contrived one-dimensional mapscharacterized by multiple critical points. Certainly similarconsiderations of maps of higher dimensionality will beinvestigated. Bolotin, Gonchar, Tarasov and Chekanov [1989]recently have studied the correlations between thecharacteristic properties of the classical motion andstatistical properties of energy spectra for two-dimensionalHamiltonian with a localized region having a negativeGaussian curvature of the potential energy surface. Theyhave found that at such potential energy surface withnegative Gaussian curvature, the transition ofregularity-chaos-regularity occurs. Further studies of amore general nature will be of great interest. It should bepointed out that the establishment of order out of chaosimplies the imposition of rules on an otherwise unspecifiedsituation. This is the basis of evolutionary processes innature [Farmer, Lapedes, Packard and Wendroff 1986].

One can easily show or construct nonlinear dynamicalsystems with multiple attractors and consequently requirevery accurate initial conditions for a reliable predictionof the final states. This is the issue raised by Grebogi,McDonald, Ott and Yorke [1983]. We will leave this section

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with this issue as food for thought.

6.8 A new definition of stabilityAs we have discussed earlier, there are several notions

of stability, and structural stability is intuitively easyto understand and has some nice attributes. Yet we have alsonoted that structural stability is in some sense a failurebecause structurally stable systems are not dense fordimensions >- 3. It is true that Anosov systems, AS systems,and Morse-Smale systems (or hyperbolic strange attractors)are structurally stable, but they are rather special, andmost strange attractors and chaotic systems which appear inapplications (such as the Lorenz model) are not structurallystable.

We have also pointed out that there have been severalattempts to define stability such that the stable systemsare dense, but they have failed. Recently, Zeeman [1988]proposed a new definition of stability for dynamical systemswhich is particularly aimed at nonlinear dissipativesystems. There are several advantages of this new definitionof stability than that of structural stability. Forinstance, the stable systems (in this new definition) aredense, therefore most strange attractors are stable,including non-hyperbolic ones. This approach offers analternative to structural stability. As we shall see, to acertain extent, it is a complement to structural stability.

In the following, we shall briefly discuss Zeeman'sstability (for simplicity, we shall simply call itstability), its advantages over and its differences fromstructural stability, some examples, and some difficulties.To limit our discussion, we shall only discuss the stabilityof vector field and flows, we shall leave thediffeomorphisms for the readers to read the paper by Zeeman.

Let M be a smooth oriented n-dimensional Riemannianmanifold. Let R. denotes the non-negative reals. Given asmooth vector field v on M, and given e > 0, the

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Fokker-Planck equation for v with e-diffusion is the partialdifferential equation on M

ut = euxx - div(uv), (6.8-1)

where u: MXR+ -+ R+, u(x,t) -> 0, x e M, t >- 0 and theintegral of u over M equals to unity. Here uxx is theLaplacian of u, and the divergence are determined by theRiemannian structrue (see Section 2.4). The function urepresents the smooth probability density of a population onM driven by v and subject to a-small perturbation. Forfurther details on Fokker-Planck equation, see anyprobability theory or statistical mechanics books, see alsoRisken (1984).

Let u"-`: M -+ R+ denote the steady-state solution of

Eq.(6.8-1). If M is compact (for noncompact M a suitableboundary condition on v will be required for the existenceof u), it has been established for the existence,uniqueness, and smoothness of the steady-state solution ofEq.(6.8-1). As before, next we shall define the equivalencerelations. Two smooth functions u, u': M -+ R are said tobe equivalent, u - u', if there exist diffeomorphisms a,of M and R such that the following diagram commutes:

M " I R

c 1 1 QM ".-+ R.

We say a function is stable if it has a neighborhood ofequivalents in the space of all smooth functions on M withCO topology. We define two smooth vector fields v, v' on Mto be e- equivalent if u"-E - u"". A vector field is said tobe e-stable if it has a neighborhood of e-equivalents in thespace X(M) of all smooth vector fields on M with etopology. A vector field is stable if it is e-stable forarbitrarily small e > 0. Zeeman [1988] has shown thate-stable vector fields are open and dense, and stable vectorfields are dense (but not necessarily open).

As an example, let M = R, v(x) = -x. Then the

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Fokker-Planck equation becomes ut = euXX + (ux)x. Then we cansolve the steady- state equation directly and we obtain

u = (Af ey'/2'dy + B)e-X'i2ewhere A and B are constants. If A < 0 then the bracket tendsto -co as x - co, and so u < 0 for sufficiently large valuesof x. And if A > 0 then u < 0 for sufficiently largenegative values of x. Therefore, u >_ 0 implies A = 0. Andthe condition of unit probability determines the value of B,and we have u = e'X'/2`/27re. Clearly, the steady-statesolution is just the normal distribution. Intuitively we cansee that if there were no diffusion (e = 0), then the wholepopulation would be driven towards the origin, approachingthe Dirac delta function as t - -. When c > 0, the diffusionterm pushes the population away from the origin, opposingthe incoming drive term until they reached a balance in thenormal distribution.

Since the circle is the simplest nontrivial compactmanifold, let us look at some examples of vector fields onS1. Here we shall not get into the detailed construction ofthe probability density u(x), but we shall consider somespecial cases qualitatively.

Let v be a vector field on S' given by v: S' - R. If v >0 the flow of v is a cycle. The invariant measure of theflow is k/v, where k is a constant, chosen in such a waythat k/v = 1. Furthermore, if v has only two criticalpoints, a minimum at p and a maximum at q. In the situationof the pendulum, p is the position where the pendulum standsvertically upward and nearly at rest (highest potentialenergy but lowest kinetic energy), and q is the positionwhere the pendulum hangs vertically downward with maximumvelocity (lowest potential energy but highest kineticenergy). Then the invariant measure will have a maximum at pwhere the flow lingers longest and a minimum at q where theflow is the most rapid as shown in Fig.6.8.1(a). The steadystate of the Fokker-Planck equation is an e-approximation ofthe measure and qualitatively the same, with maximum andminimum occurring slightly before the flow reaches p and q

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respectively. We have discussed this in Ch.l in detail.

max

U

(a)

U

max

(b)

Fig.6.8.1 [Zeeman 1988]Now consider the case where v has an attractor at p and

a repellor at q, as shown in Fig.6.8.1(b). The invariantmeasure is the delta function at p, and the steady-state ofthe Fokker-Planck equation will resemble a normaldistribution near p. If v is a gradient field, then themaximum and minimum will be at p and q, but if v isnon-gradient, then the maximum will be near p but theminimum may not be anywhere near q.

Note that the two steady-states shown in Fig.6.8.1 arequalitatively similar, and by the (new) definition ofequivalence, they are equivalent. And indeed, they are e-equivalent. It is then nature to ask: if we take aparametrized family of vector fields going from one to theother, are all the members of this family equivalent? If so,does this mean that the resulting bifurcation must bestable? Surprisingly, both answers are affirmative.

Fig.6.8.2 shows two fold bifurcations, in each case asink (attractor) coalesces with a source (repellor) r at afold point f and disappears (see Section 6.3). Note that,

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suppose one of the zeros p is non-hyperbolic, then aC°-small perturbation of the vector field near p can alter pto a source or to a sink. But since we leave all the otheroriginal zeros of v unaltered, we have changed the number ofsources in either case. That was the reason behind Theorem6.3.1. Thus, this type of system is not structurally stable.Zeeman (1988) gives a slightly different argument.

Fig.6.8.2 [Zeeman 1988]It appears that any two limit cycles in the plane are e-

equivalent. In fact, the flows on any two limit cycles aretopologically equivalent. The following example willillustrate that two limit cycles in the plane may not bee-equivalent.

Fig.6.8.3(a) shows a limit cycle on which the flow is asin Fig.6.8.1(a), with a source inside. This situation wouldarise from the following dynamical equations in polarcoordinates [Zeeman 1981]:

dr/dt = r(1 - r), dO/dt = 2 - rcosO.The resulting steady-state resembles a volcano crater, withthe rim of the crater above the limit cycle, with a maximumat p and a saddle at q on the lip of the crater and aminimum at the source. Let us compare this with thehysteresis cycle on a cusp catastrophe given by theequations in Cartesian coordinates:

dx/dt = -y/k, dy/dt = k(x + Y - Y3), where k >> 0.This is the same as the van der Pol oscillator with largedamping: d2x/dt2 + k(3 X2 - 1) dx/dt + x = 0.Here the limit cycle has two branches of slow manifoldseparated by the catastrophic jumps between the two branches

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of slow manifold as shown in Fig.6.8.3(b). The resultingsteady-state resembles two parallel mountain ridges withmaxima at p, and p2 near the fold points of thecatastrophes, saddles at q, and q2, and a minimum at thesource inside.

(a) (b)

Fig.6.8.3 (Zeeman 1988]It is obvious that the two flows are topologically

equivalent but differ in character. But the twosteady-states are not equivalent, thus the two vector fieldsare not e-equivalent.

The above three examples illustrate the differencebetween stability and structural stability. Fig.6.8.1illustrates two flows that are e-equivalent but nottopologically equivalent, while Fig.6.8.3 illustrates twoflows that are topologically equivalent but note-equivalent. The second bifurcation in Fig.6.8.2 is an

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example of a flow that is stable but not structurallystable. Conversely, in a one-parameter family of flowsjoining those in Fig.6.8.3, there will be a bifurcationwhere the flow is structurally stable but not stable.

Roughly speaking, structural stability captures thetopological properties of the system like basins ofattraction, separatrices and saddle connections, whileZeeman's stability criteria captures qualitative propertiesof the measure. Putting it differently, structural stabilityretains the orbit structure but ignores the speed of flowand the smoothness, and loses touch with the experimentaldata, while Zeeman's stability retains the smoothness andkeeps in touch with the data but ignores the dynamics andthe direction of flow. In this sense, we say that theycomplement each other.

As we have mentioned earlier, structurally stablesystems are not dense for dimensions >_ 3 in general, whileZeeman's stable systems are dense. Thus the qualitativeproperties of Zeeman's stable models are robust, i.e., theyare preserved under perturbations. Although the phenomenonmay be a perturbation of the model, the robustness of themodel will remain to be a valid description of thephenomenon.

Another serious criticism of structural stability is itslack of smoothness. As we have noticed before, in itsdefinition, it is necessary that the equivalence relation bea homeomorphism (for topological conjugacy and equivalence,see, Section 4.2) rather than a diffeomorphism. For example,Fig.4.1.1(a) is topologically equivalent to it with arotation so that it spirals out. Clearly the equivalence isonly subject to homeomorphism but not diffeomorphism. Theimportance of smooth equivalence relations is that if we canprove a given situation is smoothly equivalent to a standardmodel, then all the smooth qualitative properties of thestandard model are automatically carried over to the givensituation. But if a given situation is only topologicallyequivalent to a standard model, then any smooth qualitative

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properties of the given situation has to be stated andproved separately.

It should be noted that Zeeman's stability can notdistinguish between volume-preserving flows. This is becausethe theory is designed for dissipative systems, and is notsuitable for Hamiltonian systems. But nor is structuralstability useful to study Hamiltonian systems because allHamiltonian systems are structurally unstable.

Section 4 of Zeeman's paper develops an analogous theoryfor the stability of diffeomorphisms. It has been pointedout that this theory is not as elegant as that for flows.The next two sections of Zeeman's paper prove the existence,uniqueness, and classification theorems.

It is clear that these results are the beginning of avery interesting and fruitful undertaking. At the end of thepaper, Zeeman suggested some open questions for furtherresearch:

(a) Extend the results for flows to non-compcat manifoldswith suitable boundary conditions;(b) Prove the analogous density and classification theoremsfor diffeomorphisms;

(c) Develop a theory of unfoldings of unstable vector fieldsof finite codimensions;

(d) Investigate the stability of specific strangeattractors, both Axiom A and non-Axiom A, such as Lorenz andHenon;

(e) Extend the results from C' to C", r eIn these broad classes of problems, the last two classes

are of immediate impact on applications. Indeed, somestrange attractors, such as Lorenz, Cantor-like, Henon, vander Pol, Duffing, coupled-attractors, etc. must be studiedwith Zeeman's criteria for stability and classification.

Of course, with better understanding of this stabilitycriteria, a even more general and more satisfying definitionof stability may evolve. Examples and practical problems mayprovide enough imputus for mathematicians to construct suchmore satisfying criteria for stability.

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Appendix

In this appendix, we shall use the Lorenz model toillustrate the center manifold theorem and its applicationsfor bifurcation calculations. For more detail, see, e.g.,Guckenheimer and Holmes [1983], Carr [1981].

We shall study the bifurcation of the Lorenz equation,Eq.6.7.1, occurring at the origin and a = 1. The Jacobianmatrix at the origin is

-a a 0

a -1 0

0 0 -$When a = 1, this matrix has eigenvalues 0, -a - 1, and -Qwith eigenvectors (1, 1, 0), (a, -1, 0), and (0, 0, 1). Byusing the eigenvectors as a basis for a new coordinatesystem, we have

u 11/(1 + a) a/(1 + a) 0 xv = 1/(1 + a) -1/(1 + a) 0 yw 0 0 1 z1 1I

Then the Lorenz equation becomesdu/dt = -a(u + av)w/(1 + a),dv/dt = -(1 + a)v + (u + av)w/(1 + a),dw/dt = -pw + (u + av)(u - v),

and in matrix form we have

(A-1)

du/dt 0 0 0 u -a(u + av)w/(1 + a)dv/dt = 0 -(l+a) 0 v + (u + av)w/(1 + a)

dw/dt 0 0 -Q w (U + av)(u - v)

Now the linear part is in standard form. In the newcoordinates, the center manifold is a curve tangent to theu-axis. The projection of the system onto the u-axis yieldsdu/dt = 0. Nonetheless, the u-axis is not invariant becausethe equation for dw/dt contains u2 term. If we make anonlinear coordinate transformation by setting w' = w -

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u2/,8, we have

dw'/dt = -pw' + (a - 1)uv - av2+ 2au(u + av)(w' + u2/Q)/('6 (1 + a)).

Thus, in the new coordinate system (u, v, w'), we havedu/dt = -a(u + av)(w' + u2/p)/(1 + a).

Now the projection of the system onto the u-axis in the newcoordinate system gives du/dt = (-a/Q(1 + a))u3. Note alsothat there is no u2 term in any of the equations for v andw'. Thus, the u-axis in the new coordinate system isinvariant up to second order. Further efforts to find thecenter manifold can proceed by additional coordinatetransformations which serve to make the u- axis invariantfor the flow iteratively by changes in v and w'.

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Chapter 7 Applications

7.1 IntroductionFollowing the discussion at the end of Section 6.5, we

would like to point out the interaction of spatial andtemporal patterns, then follow with specific illustrations.Suppose that the dynamics of a physical system is governedby the system of evolution equations

av/at = F(v,µ) (7.1-1)

where v is an element of an appropriate Banach space, g e Ris a real bifurcation parameter, and F is a nonlinearoperator defined on a neighborhood of the origin satisfyingF(0,0) = 0. We assume that the linearized operator

A = D,F(0,0) (7.1-2)

has a simple zero eigenvalue and, addition, a simple pair ofimaginary eigenvalues ±ino (where no > 0). The remainingspectrum of A is assumed to be to the left of the imaginaryaxis. Eq.(7.1- 1) has the stationary solution v = 0 for µ =0. When the externally controllable bifurcation parameter µis varied away from zero, then due to the nonlinearity of F,two basic types of solutions bifurcate from the trivial one.They are the steady-state solutions associated with the zeroeigenvalue, and time-periodic or Hopf solutions associatedwith the eigenvalues ±ino of A. The nonlinearity of F causesthese two solutions to interact and since they tend to thetrivial solution v = 0 for µ - 0, F has a degeneratebifurcation at (0,0). The degeneracy can be removed bysubjecting F to small perturbations, representable byadditional imperfection parameters a in F itself, F - Fo.

This is achieved by stably unfolding the algebraicbifurcation equations to which Eq.(7.1-1) will be reduced.Then, as the unfolding parameters (functions of a) arevaried, zero and imaginary eigenvalues occur for differentvalues of µ and, with the degeneracy so removed, newbifurcation phenomena which are structurally stable springup.

Since the linearization of Eq.(7.1-1) at (0,0) has

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(2w/flo)-periodic solutions, we seek periodic solutions ofEq. (7.1-1) near (0,0) with period 2w/fl where fl is close tono. Setting s = fit, u(s) = v(s/fl) so that u has period 2w ins. We can rewrite Eq.(7.1-1) as

N(u,µ,r) = rdu/ds + Lu - R(u,g) = 0, (7.1-3)

here r = n - flo, L = flod/ds - A and R(u,µ) = F(u,µ) - Au. Inthe space of (27r)-periodic vector-valued functions u = u(s)the linear operator L has a three-dimensional nullspacespanned by the eigenfunctions f = (f1, f2, f3). We can reducethe bifurcation problem Eq.(7.1-3) to an algebraic one,i.e., the degenerate algebraic system o f bifurcationequations

G(x,µ,y) =f

a(x,µ,y')

L yb(x,µ,y') = 0 (7.1-4)

with a(0,0,0) = b(0,0,0) = 0, ax(0,0,0) = 0. Eq.(7.1-4)describes the Z2-covariant interaction between the Hopf andsteady-state solutions of Eq.(7.1-1).

The multivalued solutions of Eq.(7.1-4) are thebifurcation diagrams in (x,µ,y)-space. We classify themtogether with their stable perturbations by means ofimperfect bifurcation theory. Eq.(7.1-4) possess two coupledtypes of solutions, viz., pure steady-state solutions withamplitude x determined by (a(x,µ,0) = 0, y = 0), andperiodic solutions with y + 0 obtained by the simultaneoussolution of the equations a(x,µ,y') = 0, b(x,µ,y2) = 0. Theperiodic solutions branch from the steady-state at asecondary Hopf bifurcation point and may further undergotertiary bifurcations to tori [Armbruster 1983]. By changingcoordinates so that the qualitative topology of thebifurcation diagram G = 0 is preserved, the special role ofthe externally controllable bifurcation parameter µ isrespected, and G takes the simple polynomial forms given inArmbruster [1983], from which the solutions of G = 0 mayeasily be determined.

To classify all the possible stable and inequivalent,i.e., qualitatively different, bifurcation diagrams that mayarise when a given G(x,µ,y) is subjected to small

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perturbations which correspond to imperfections in F, theuniversal unfoldings of G are determined. The unfoldingparameters of a universal unfolding of G are functions ofthe imperfection parameters a in F.(v,A). Their number, thecontact codimension of G, is a measure of the degree ofcomplexity of the singularity. Hence, unfolding G displaysthe effects of all imperfections. The result is a finitelist of generic perturbed bifurcation diagrams describinginteracting Hopf (H) and steady-state problems (S). Of majorinterest for applications are special points in thebifurcation diagrams, viz., limit points and secondarybifurcation points (SB) which are here all Hopf bifurcationpoints, and tertiary bifurcations (T) from the Hopf branchto a torus. The stability properties are indicated byassigning to each branch of a diagram its stability symbol(-- is stable, etc.), i.e., the signs of the real parts ofthe eigenvalues of the Jacobian DG. Fig.7.1.1 shows thesimplest secondary bifurcation of a Hopf branch (H) from asteady-state in the (x,µ)-plane associated with the normalfrom a=x2 +ezy' +µ=0, b=y(x-a) = 0. Fig.7.1.2shows two Hopf branches bifurcating from a steady-state withbistability. In Fig.7.1.3 a tertiary bifurcation point Tappears where a transition to a double-periodic solutionoccurs. There are many more diagrams exhibiting a variety ofnew phenomena such as gaps in Hopf branches, hysteresisbetween Hopf and steady-state branches, periodic solutionscoming out of nowhere, i.e., not connected to steady-states,and so on. As an application we shall discuss the problem ofoptical bistability in Section 7.3.

Fig.7.1.1. Secondary Hopf bifurcation (SB) emerging from asteady state

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Fig.7.1.2. Simultaneous Hopf bifurcations in y-directionoriginating at a hystersis branch in the(x,),)-plane

A

Fig.7.1.3. Hopf-steady-state interaction leading to a toruspoint T

7.2 Damped oscillators and simple laser theoryFor a damped anharmonic oscillator with mass m, damping

constant r, restoring forces - ax - /3x3, the equation ofmotion (non-autonomous system) is

mdz x/dt2 + rdx/dt = - ax - /3x3. (7.2-1)The restoring force k(x) = - ax - Px3 possesses a potential

k(x) = - aV/ax, (7.2-2)

where V(x) = axe/2 + 6x4/4. (7.2-3)

The potential can be plotted as a function of x fordifferent values of a and P. Fig.7.2.1 compares the usualquadratic potential with the fourth order potential for agiven pair of a and P. Because of this potential, we can

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easily discuss its stability.

P. E.

Fig. 7.2.1

In the case p > 0, clearly from the graph of V vs x, thesystem is globally stable. Wherever the particle starts, itcomes to rest at a finite value of x. On the other hand, wehave global instability for P < 0. Now let us look at thelocal stability. It is suffice to consider only fi > 0 case,as we notice the "symmetry" between f > 0 and P < 0. Let usfirst consider the steady state which is characterized by:

d2 x/dt' = dx/dt = 0. (7.2-4)The states of stable and unstable equilibria are defined by

ax + Qx3 = 0. (7.2-5)

For a > 0, we have the stable solutionxo = 0. (7.2-6)

Mathematicians call this an "attractor". For a < 0, thestate xo = 0 becomes unstable ("repeller"), and instead, wehave two stable solutions

x1 = ±(Ic/RI)'. (7.2-7)

With the transition from a > 0 to a < 0 the system passesthrough an instability and the particle is now either at x,or x2. In fluid dynamics, the change of one stability to

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another is called an exchange of stabilities. Inmathematics, the passing over from one stability to two newones is known as Hopf bifurcation. The state where a = 0, x= 0 is a marginal state, i.e., bifurcation point.

As one knows from experience in all areas of physical,biological, and social sciences, one finds that fluctuationmust usually incorporate into the parameter equations. Letus assume the particle gets perturbative forces of equalmagnitude but random in forward or backward directions. Letus add to Eq. (7.2- 1) the random force

0(t) = n Ei (-1)" 6 (t - tj) , (7.2-8)where n stands for the magnitude of the force, ni is arandom variable can either be 0 or 1 so that it gives thedirection of the random force, t, is a random time sequence,and 6 is the usual Dirac 6-function. Thus Eq.(7.2-1) becomes

md2x/dt2 + rdx/dt = - ax - px3 + 0(t). (7.2-9)

As we have pointed out earlier, one can transform Eq.(7.2-9)into a system of two first order equations. To illustratesome of the features we would like to display, forsimplicity we assume that the oscillator is heavily damped.In this case, we can formally set m = 0 in Eq.(7.2-9) whichbecomes

dx/dt = ax - bx3 + F(t) (7.2-10)

where a = a/r, b = a/r, F(t) = 0(t)/r. One can discuss thisequation from various viewpoints. If we assume thefluctuation is of the form Eq.(7.2-8), i.e.,

F(t) = Fo Ej (-1) j6 (t - tj) , (7.2-11)then we have <F(t)> = 0. One can also show that thecorrelation function

<F(t)F(tI)> = (F0=/to) 6(t-t') = C 6(t-t'), (7.2-12)

where to is the mean time between perturbative impulses. Toanalyze Eq.(7.2-12), we first discuss Eq.(7.2-10) with F(t)= 0. Then the time-dependent solutions of dx/dt = -ax - bx3are

x(t) = ± ay'{exp[2a(t - t')] -b)",', for a > 0 (7.2-13)and

x(t) = ±IaI''{exp(-2IaI(t - t') for a < 0. (7.2-14)

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Here a has the meaning of an inverse relaxation time for thesystem. In either cases a > 0 or a < 0, x tends to itsequilibrium value. Because an explicit solution can not befound for more general cases of order-parameter equations,as usual, we must discuss local stability by linearization.Define the steady state coordinate xs by dxs/dt = 0 andassume x = xs + 6x. First linearize Eq. (7.2-10) about xs = 0which yields

6(dx/dt) = -a6x, (7.2-15)

with the solution 6x = A exp(-at). (7.2-16)

For a > 0 the system is stable, a < 0 unstable, and for a =0 it is a marginal state. When the relaxation constant aapproaches to 0, we have the critical slowing downphenomena. If a < 0, the coordinate of the stable point is

Ixis = (IaI/b)". (7.2-17)

By inserting x = xs + 6x (7.2-18)

into dx/dt = -ax - bx3, we havedx/dt = dxs/dt + d(6x)/dt

_ - aft - 3bIaI6x/b - xs(a + bIaI/b)- 2Ial6x, (7.2-19)

this is because a < 0, a + Ial = 0. Thus yields therelaxation time

r = (2Ial)-'. (7.2-20)

Now consider the fluctuation of x in the linearized theoryand solve the equation

d(6x)/dt + aft = F(t) (7.2-21)

which yields6x = exp(-at)fto exp(ar)F(r) dr. (7.2-22)

The correlation function of the coordinate gives a measurefor the temporal behavior of the system. SubstitutingEq.(7.2-22) into <6x(t)6x(t')> yields (with Eq.(7.2-12))<6x(t)6x(t')> = C exp[-a(t - t')]/2a, t >- t' (7.2-23)

for to - -oo . From Eq.(7.2-23) it is clear that as a - 0 notonly the relaxation time r , but also the coordinatefluctuation become infinite. It is important to point outthat the divergence of Eo.(7.2-23) for a - 0 is caused bylinearization procedure. In other words, while the

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fluctuation 6x for a < 0 or a > 0 are finite and can beneglected in most cases, the linearization procedure breaksdown near the point a = 0. However, in the exact theory, itis still true that at the critical point a = 0 thefluctuation in 6x becomes very large.

At any rate, the concepts of critical fluctuation,critical slowing down, symmetry breaking, etc. are part ofthe standard repertoire of phase-transition theory.

In statistical mechanics, Eq.(7.2-10) can be consideredas an extension of the Langevin equation of Brownian motion

dv/dt = - av + F(t). (7.2-24)

It is quite simple to solve the linearized form ofEq.(7.2-10), i.e., Eq.(7.2-24), but the solution ofEq.(7.2-10) becomes a formidable task even for this simplecase with nonlinearity kept. One may want to proceed to theFokker-Planck equation given by:

df(q,t)/dt = -a(K(q)f)/aq + /a=(Q(q)f)/aq2. (7.2-25)

Here f(q,t)dq is the probability of finding the particle inthe interval (q, q+dq) at a time t. The drift coefficientK(q) and diffusion coefficient Q(q) are defined by:

K(q) = limt,0 (1/t)<q(t) - q(0)> (7.2-26)

and Q(q) = limt.a (1/t)<(q(t) - q(0))2>. (7.2-27)

One has to imagine that Eq.(7.2-10) in this context issolved for a time interval which comprises many pushes ofF(t) but small compared with the overall motion of thesystem. In the present case one readily finds that K(q) isidentical with the force k(x) in Eq.(7.2-2), i.e.,

K(q) = - aq - bq3 = - aV/aq (7.2-28)

and Q(q) = C, (7.2-29)

where C is defined as the coefficient in the correlationfunction Eq.(7.2-12). The Fokker-Planck Eq.(7.2-25) thenreads

df(q,t)/dt = - a[(- aV/aq)f - /C of/aq]/aq. (7.2-30)

This equation has the form of a conservative law. Let theprobability current be denoted by j, then we have

df/dt = - aj/aq. (7.2-31)

In the stationary case, f = 0 and we find the solution

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f = qexp(-2V/C) = qexp[-(2/L) (aq2/2 + bq4/4) ], (7.2-32)here we have taken into account that f vanishes at infinity.The distribution function f is of great importance becauseit governs the stability, the fluctuation, and the dynamicsof the system.

(a) For stability, a simple comparision with ourprevious considerations reveals that these systems areglobally stable whereby f is normalizable. V in Eq.(7.2-32)serves as a Liapunov function VL (let VL = V(q) - V(qo))which satisfies the following conditions:(i) V1(q) is C' in a region surrounding qo a D,(ii) V1(go) = 0,(iii) V1(q) > 0 in D,(iv) dVL/dt = k(q) aV/aq <_ 0.

Liapunov's theorem then states that: If there exists such aLiapunov function VL in D, then qa c D is stable.

(b) For instance, by expanding the exponent V(q) in f inEq. (7.2-32) about the steady state, using q = qs + &q, oneget an expression for the probability of finding afluctuation of the size 5q,f(q) = nexp{-2[V(q,) /C + 92V(q)/ag2Iq (6q)2/2C]. (7.2-33)

(c) With V(q), one can develop the dynamics of thesystem. We shall not go into this in detail.

With proper replacement of variables, one can study thethermodynamics of the system, for instance, one can obtainthe relation between fluctuation and dissipation, Einstein'srelation for the probability W(q) for a fluctuation of size6q for Brownian motion, as well as demonstrate the Landautheory of phase transitions. So far we have discussed asimple system in "thermal" equilibrium. One can alsodemonstrate by explicit examples which lead to equations ofthe form Eq.(7.2-10) or their generalizations to manydegrees of freedom for systems which are far from thermalequilibrium.

In different cases, x (or generalized coordinates q) mayrepresent very different quantities, e.g., the laser lightfield, electric current, velocity field for fluids,

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concentrations of chemical reactions, etc. Now the steadystates q = qs + 0 are now maintained by a balance betweenenergy input and dissipation. It should be emphasized that agreat deal of the analysis illustrated above applies equallywell to those more general cases in diverse fields ofscience, engineering, and social science.

In the following, we shall briefly discuss a simplelaser theory which is a system far from thermal equilibrium.This allows one to study cooperative effects in greaterdetail.

Simple laser theoryWe shall describe the laser field either quantum

mechanically or classically, and our formulation will be insuch a way that the equations can be understood asclassical ones. Nonetheless, in most cases they possessexact quantum mechanical analogue.

The usual treatment of lasers consider the electricfield strength E(x,t) and decomposes it into spatial modesb1(x), and b, (x) are determined by the usual resonatortheory. Thus we assume that these spatial modes aredetermined completely and they form an orthonormal set. Thenthe expansion of E(x,t) can be written as

E(x,t) = Ei E1(t)b1(x), (7.2-34)

where E,(t) are time-dependent amplitudes. One can decomposethese amplitudes into positive and negative frequency partsaccording to

E1 (t) = E1+ (t) a-'2.t + Ej- (t) ei°it, (7.2-35)where Il, is the frequency of mode i in the unloaded cavity,i.e., without the presence of laser active atoms, and E,=are slowly varying amplitudes. We want to derive equationsof motion for these slowly varying amplitudes except forultrashort pulses. We shall use dimensionless units for E1:by putting

Ei- = -i(r1tlj/2Eo)Y`aj+,Ei+ = i(Xif1/2e0)'hai. (7.2-36)

Quantum mechanically, a,+, a; are creation and annihilation

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operators of photons of the mode i, and in classicaltreatment they are merely c-number time-dependent complexamplitudes. Of course, in some cases, it is preferable touse the field instead of the mode decomposition. Also, dueto the quantum-classical correspondence, one can replace thequantum correlation by classical averages.

For simplicity, let us consider single mode lasers, sothat we can drop the index i. In a region not too far aboveand below the laser threshold, the mode amplitude a+(t)obeys the simple equation

da+(t)/dt = - Ka+(t) + Ga+(t) + F+ (t) , (7.2-37)where K accounts for the losses by the mirrors, refraction,absorption due to impurities, etc., G describes the gain bythe stimulated emission, and F+(t) represents thefluctuation or noise of the amplitude. This noise term canstem from the spontaneous emission of the atoms into allmodes, the interaction of the atoms with lattice vibrations,the pumplight, etc. For more detail, see Haken [1970] orSargent, Scully and Lamb (1974].

As before, we assume that the statistical average of thefluctuation vanishes, i.e.,

<F+(t)> = <F-(t)> = 0 (7.2-38)

and they have correlation function<F+(t)F"(t')> = C 6(t-t'), (7.2-39)

as in Eq.(7.2-12). Eq.(7.2-39) expresses the fact that thefluctuations have a very short "memory" compared to othertime constants in the systems. The constant C depends on thecavity width, the number of thermal photons, dampingconstants, occupation numbers of the individual atoms, etc.

The gain function G is proportional to the number ofexcited atoms N2 minus the number of atoms in the groundstate N,. Furthermore, the gain depends on the line shape ofthe atoms. The closer the laser frequency fl to the atomicresonance v, the larger the gain. For the homogeneousLorentzian linewidth with half-width r, the real part of thegain is [Haken 1970; V. Arzt et al 1966; Fleck 1966]:

ReG = (N2 - N1)r gj2/[r, + (fl - v)2 ], (7.2-40)

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here constant g contains the optical matrix element. It isimportant to notice that the inversion N2 - Ni is lowereddue to the process of stimulated emission. That is, for nottoo high laser amplitudes, we have the instantaneousinversion

N2 - N, = (N2 - N1)0 - constant-a+(t)a(t),(N2 - N1)0 = Do. (7.2-41)

The first term on the right hand side, (N2 - N1)0, is the

unsaturated inversion, the second term describes thecowering of the inversion due to laser action. InEq.(7.2-41) we assume that the atomic inversion respondsimmediately to the field. Substituting Eq.(7.2-41) intoEq.(7.2-40) we obtain the saturated gain:

GS = G' - f3a'a. (7.2-42)Introducing Eq.(7.2-42) into Eq.(7.2-37) we have the basiclaser equation derived previously [Haken & Sanermann 1963;Lamb 1964; Haken 1964].

da'/dt = - Ka' + Gua' - Qa'(a'a) + F'. (7.2-43)This is the familiar equation of Eq.(7.2-10). It should alsobe mentioned that without the noise term F% Eq. (7.2-43) issimilar to the Duffing equation which describes the behaviorof the hardened spring with cubic stiffness term. As we havepointed out in Chapter 4, this system has a unique closedorbit which is the f2-set of all the orbits except the fixedpoint. Furthermore, this system is auto-oscillatory sinceall solutions (except one) tend to become periodic as timeincreases.

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Once again, we may interpret Eq.(7.2-43) as the equationof a strongly overdamped motion of a particle in thepotential field V = (K - G") I a` I : /2 + $ I a` 14/4 . In addition,the particle experiences random pushes by the fluctuatingforce F'(t).

Let us first discuss the situation below threshold.Fig.7.2.2a shows the potential V in one dimension for G" - K5 0. After each push excerted by the fluctuating force onthe particle, it falls down the slope of potential hill.When we multiply E (or a') by exp(iflt) to obtain E(t) ofEq.(7.2-34) and consider a sequence of random pushes,Fig.7.2.2b results. As was shown by Mandel & Wolf [1961],the field amplitude is Gaussian distributed as isrepresented by Fig.7.2.2c.

Fig.7.2.2 Laser below threshold, G - K < 0.

V

q,E

a) Potential.

E - q exp(i(ot)

b) Real part of field amplitude vs. time.

336

f (q)

_> q

c) Gaussian distribution of field amplitude.For above laser threshold, G"- K > 0, the potential

curve of Fig.7.2.3a applies. The state E = 0 has becomeunstable and is replaced by a new stable state Eo + 0. Forthe moment, if we ignore the fluctuations, a coherent waveemerges as in Fig.7.2.3b. When we take into account theimpact of fluctuations, we must resort to higher thanone-dimensional potential, Fig.7.2.3c. The random pushes inthe central direction will result the same effect as in theone-dimensional situation, but the random pushes in thetangential direction will cause phase diffusion. Indeed,this is the basis leading to the prediction of Haken [1964]that laser light is amplitude stablized with smallsuperimposed amplitude fluctuations and a phase diffusion.Furthermore, if the corresponding Langevin-type equation,Eq.(7.2-10 or 7.2-43) is converted into a Fokker-Planckequation, Eq.(7.2-25), the steady-state distributionfunction can be easily obtained by using the adiabaticelimination method [Risken 1965]. These and furtherproperties of laser light were further studied boththeoretically and experimentally by a number of authors, forinstance the references cited in Haken [1970], Sargent,Scully and Lamb [1974], and the recent book by Milonni andEberly [1988].

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(a)V

(b)

q,E

t

Fig.7.2.3 Laser above threshold, G - K > 0. a)Potential in one dimension. b) Real part of field withoutfluctuations.

f(q,, qz)

(C)

1

Fig.7.2.3 c) Potential in two dimensions. d) Laser lightdistribution function. [Haken 1986]

It has to be said that the laser was recognized as thefirst example of a nonequilibrium phase transition and aperfect analogy to the Landau theory of phase transitioncould be established [Graham and Haken 1968,1970; Haken1983]. When we plot the stable amplitude E. vs. (Gu - K), weobtain the bifurcation diagram of laser light, Fig.7.2.4.

q0, Eo

G-K

Fig. 7.2.4 Bifurcation diagram of laser light

E - q exp(imt)

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It should also be pointed out that if one considers theclassical dispersion theory of the electromagnetic waves,then the electric field satisfies the wave equation,

- V2 E + (1/c2 ) a= E/cat= + 4,racaE/at/c2= - 4,ra' P/ate /C2 , (7.2-44)

where ac is the conductivity which describes the damping ofthe field, and P is the macroscopic polarization. One canthink of atoms dispersed in a medium and we may representthe polarization as a sum over the individual atomiccontributions at xi by

P(x,t) = E1 6(x - x1)pq(t), (7.2-45)

where pi is the dipole moment of the i-th atom. Then thefield equation (7.2-44) is supplemented by the equation ofthe atom i,

a=p;/at2 + 2aap,/at + pep; = e2E(x,t)/m, (7.2-46)where a is the damping constant of the atoms. Once again, wecan see that Eq.(7.2-46) is a special case of Eq.(7.2-9).Indeed, if one pursued this further by decomposing thedipole moments in terms of raising and lowering operator foratomic levels, and using the interacting Hamiltonian, onecan get field equations, the equation for the atomic dipolemoments, and the equation for the atomic inversion. We shallnot get into any more details here. Interested readersplease see, e.g., Haken (1970], Sargent, Scully and Lamb[1974].

Statistical concept of physical processes, not too farfrom equilibrium, was first established in the theory ofBrownian motion [Langevin 1908]. The theory of Fox andUhlenbeck [1970], resulted in a general stochastic theoryfor the linear dynamical behavior of thermodynamical systemsclose to equilibrium, which includes the Langevin theory ofBrownian motion, the Onsager and Machlup theory [1953] forirreversible processes, the linearized fluctuating equationsof Landau and Lifshitz for hydrodynamics (1959], and thelinearized fluctuating Boltzmann equation as special cases.For more detailed discussions of the general theory ofstochastical processes, see, for instance, Gardiner (1985].

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As has been pointed out by Fox [1972], in the general theoryas well as in each of the above special cases, themathematical descriptions involve either linear partialintegro- differential equations or set of linearinhomogeneous equations. These inhomogeneous terms are theusual stochastic (or fluctuating) driving forces of theprocesses frequently termed the Langevin fluctuations. Theseterms are being referred to as additive fluctuations foradditive stochastic processes. Fox [1972] has sincesystematically introduced the stochastic driving forces forhomogeneous equations in a multiplicative way, and suchprocesses are called multiplicative stochastic processes.

Multiplicative stochastic processes arise naturally inmany disciplines of science. Fox [1972] has established somemathematical groundwork for multiplicative stochasticprocesses (MSP), and pointed out the relevance of MSP innonequilibrium statistical mechanics. Shortly after laserswere invented, it became clear that both lasers andnonlinear optical processes were examples of dynamicalprocesses far from thermodynamical equilibrium. Later,Schenzle and Brand [1979], motivated by laser theory andnonlinear processes, demonstrated that: (i) an ensemble oftwo-level atoms interacting with plane electromagneticwaves, described by the well-known Maxwell-Block equations;(ii) parametric down conversion as well as parametric upconversion; (iii) stimulated Raman scattering; and (iv)autocatalytic reactions in biochemistry, are examples ofnonlinear processes whose stochastic fluctuations aremultiplicative. They have obtained some very interestingresults. It is natural to expect that many real systems haveboth types of fluctuations, that is, there may be a mixtureof both processes. Indeed, they have discussed such systemswith mixture of both fluctuations [1979]. Some furtherdiscussions on multiplicative fluctuations in nonlinearoptics and the reduction of phase noise in coherentanti-Stokes Raman scattering have been discussed by Lee[1990]. There are many unanswered questions on the

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fluctuations of nonlinear phenomena are awaiting for furtherstudy.

In general, relaxation oscillations characterized by twoquite different time scales can be described by xt =f(x,y,g,e) and yt = eg(x,y,µ,e), where at is the partialderivative of a with respect to t, µ << 1 and g is thecontrol parameter. Baer and Erneux [1986] have shown how theharmonic oscillations near the bifurcation pointprogressively change to become pulsed, triangularoscillations.

There are several classics which deal with nonlinearoscillations. One of them is by Minorsky [1962]. It has agreat deal of information and results worked out. Indeed,many recent results for applications of nonlinearoscillations can be found in Minorsky (1962]. Another bookby Krasnosel'skii, Burd and Kolesov [1973] is also a veryuseful source. Another classic is the one by Nayfeh and Mook[1979]. This one provides many details of low dimensionalnonlinear oscillations and their dynamical evolutions. Inthe 1950' and 1960's, Lefschetz edited a seriers of volumeson the contributions to the theory of nonlinear oscillations[1950, 1950, 1956, 1958, 1960].

7.3 Optical instabilitiesOne of the unexpected features of early development of

laser systems was the presence of output pulsations evenunder steady pumping conditions. It did not take long torecognize this result as an important aspect of lasers. Infact, spiking was observed in some masers even before thediscovery of lasers [Makhov et al, 1958; Kikuchi et al,1959; Makhov et al, 1960]. As early as 1958, Khaldre andKhokhlov [1958], Gurtovnick [1958], and Oraevskii [1959]linked the output pulsations to the emergence of dynamicalinstabilities. Uspenskii [1963; 1964], Korobkin andUspenskii [1964] already in the early 1960's studied theinstability phenomena for homogeneously broadened laser

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systems. The current efforts in optical instabilites seek toanswer the same overall questions as the ones whichmanifested themselves in other disciplines, namely: What arethe origins and the functions of the evolutionary structuresof the dynamical systems? Are there universal laws whichdemand the growth of certain instabilities and structures?What differentiates among the many possible routes of asystem to certain macroscopic behavior?

The earliest theoretical models of laser action werebased on the description of the energy exchanges between acollection of inverted two-level atoms and the cavity field.Later on, we have found that the rate equation approach isinadequate to provide a faithful description of the observedinstabilities [Hofelich-Abate and Hofelich, 1968]. A verysignificant advance in the field of laser instabilities wasthe one by Haken [1975], who established the homeomorphismbetween the single-mode laser model and the Lorenzequations. Such a homeomorphism between the single-modelaser and the Lorenz equations establishes thatdeterministic chaos is also a part of chaotic laser behavioras long as the single-mode approximation is sufficientlyaccurate for the laser system. More importantly, such ahomeomorphism unifies seemingly different phenomena fromdifferent disciplines, and Synergetics provides themotivation and the guidance for an organized approach to theproblem of dynamical systems and chaotic behavior ingeneral, and laser instabilities in particular.

In the following, we shall discuss briefly theMaxwell-Bloch equations for a simple ring cavity anddemonstrate some interesting features of a ring cavity, suchas: (i) if the detuning of the incident light with theabsorber introduced, in a stationary situation thetransmitted field becomes a multi-valued function of theincident field; (ii) the stationary solution is not alwaysstable even when it belongs to the branch with positivedifferential gain, in fact, in some cases the transmittedfield exhibits a chaotic behavior.

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Let the simple ring cavity be the following:

where E1, ET and E. are the incident, transmitted, andreflected field respectively. L is the length of the samplecell containing a two level absorber (for simplicity,homogeneously broadened) and LT the total length of theoptical path in the ring cavity. Also assume thereflectivity of MT and Mz be R and 1 for M3 and M4. LetE(t,z) be the complex envelope of the electric field, thenwe have the following boundary conditions

E(t,0) = T E1(t) + R E(t - 1/c,L)exp(ikLT),ET M = T E(t,L)exp(ikL),

where T = 1 - R and 1 = LT - L. The propagation of theelectric field in the non-linear absorber can be describedby the Maxwell-Block equations [Sargent, Scully and Lamb1974]:

aE/az = 47ringko, (7.3-1a)

aN/ar = - r1(N + 1/z) + iµ(a*E - aE*)/z, (7.3-1b)

as/ar = (iaft - r1)a - iµNE,here T = t - z/c is the retarded time, a is the

(7.3-1c)

dimensionless polarization and N = /(N1 - N2), µ thetransition dipole moment, and Aft = w - ft (where ft is thetranstion frequency of the two level atom) is the detuning

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frequency, r1 and r, are the transverse and longitudinalrelaxation rates respectively, and n the density of theatoms.

Let us limit our considerations to fast transverserelaxation, which means that the polarization follows theelectric field adiabatically (adiabatic elimination method)i.e., as/ar = 0, we have

a = iµNE/(ien - r1). (7.3-2)

Substituting Eq.(7.3-2) into Eq.(7.3-1a), then the electricfield can be written in the integral form.E(r+z/c,z) ( i , &

(7.3-3)where9 = 2Trnkµ2 and W(r,z) = f0z N(r + z'/c,z')dz'. (7.3-4)Substituting Egs.(7.3-3) and (7.3-4) into Eq.(7.3-1b) andthen integrating over z, one has

aw(r,z)/ar = - r,(W + z/2)- µ2 IE(r,O) 12 (exp[4Ar1W/(en2 + rL2) ] -1)/40. (7.3-5)

Introducing the dimensionless quantitiese(t,z) = µ E(t,z)/z,/r1r1 + e2 ),xtry,

0(t) a W(t - LT/c,L)/L, (7.3-6)

A = en/r1.Combining Egs.(7.3-3) and (7.3-5) together with the boundaryconditions with the dimensionless quantities in (7.3-6) wehave the following set of equations which do not involve theoptical coordinates:e(x,0) =

Re(x-k,0)exp(aLO(x))exp(i(aLe(O(x)-/)-60), (7.3-7a)

do(x)/dx = -(0(x)+/) - 21e(x-k,0)12[exp(2aLp(x))-1]/aL,(7.3-7b)

ET = ,/e(x-k,0)exp(aLO(x))exp(i(aLe(0(x)+/)-(6o kl)

(7.3-8)where eT M = hET(t - 1/c)/2(rlri (1 + e2)

e,(x) µE1(t)/2(r1r1 (1 + e2and a = 2Or1/ (en2 + r12 ) ,is the effective absorption coefficient, and

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60 = - k(,/61L + 1) + 27rM,(where Ei' = 1 - 4Trnµ2 Aft/ (efl2 + I12) is the lineardielectric constant, and 2wM is the multiple of 2v nearestto k( 6,'L+1)) being the mistuning parameter of the ringcavity and k = IjLT/c being the dimensionless round triptime. Egs.(7.3-7a,b) and (7.3- 8) can be interpreted asdifference - differential equations and whose solutions areuniquely determined by the initial value 0(0) and theboundary condition 6(x,0) for - k <- x < 0.

For the stationary case, we can set do(x)/dx = 0 andE(x,0) = constant. By eliminating E(x,0) from Egs.(7.3-7a,b)and (7.3-8), one obtains the relationship between thestationary solution of the transmitted field and theincident field:

2= IATI2(exp(-aL) - R]2+ 4Rexp(-aL4)sin2 [6(IET12)/2]/T2 ). (7.3-9)

where 6(kET12) = 60 - aL (0 + 1/2), n (7.3-10)and ^ denotes the stationary solution and 0 is related toI 1 2ET by

(¢ + 1/2)/[exp(-2aL4) - 11 = 2IETI2/TaL. (7.3-11)Here $ is a monotonic increasing function of from -1/2to zero. Also, note that 6(1^12) denotes the intensitydependent mistuning parameter, which is originated from thenonlinear shift of wavenumber (or frequency) in theabsorber.

In the limit of aLA - 0 and 60 - 0, Eqs. (7.3-9) and(7.3-11) reduce to the absorptive bistability obtained byBonifacio and Lugiato [1978]. On the other hand, if theconditions aLA << 1, aL << 1, 1601 << 1 and

I

ET 1-/T << 1 aresatisfied, then Eq.(7.3-9) reduces to

1

1 2 = IETI2 [1 + 60/2)2/T2 (7.3-12)by the approximation 6 (I 6T 12 ) 60 - 2aLe I ET 12 /T. Eq. (7 . 3-12 )agrees with the relation obtained by Gibbs et al [1976]exhibiting the dispersive bistability experimentally.

If the parameter aLA is sufficiently large, thetransmitted field oscillates as a function of the incidentfield due to the factor sin 2(6(IETI2)/2]. Consequently, the

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ordinary bistable behavior will be drastrically modified.For fixed parameter aL and reflectivity R to be 4.0 and 0.95respectively, the relations between the transmitted fieldand the incident field are shown in Fig.7.3.2 for variousvalues of aLa.

0.5 1 \(a) (b) (c) (d) j

5 10

Fig. 7.3.2. Relations between the transmitted field and theincident field for (a) aL = 0.0, (b) aLa = 2 , (c) aLa = 4,and (d) aLe = 6 [Ikeda 1979].

As aLa increases, the "ordinary" bistable relation (a)for aL = 0 changes to the ones typified by (c) and (d),i.e., new branches appear in the lower tensity of IETI andtheir number increases with aLa. Indeed, when the magnitudeof aL is small enough, then a pair of new branches withnegative and positive differential gains is generated whenaLe is increased by 2w. Such multiple-valuedness is due tothe intensity dependent mistuning of the cavity with theincident light. The possibility of multiple-valued responseof the transmitted light has also been discussed by Felberand Manburger [1976] for F-P cavity system containing a Kerr(cubic) medium. They have also pointed out that the physicalorigin of the bistable behavior is the nonlinear increase ofoptical path length at high cavity energy density whichbrings the initially detuned cavity into resonance with the

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field. Once the resonance is achieved the transmissivity ishigh.

For the discussion of the stability of themultiple-valued stationary state, we shall consider thelimiting case k << 1, i.e., very short cavity round triptime. In this limit, we may set do(x)/dx in Eq.(7.3-7b)equal to zero. Then Egs.(7.3-7a,b) and (7.3-8) reduce to thefollowing difference equations:Eon = 'r-TEln + REen_lexp(aLOn)exp{i(aLA(On+') - 60)),

(7.3-13a)ETn = TTEon_lexp(cLOn)exp(i[aLe(On+/) - (60 + kl)]),

(7.3-13b)where Eon, ETn, and EIn denote E (xo nk, 0) , ET (xo nk) andE1(xo nk) respectively, and On relates with E0n_1 by

(On + 1/2)/[1 - exp(aL$n)] = 2IEO.n_1l2/a (7.3-14)It should be noted that when aL = 60 = 0 one can choose Eo,nas real, thus Eq.(7.3-13a) can be solved by graphicalmethod. It can be shown that in the case of pure absorptivebistability, the branch with the positive differential gainis always stable. The stability problem in general has notbeen properly analyzed as yet. Note that as we havediscussed earlier in Chapter 6, in particular, the lastsection, the proper mathematical setting and theirtechniques are readily available. See elsewhere in theseNotes and references therein.

For more detail discussion of the Maxwell-Bloch theoryof ring laser cavity, see, for instance, Lugiato et al[1987], Risken [1986; 1988], Milonni et al [1987]. Itsuffices to say that from Egs.(7.3-1) one can identify,formally and intuitively, that the Maxwell-Bloch equationsare closely related to the Lorenz equations, andconsequently the unstable region of the parameter space.Some detailed derivations of this relationship can be foundin Miloni et al [1987]. The Lorenz model has beenwell-studied, and we shall not go into any detail here. Onewould expect that the projection of the phase spacetrajectory in the E-N-plane will produce the "butterfly"

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figures as in the Lorenz model. Indeed, one would.Since the electric field amplitude is not directly

observable, it is interesting to know how to distinguish asymmetry breaking transformation using traditionalmeasurements. The solution is the application of heterodynetechniques using a stable reference source. For heterodynetechniques in infrared and optical frequencies, see forinstance, Kingston [1978]. It suffices to say that the maindiffernce between a symmetric and an asymmetric electricfields is that the symmetric one has a zero average electricfield, while the asymmetric one has non-zero averageelectric field. Thus, the heterodyne spectrum of a symmetricsolution has symmetric frequency components around In - notbut with zero spectral power, while an asymmetric solutiondisplays a distinguishing signature of having a line at thebeat frequency in - n0I and with symmetric sidebands.

It should be noted that a survey of the availableexperiments indicates that significant areas of disagreementstill exist between the theoretical predictions of theMaxwell-Bloch model and the experimental data. Indeed,nobody really knows how to model the active medium withsufficient accuracy, in particular, for solid state lasers.Even for gas lasers, which can operate with a homogeneousbroadened gain and yet show behavior which is not compatiblewith the usual descriptions. In fact, Lippi et al [1986]have shown that the CO2 laser whose unstable behavior nearthreshold is in striking difference with the theoreticaldescriptions.

Hendow and Sargent [1982; 1985] and Narducci et al[1986] have discovered a new type of instability, the phaseinstability, and its dynamical origin can be attributed bythe loss of phase stability instead of its amplitude. Theexperimental confirmation showed good qualitative agreementbetween the theoretical predictions and the experimentalobservations [Treducci et al 1986]. Experiments have alsoshown that frequent appearance of regular and chaoticpulsations in inhomogeneous broadened lasers [Casperson

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1978; 1980; 1981; 1985; Bentley and Abraham 1982; Maeda andAbraham 1982; Abraham et al 1983; Gioggia and Abraham 1983;1984].

An important warning about the adequacy of theplane-wave approximation came from the lack of quantitativeagreement between the predictions of the plane-wavestationary theory of optical bistability [Bonifacio andLugiato 1976; 1978; Lugiato 1984], and the failure of thetime-dependent plane-wave calculations to match the observedpulsing pattern [Lugiato et al 1982]. Indeed, a growingnumber of experimental and theoretical results support theview that transverse effects play a significant role andmaybe even more influential when the optical resonatorcontains an active medium [LeBerre et al 1984; Valley et al1986; Derstine et al 1986; McLaughlin et al 1985; Moloney etal 1982; Moloney 1984].

There is another type of optical instability which isvery distinct from the optical chaos mentioned aboverelating to the Lorenz model of a nonlinear dynamicalsystem. This other type of optical instability is due tononlinear delay feedback, which is related to the iterativemaps.

In the following, the stability results of linearizedequations due to Ikeda [1979] is presented. The linearmotion of eo,, around its stationary solution ischaracterized by two eigenvalues of a 2x2 evolution matrix,and the stationary solution is stable only if each of theeigenvalues has an absolute value less than unity. Thestability has also been studied numerically. As expected,the branches with negative differential gain dIeTI/dIelI < 0are always unstable. It is interesting to note that even thebranch with positive differential gain is not always stable.The stationary solution becomes stable when 1e,j is set inthe vicinity of the supremum or infimum of the branch, asillustrated in the following figure.

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IE1I

A I II I

E,1E11

E2

Fig.7.3.3 Stable, .... unstable. A & D branches arestable. Note that all stationary solutions in the region el< If, I < E2 are unstable.

Unstable positive differential gain region can lead toregenerative oscillation, for more details, see e.g.,Goldstone & Garmire [1983]. As we have noticed that in theregion el < 1 61 1 < E2 all stationary solutions are unstable.What happens in this region? From an appropriate initialvalue eo.0, one can iterate Egs.(7.3-13a,b) and it results inan erratic behavior of the transmitted field. It has beenfound that as the iterated step is advanced the plottedpoint tends to be "attracted" into a figure which appears toconsist of an infinite set of one-dim curves. It is alsointeresting to know that almost identical figures areobtained when the initial value eo,o is changed over a widerange. This suggests that the figure represents the "strangeattractor" of the difference equation (7.3-13a,b). For

350

details, see, Ikeda [1979]; Ikeda, Daido and Akimoto [1980,1982].

For more general discussion on iterate maps, see, Colletand Eckmann [1980]. For more detailed discussions andfurther references on optical bistability, see, Bowden,Ciftan and Robl [1981]; Bowden, Gibbs and McCall [1984];Gibbs [1985]; Gibbs, Mandel, Peyghambarian and Smith [1986];Goldstone [1985]; Lugiato [1984]; Zhang and Lee [1988].Gibbs [1985] provides the most comprehensive treatment ofoptical bistability up to 1984-5, including references.Goldstone's review article is also recommended. For recentresults, see those conference proceedings. There are somereview articles on optical instabilities, e.g., Boyd, Raymerand Narducci [1986]; Narducci and Abraham [1988];Orayevskiy [1988]; Chrostowski and Abraham [1986], Milonniet al [1987], and forthcoming conference proceedings.

7.4 Chemical reaction-diffusion equationsThe usual study of chemical processes is concentrated in

that several chemical reactants are put together at acertain time and then processes taking place are thenstudied. In equilibrium thermodynamics, one usually comparesonly the reactants and the final products and observes therate as well as the direction of a process taking place.Here we would like to consider the following situation whichcan be served as a model for biochemical processes. Let ussuppose several reactants are continuously put into areactor vessel where new chemicals are continuously producedand the end products are then removed in such a way thatthey satisfy steady state conditions. Certainly, theseprocesses can be maintained for any finite amount of timeonly under conditions far from thermal equilibrium. Indeed,a number of interesting questions, such as, under whatconditions can we get certain products in a well-controlledlarge concentration? Can such processes produce some spatialor temporal patterns? Some of the answers may have some

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bearing on the formation of structures in biological systemsand on the theories of evolution.

Just to illustrate the applications, in the following weshall discuss deterministic reaction equations withdiffusion, leaving the stochastic point of view as an "afterthought". Before we get into any specific reactionprocesses, it suffices to say that the time rate of changeof the concentration of a given species can be related tothe rate constants, concentrations of other species, reversereaction rate constants, losses, and diffusion terms, whichcan be written as

dn/dt = D"V2n + g(n). (7.4-1)

For some situations, g(n) may be derived from a potential V,i.e., g(n) = - aV/an. (7.4-2)

When studying the steady state, dn/dt = 0, we want to derivea criterium for the coexistence of two phases, i.e., weconsider a situation in which we have a change ofconcentration within a certain layer. To study the steadystate equation, we may invoke an analogy with an oscillator,or more generally, with a particle under the influence ofthe potential V(n) by noting the following correspondence(just for one-dimension):

x time, V potential, n -+ coordinate.

Now let us consider a reaction-diffusion model with two orthree variables:

A - X, B + X - Y + D, 2X + Y - 3X, X - E,between molecules of the species A, B, D, E, X, Y. Only thefollowing corresponding concentrations enter into theequations of the chemical reaction, a, b, n1, and n2 for A,B, X, and Y respectively. We shall treat a and b fixed,whereas ni and n2 are treated as variables. Thus thereaction-diffusion equations read as:

ant/at = a - (b + 1) n, + n12 n2 + D1V2 n1, (7.4-3)ant/at = bn1 -n12 n2 + D2V2 n2, (7.4-4)

where Di and D2 are diffusion coefficients. For simplicity,let us only consider one spatial dimension. We shall subjectthe concentrations n1 and n2 to two kinds of boundary

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conditions; eithern1(0,t) = nl(l,t) = a, n2(O,t) = n2(l,t) = b/a, (7.4-5)

or n1 and n2 remain finite for x - ± oo. (7.4-6)

One can easily verifies that the stationary state ofEgs.(7.4- 3,4) is given by

nis = a, n2s = b/a. (7.4-7)In order to see whether any new solution classes can occur,i.e., if any new spatial and/or temporal structure mayarise, we can perform a linear stability analysis of thereaction-diffusion equations, Egs.(7.4-3,4). Let

nt = nts + q1; n2 = n2s + q2, (7.4-8)and linearize Egs.(7.4-3,4) with respect to q, Is. It isstraight forward that the linearized equations are

aq1/at = (b - 1) q, + a' q2 + D1a' ql/ax' , (7.4-9)aq2/at = - bq1 - a' q2 + D2a' q2/ax' . (7.4-10)

The boundary conditions (7.4-5,6) require that:q,(0,t) = q,(1,t) = g2(O,t) = g2(l,t) = 0, (7.4-11)

and q1's remains finite for x - ± o.Treating q as a column matrix, then Egs.(7.4-9,10) can

be written asaq/at = Lq, (7.4-12)

where D1 a' /ax' + b - 1 a'L = (7.4-13)

-b D2 a2 /ax' - a'One can satisfy the boundary conditions (7.4-11) by

settingq(x,t) = q exp(git)sin(lrx), with 1 = 1,2,... (7.4-14)

As usual, solving the characteristic equationµ' - CZµ + Q = 0, (7.4-15)

where a _ (-D, ' + b - 1 D2' - a') ,Q = (-D,' + b - 1) (-D2' - a' ) + ba' ,

and Di' = D;1' a' , i = 1, 2.One can easily show that an instability occurs if Re(A)

> 0. For fixed a but changing the concentration b beyond thecritical value b., one can find oscillating solutions aswell as bifurcating solutions.

The above equations may serve as a model for a number of

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biochemical reactions and can also provide someunderstanding, at least qualitatively, theBelousov-Zhabotinski (B-Z) reactions where both temporal andspatial oscillations have been observed [Hastings and Murray1975]. Kuramoto and Yamada [1976] were first to propose thepossibility of turbulence-like behavior of reactantconcentrations in oscillatory chemical reactions. Theirdiscussion was based on the reduced form of B-Z equations.Schimitz, Graziani, and Hudson [1977] recently publishedexperimental data obtained with the B-Z reaction whichshowed evidence of chaotic states. Olson and Degn [1977]presented results on chaos in a biochemical system, thehorseradish peroxidase reaction. Indeed, these studies wereguided by the pioneer work of Rossler who had shown that asimple set of three ordinary differential equations couldproduce chaos [1976]. Consequently, chaos is most likely tobe found in laboratory chemical reactors.

Subsequently, Hudson et al [1979] have shown that anentire sequence of states, some periodic and some chaotic,could be obtained by varying a single parameter, the flowrate or residence time. Similar behavior has since beenconfirmed by Turner et al [1981]. Experimental studies onthe B-Z reactions continue, and the investigation of chaoticbehavior in chemical reactors is quite active. Indeed, evenhigher forms of chaos, i.e., more than one positive Liapunovcharacteristic exponent, is likely to be found in chemicalsystems [Rossler and Hudson 1983]. There have been severalstudies in heterogeneous systems governed by partialdifferential equations.

The experiments were carried out in an isothermalcontinuous stirred tank reactor (CSTR). The reactants arefed more recently by means of precise constant volume pumps.And data are taken with a platinum wire electrode and abromide ion electrode which are connected to a digitalcomputer. Data have been obtained as a function of flowrate, temperature, and feed concentration. We shall limitour discussion to results obtained at a single temperature

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and mixed feed concentrations. A portion of the series ofoscillations with the bromide electrode is shown inFig.7.4.1. The oscillations in la (with the residence time r= 6.76 min) are alternating single and double peaks andthose in lb (r = 6.26 min) are double peaks. Bothoscillations are periodic and stable. Chaotic behavior isobserved at r = 5.89 min in lc. This behavior isreproducible and continues until the external conditions arechanged. This chaos is primarily an irregular mixture of twoand three peaks. Two other regions of chaotic behavior werefound in le (r = 5.63 min.) and lg(r = 5.34 min.). It shouldbe pointed out that the ability of realistic B-Z reactionmodels to generate chaos is not yet completely clear.Indeed, Noyes and his coworkers [1978] have seen onlyperiodic solutions eventhough they have analyzed carefullyat such models. On the other hand, by modifying theseequations, Tomita and Tsuda [1979] and Turner et al [1981]have obtained chaos. Recently Hudson et al [1984] havepresented simulated results of chaos in two single,irreversible, exothermic reaction whose reactors coupledthrough the heat transport. Such a single reaction has beenshown to produce sustained oscillations in a non-adiabaticcontinuous stirred reactor by both experiments andsimulations. It is shown that for two almost identicalreactors, if only one parameter, the heat transfercoefficient governing heat flow between the two reactors, isvaried, the system changes from periodic, throughquasiperiodic, and finally becomes chaotic. Itis also interesting to note that the flow appears to havethe topology of a folded torus such as that found with thedriven van der Pol osicillations [see, e.g., Guckenheimerand Holmes 1983]. The fact that chaos is found in twocoupled tanks indicates that complex behavior may beprevalent in many other systems involvingreaction-diffusion.

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L

0180

e E

I 1 I I I I t I I I I I I I I

270 280 50 60

L

II140 1 1 1 1 1 1 1 1

It 10 min

L

16

0CL

14

12

I"YI

I I I I I I I

1470 1480

Time (minutes)

L

I I I I I I I I

I

I I I I I I I I I I I I I I I 1

N- 10 min +{ 90 100Time (minutes)

Fig. 7.4.1 Summary of the behavior of the Belousov-Zhabotinski reaction with variation of a single parameter; T= 25°C; (a) r = 6.76 min; (b) r = 6.26 min; (c) r = 5.89min; (d) r = 5.85 min; (e) r = 5.63 min; (f) r = 5.50 min;(g) r = 5.34 min; (h) r = 5.28 min. [Hudson et al 1979].

I I I I I I

000180

t E

160

0a140

10 min - Ir- 10 min -+

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Recently, Argoul et al [1987] have done experiments onthe B-Z reaction in a continuous flow reactor which reveal aspiraling strange attractor arises from the interaction of alocal subcritical Hopf bifurcation with a global homoclinicbifurcation. They further point out that the proximity ofthese two bifurcations justifies the application of atheorem by Sil'nikov [1965,1970] which ensures the existenceof chaos. (See Section 6.3). Tam et al [1988] reported thefirst experimental observation of a new type of spatiallyextended open chemical system, the Couette reactor. This isan effectively one- dimensional reaction-diffusion systemwith well-defined boundary conditions. The experimentreveals steady, periodic, quasiperiodic, frequency-locked,period-doubled, and chaotic spatiotemporal states, andqualitatively agrees with the model, and provides someinsight into the physical mechanism for the observedbehavior.

In a series of papers, Gray and Roberts [1988a,b,c,d]and Gray [1988] have developed a complete analysis ofchemical kinetic systems describable by two coupled ordinarydifferential equations and contain at most three independentparameters. They considered the thermally coupled kineticoscillators studied by Sal'nikov.

It should be pointed out that Ohtsuki and Keyes [1987]have utilized a field-theoretic renormalization-group methodto investigate crossover behavior in nonequilibriummulticritical phenomena of one-component reaction-diffusionsystems. An expression for crossover exponents is derivedand mean-field values of them are obtained as a function ofn.

As an application to the reaction-diffusion equation,recently Conrad and Yebari [1989] studied a simple model ofthe dissolution-growth process of a solid particle in anaqueous medium in the stationary case. The resultingnonlinear eigenvalue problem consists of areaction-diffusion equation in the aqueous medium limited byan unknown interface. The various types of bifurcation

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diagrams depending on the nonlinear reaction term aredescribed, and it is also found that more than one solutionexist in general. And chemical dissolution of atwo-dimensional porous medium by a reactive fluid whichproduces a fractal pattern is studied by Daccord [1987]. Aninterpretation of the evolution of the injection pressurewith time which yields the fractal dimension is alsopresented.

A reaction-diffusion equation related to gaslesscombustion of solid fuel has been studied. A formalbifurcation analysis by Matkowsky and Sivashinsky [1978] hasshown that solutions demonstrate behavior typical for theHopf bifurcation. A regorous tretment of this problem isdeveloped by Roytburd [1985]. In order to circumventdifficulties involving a possible resonance with thecontinuous spectrum, appropriate weighted norms areintroduced. A suitable version of the Hopf bifurcationtheorem is developed and the existence of time periodicsolutions is proven.

Under general assumptions, Kopell and Ruelle [1986]studied the temporal and spatial complexity of solutions tosystems of reaction-diffusion equations. The time averagedversions of complexity give upper bounds on entropy andHausdorff dimension of any attracting set.

Recently, Parra and Vega [1988] have considered a first-order, irreversible, exothermic reaction in a bounded porouscatalyst with a smooth boundary of one, two or threedimensions. They considered the cases for small Prater andNusselt numbers, and a large Sherwood number; two isothermalmodels are derived. Linear stability analysis of the steadystates of such models shows that oscillatory instabilitiesappear for appropriate values of the parameters. They havealso carried out a local Hopf bifurcation analysis toascertain whether such bifurcation is subcritical orsupercritical.

For those readers who are interested in the dynamics ofshock waves and/or reaction-diffusion equations, Smoller

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[1983] provides a comprehensive study of these subjects. Amore chemically oriented discussion can be found in Vidaland Pacault [1984].

Numerically, a variety of time-linearization, quasi-linearization, operator-splitting, and implicit techniqueswhich use compact or Hermitian operators has been developedfor and applied to one-dimensional reaction-diffusionequations by Ramos [1987]. It is shown that time-linearization, quasi-linearization, and implicit techniqueswhich use compact operators are less accurate thansecond-order accurate spatial discretizations if first-orderapproximations are employed to evaluate the timederivatives. Furthermore, quasi-linearization methods arefound to be more accurate than time-linearization schemes.Nonetheless, quali-linearization methods are less efficientbecause they require the inversion of block tridiagonalmatrices at each iteration. Comparisons among the methodsare shown in terms of the L2-norm errors. Some improvementsin accuracy are also indicated.

7.5 Competitive interacting populations, autocatalysis,and permanence

As we have seen in Section 1.1, the predator-prey modelof interacting populations, in terms of Lotka-Volterraequations, is rich in structures. In this section, we shallfirst discuss the effect of crowding, then we shall discussin some detail autocatalysis and permanence. Multispeciesand their applications in biochemical reactions will also bediscussed briefly.

Before we discuss any specific situations, it isinteresting to note that many systems of nonlineardifferential equations in various fields are naturallyimbedded in a new family of differential equations. Eachequation belonging to that family can be brought into afactorized canonical form for which integrable cases can beidentified and solutions can be found by quadratures.

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Recently, Brenig (1988) has developed such a technique, andgeneralized multi-dimensional (multi-species) Volterraequations are used as examples to illustrate the power ofsuch an approach.

(i) The effect of crowdingLet us introduce a term representing retardation of

growth due to crowding in the predator-prey problem ofVotka-Volterra equations in Section 1.1. In particular, weconsider the equations

dN,/dt = aN, - bN,N2 - eN12 (7.5-1)dN2/dt = -cN2 + dN1N2.

Again, all the constants a, b, c, d, e are positive.One equilibrium point is N1 = N2 = 0. Another one is N, _

a/e, N2= 0, corresponding to the equilibrium of the logisticgrowth of the prey in the absence of predators. Anyequilibrium point with nonzero values of bothsatisfy:

a - bN2 - eN1 = 0, - c + dN1 = 0.Thus the unique equilibrium solution is

N, = c/d, N2 = (da - ec)/bd.If a/e > c/d, this is a positive equilibrium.

N, and N2 must

Applying the same procedure as in Section 1.1, we canchange the variables

x, = dN1/c, x2 = bdN2/(ad -ce) . (7.5-2)Then it converts Eq.(7.5-1) to

dx1/dt = axe (1 - x2) + fix1(1 - x1) ,dx2/dt = - cx2(1 -x1),

where a = a - ce/d, p = ec/d. For xt > 0, x2 > 0, let usdefine V(x1,x2) = cx1 - c logxl + axe - a logx2.It has a minimum at the equilibrium point (1,1). Indeed, wefind that dV(x1,x2)/dt = - c fi(1 - x1)2 _< 0.Thus, V is a Liapunov function of the system. UsingCorollary 5.2.15, one can show that (1,1) is asymptoticallystable over the interior of the positive quadrant.

Hainzl [1988] studied the predator-prey system a laBazykin [1976) whcih depends on several parameters,

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including the stability of equilibria, the Hopf bifurcation,the global existence of limit cycles, the globalattractivity of equilibria, and the codimension twobifurcations.

Recently, Hardin et al [1988] analyzed a discrete-timemodel of populations that grow and disperse in separatephases, where the growth phase is a nonlinear process thatallows for the effects of local crowding, while thedispersion phase is a linear process that distributes thepopulation throughout its spatial habitat. The issues ofsurvival and extinction, the existence and stability ofnontrivial steady states, and the comparison of variousdispersion strategies are discussed, and the results haveshown that all of these issues are tied to the global natureof various model parameters.

Recently, Tucker and Zimmerman [1988] have studied thedynamics of a population in which each individual ischaracterized by its chronological age and by an arbiraryfinite number of additional structural variables, and thenonlinearities are introduced by assuming that the birth andloss processes, as well as the maturation rates ofindividuals, are controlled by a functional of thepopulation density. The model is a generalization of theclassical Sharpe-Lotka-McKendrick model of age-structuredpopulation growth [Sharpe and Lotka 1911, McKendrick 1926],the nonlinear age-structured model of Gurtin and MacCamy[1974], and the age-size-structured cell population model ofBell and Anderson [1967]. Weinstock and Rorres [1987]investigated the local stability of an equilibriumpopulation configuration of a nonlinear, continuous,age-structured model with fertility and mortality dependenton total population size. They introduce the marginal birthand death rates, which measure the sensitivities of thefertility and mortality of the equilibrium populationconfiguration to changes in population size. They have foundthat in certain cases the values of these two parameterscompletely determine the stability classification.

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(ii) Autocatalytic reaction:Recently, some experimental investigations were

undertaken in studying competition, selection and permanencein biological evoluation and molecular systems far fromequilibrium. Polynucleotides, such as RNA and enzymes ofsimple bacterio- phages are used in these experiments[Biebricher 1983]. Polynucleotides, strangely enough, havean intrinsic capability to act as autocatalysts built intotheir molecular structures. A combination of anautocatalytic reaction, a degradation reaction and arecycling process was found to be adequate in representingan appropriate mechanism for modeling such system. Weconsider the open system

A + X ,-c 2X, X d,"d B, B r(E)-+ A. (7.5-3)The rate constant of the recycling reaction r(E) isdetermined by an external energy source E. A simple exampleof such a process is a photochemical reaction using a lightsource. The dynamics of the mechanism (7.5-3) can bedescribed by

da/dt = rb + c'x' - cax, (7.5-4a)

db/dt = dx - (d' + r)b, (7.5-4b)

dx/dt = cax + d'b - c'x' - dx, (7.5-4c)

where we denote the concentrations of A, B and X by a, b, xrespectively. A trivial constant of motion is the totalconcentration of all substance, i.e.,

a + b + x = co = constant. (7.5-5)

Thus we are left with a two-dimensional system defined onthe state space:

S = ((a,b,x) a R+3: a+b+x=co).Clearly the fixed points of Egs.(7.5-4) are

P. = (bo,x,) _ (0,0), (7.5-6a)

P, _

_(b,,x1)

[cco(d'+r)-dr]/[(c+c')(d'+r)+cd] (d/(d'+r),1).(7.5-6b)

If co < Ccrit = dr/ [c (d'+r) ] , then P, it R+3, i.e., outside thephysically relevant state space S. At co = ecrit, P, = PO andenters S for co > ccrit. Local stability analysis shows that

362

Pi is asymptotically stable and PO is a saddle for co > cor;t,and P. is a sink for co < ccr;t

In order to find global stability, we apply the Dulac-function, x'1, to the vector field, Egs.(7.5-4), and obtain:X-' db/dt = (db/dt) = d - (d'+r)b/x, (7.5-7a)

X-1 dx/dt = (dx/dt) = c(co - b - x) + d'b/x - c'x - d.(7.5-7b)

These equations (or vector field), Egs.(7.5-7), have astrictly negative divergence on S. Thus the flow (7.5-7) isarea contracting, consequently, periodic orbits are notpossible. It is appropriate to recall and restate the famoustheorem of Poincare-Bendixson (Theorem 4.3.1): A nonemptycompact fl- or a- limit set of a planar flow, which containsno fixed points, is a closed orbit. Thus, the Poincare-Bendixson theorem implies that the stable stationarysolutions Po or Pi are indeed globally stable; everysolution starting in S converges to the stable fixed point.

If the degradation is irreversible, i.e., d' = 0, globalstability of Pi can also be proved by means of the Liapunovfunction:

V = c(b - b1)2 + 2d (x - x, log x), (7.5-8)

where (b1,xl) are the coordinates of P1, given by (7.5-6b).The autocatalytic reaction mechanism described by

(7.5-3) admits only two qualitatively different types ofdynamical behavior. The plane of external parameters (r,co)is split into two regions corresponding to the stability ofeither PO or P, (Fig. 7.5.1)

P

PO

co = ccrit

t

YFig.7.5.1

363

If all reactions are irreversible (c'= d'= 0), a conditionwhich is approximately true for most biochemical andbiological systems, the concentration of A at the stationarysolution Pi and the critical value ccrit coincide and they areindependent of r. The irreversible autocatalytic processesconsume the reactant A to the ultimate limit, i.e., Pi wouldbecome unstable if the concentration of A went below ccrit'

(iii) Multispecies - competition between autocatalyticreactions

Now consider a network of 2n+1 reactions: n first-orderautocatalytic reactions followed by n degradation processesand coupled to a common irreversible recycling reaction,which is again controlled from the outside, which allowsdriving the system far from equilibrium:

A + X( ci."°: 2X1, i = 1, 2,..., n (7.5-9a)X; d, '12d- B, (7.5-9b)

B ' r(E)- A. (7.5-9c)One may consider this reaction scheme as competition betweenn autocatalysts for the common source of material A forsynthesis. For the reversible reactions (7.5-9a,b), there isa thermodynamic restriction of the choice of rate constantsdue to the uniqueness of the thermodynamic equilibrium:

b/a = cid1/c1'di' for all i = 1,....n,and a and b are equilibrium concentrations.

The dynamics of mechanism (7.5-9) again can be describedby:

da/dt = rb + Ei c1'x,2 - Ei ac,x,, (7.5-10a)db/dt = Ei d;xi - (r + E, d1')b, (7.5-10b)

dxi/dt = x,(c1a - c,'x, - d1) + d1'b. (7.5-10c)

Of course, we still have the constant of motion,a + b + E1 xi = co = constant.

The fixed points are readily computed for the case ofirreversible degradation, i.e., d1' = 0. There are 2" fixedpoints of Egs.(7.5-10). A convenient notation whichcorrespond to the nonvanishing components is: P; is thefixed point with x, + 0 and xk = 0 for k + i; P,, is the

364

fixed point with xi 0, x + 0, and xk = 0 for k + i, j;

etc. See Fig.7.5.2.

Fig. 7.5.2

Without a loss of generality, we can arrange the indicesin such a way that d1/c1 < d2/c2 < d3/c3 < ... <d,/cn. It isstraightforward to show from Egs.(7.5-10) that the presenceof species X; at a stable stationary point implies also thepresence of species Xj with j < i. Consequently, all fixedpoints different from P0, P1, , P,, P121 I Pn-1,nfP1231...IP12 n

are unstable irrespective of the values of theexternal parameters r and co. Indeed, only one of the fixedpoints listed above is stable depending on r and c0: if ci'> 0, then a sequence of double point bifurcations of thetype P12 ; P1z ;,;+1 with increasing value of co. At thepoint of coincidence, a change in stability takes place anda new species is introduced into the system at the stablestationary state. The larger the environment, which cansustain a larger total concentration c0, the more speciescan coexist. A similar phenomenon has been found for thecompetition of self-replicating macro-molecules inLotka-Volterra food chain [So 1979; Gard and Hallman 1979],or based on Michaelis-Menten type kinetics [Epstein 1979].

When the autocatalytic reactions are also irreversible,i.e., c1' = 0, a condition usually met by ecologicalsystems, and if the values of the quotients d,/ci aredistinct, then only PO and the one species fixed points, P1,

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P2, ... , Pn exist at finite values of co. P1, the steady statecorresponding to the lowest value of d1/c1, is stable for co> d1/c1. In this case we observe the selection of thefittest species, which is characterized by the smallestquotient of degradation rate over the replication rate.

Theorem 7.5.1 For an irreversible degradating dynamicalsystem (7.5-10) (d;'= 0, for all i e [l,n]), if there is nostationary point in the interior of the state space, i.e.,P1Z n does not exist, then at least one species dies out.

Proof: Suppose the system of linear equationsx,-' dxi/dt = c1a - c,'xj - d1 = 0, i e [l,n]db/dt = E1 d;x1 - rb = 0,

has no solution (a, b, x1,..., xn) with positive entries,then a well-known convexity theorem ensures the existence ofreal numbers q, (i a [O,n]), such that

go(E1 d1xi - rb) + E, q,(c,a - c,'x, - d-) > 0,

for all a, b, x; > 0, i e [l,n]. A Liapunov functionV = q,b + E1 q; log x1, for x, > 0,

satisfies dV/dt = q db/dt + E1 xi"1 q1 dxj/dt > 0.Thus, all orbits converge to points where at least one ofthe concentrations xi vanishes.

May and Leonard [1975] have shown that for threecompetitors, the classic Lotka-Volterra equations possess aspecial class of periodic limit cycle solutions, and ageneral class of solutions in which the system exhbitsnonperiodic population oscillations of bounded amplitude butever increasing cycle time. As another example of anapplication of a theorem by Sil'nikov [1965], a criterion isobtained which allows one to construct one-parameterfamilies of generalized Volterra equations displaying chaos(Arneodo, Coullet and Tresser 1980].

Another example of a Lotka-Volterra time dependentsystem is the parametric decay instability cascading withone wave heavily damped [Picard and Johnston 1982]. This candisplay Hamiltonian chaos even without ensemble averaging.It seems likely that many such systems can be nearly ergodicwithout the necessity of ensemble averaging.

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Recently, Gardini et al [1987] have studied the Hopfbifurcation of the three population equilibrium point andthe related dynamics in a bounded invariant domain. And thetransitions to chaotic attractors via sequences of Hopfbifurcations and period doublings are also discussed.

Recently, Beretta and Solimano [1988] have introduced ageneralization of Volterra models with continuous time delayby adding a nonnegative linear vector function of thespecies and they have found sufficient conditions forboundedness of solutions and global asymptotic stability ofan equilibrium. They have considered a predator-preyVolterra model with prey-refuges and a continuous time delayand also a Volterra system with currents of immigration forsome species. Furthermore, a simple model in which twopredators are competing for one prey which can take shelteris also presented.

Goel et al [1971] studied many species systems andconsidered the population growth of a species by assumingthat the effect of other species is to introduce a randomfunction of time in the growth equation. They found that theresulting Fokker-Planck equation has the same form as theSchrodinger and Block equations. They have also shown thatfor a large number of species, statistical mechanicaltreatment of the population growth is desirable. They havedeveloped such treatment.

(iv) Permanence:In 1975, Gilpin [1975] showed that competitive systems

with three or more species may have stable limit cycles asattractor manifolds, and he argued that it would bereasonable to expect to find this in nature.

The question of whether all species in a multispeciescommunity governed by differential equations can persist forall time is one of the fundamental ones in theoreticalecology. Nonetheless, various criteria for this propertyvary widely, where asymptotic stabilty and globalasympototic stability are two of the criteria most widely

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used. But neither of these criteria appears to reflectintuitive concepts of persistence in a satisfactory manner.The asymptotic stability is only a local condition, whilethe global asympototic stability rules out cyclic behavior.Hutson and Vickers [1983] argued that a more realisticcriterion is that of permanent coexistence, whichessentially requires that there should be a region separatedfrom the boundary (corresponding to a zero value of thepopulation of at least one species) which all orbits enterand remain within.

Putting it differently, an interesting populationdynamical system is called permanent if all species survive,provided, of course, they are present initially. Moreprecisely, a system is permanent if there exists some levelk > 0 such that if xi(0) > 0 for all i = 1,..,n, then x,(t)> k for all t > T > 0. This property was calledcooperativity (e.g., Schuster, Sigmund, and R. Wolff 1979].Schuster et al [1980] has also discussed self replicationand cooperation in autocatalytic systems. It was used in thecontext of molecular evolution where polynucleotideseffectively helped each other through catalytic interactionsas we have just discussed. The notation applies equally wellin ecology, but semantically it is awkward to speak ofcooperative predator-prey communities. Moreover, Hirsch[1982] has defined a dynamical system dx/dt = f(x) ascooperative if aft/axe > 0 for all j + i, which seems to bemore an appropriate usage of this term.

Another equivalent way of defining a permanent system isto postulate the existence of compact set C in the interiorof the state space such that all orbits in the interior endup in C. Since the distance from C to the boundary of thestate space is strictly positive, this implies that thesystem is stable against fluctuations, provided that thesefluctuations are sufficiently small and rare; indeed, theeffect of a small fluctuation upon a state in C would not beenough to send it all the way to the boundary and would becompensated by the dynamics which would lead the orbit of

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the perturbed state back into C. The details of what happensin C is of no relevance in the context of permanence. In amore mathematical term, one can also define a permanentsystem as the one admitting a compact set C in the interiorof the state space such that C is a globally stableattractor.

In ecology or biology, there is a difference between thefollowing two terms. Roughly speaking, a community ofinteracting populations is permanent if internal strifecannot destroy it, while it is uninvadable if it isprotected against disturbance from without. Certainly,uninvadability is not a property per se, one has to specifywhich invaders the community is protected against.

Let x1,..., xm be the densities of establishedpopulations and xmt,,..., xm be the densities of potentialinvaders. Then the established community is uninvadable if(a) it is permanent, and (b) small invading populations willalso persist. Mathematically, the (x,,...,xm) community isuninvadable inthe (x1,...,xm) state space (where n > m), ifthere exists a compact set C in the interior of the(xi,...,xm) subspace, which is a stable attractor in the(xl,...,xn) space, and even a globally stable attractor inthe (xJ,...,xm) subspace. There are several results onpermanence.

Theorem 7.5.2 A necessary and sufficient condition foran interacting dynamical system to be permanent is theexistence of a Liapunov function [Gregorius 1979].

The above theorem was obtained by introducing theconcept of repulsivity of certain sets with respect todynamical systems defined on metric spaces.

An important necessary condition for permanence is theexistence of an equilibrium in the interior of the statespace. This is just a consequence of the Brouwer's fixedpoint theorem (Corollary 2.4.18). Hutson and Moran [1982]have proved the necessary conditions for the discretesystems, and Sieveking has proved for the continuoussystems.

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Another useful condition of the permanence of continuoussystems of the type dx1/dt = x1F1(x) was given by Hofbauer[1981]. If there exists a function L on the state space Swhich vanishes on the boundary as and is strictly positivein the interior, and if the time derivative of L along theorbits satisfies dL/dt = Lf, where f is a continuousfunction with the property that for any x on as, there is T> 1 with

1/TS0 f(x(t))dt > 0, (7.5-11)

then the system is permanent. Note that, if f in Eq.(7.5-11)is strictly positive on as, then Eq.(7.5-11) is alwayssatisfied and L is just a Liapunov function. ThenEq.(7.5-11) means that L is an "average Liapunov function".

Hutson and Moran [1982] showed that a discrete system(Tx)1 = x.F1(x) is permanent if there exists a non-negativefunction P on S which vanishes exactly on as and satisfies

supk>o limrx Y,1 inf P(Tky)/P(y) > 1 for all x e as.

Hutson and Moran [1982] have studied the discreteequations given by an analogue of the Lotka-Volterra model,

(Tx)1

= x1 exp (b1 - a11x1 - a12x2) ,(Tx)2 = x2 exp(-b2 + a21x1 - a22x2),

with all b1 and a,, > 0. They have shown that the system ispermanent iff it admits a fixed point in the interior of S.This is similar to the continuous Lotka-Volterra case.Nonetheless, for the continuous case, permanence impliesglobal stability, but the discrete case allows for morecomplicated asymptotic behavior. Sigmund and Schuster (1984]offers an exposition of some general results on permanenceand uninvadability for deterministic population models.

In a closely related discussion, Gard and Hallam [1979]have discussed the persistence-extinction phenomena and haveshown that the existence of persistence or extinctionfunctions for Lotka-Volterra systems, which are Liapunovfunctions, then the systems are persistent or extinctrespectively.

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Hofbauer and Schuster [1984], Hofbauer et al [1979,1981]have shown that in a flow reactor, hypercyclic coupling ofself- reproducing species leads to cooperation, i.e., noneof the concentrations will vanish. Yet autocatalyticself-reproducing macromolecules usually compete, and thenumber of surving species increases with the totalconcentration.

Before we end this subsection and move on to higherorder autocatalytic systems, let us briefly mention theexclusion principle, which states roughly that n species cannot coexist on m resources if m < n. This can easily beproved for the case that the growth rates are linearlydependent on the resources. But for more general types ofdependence, this principle is not valid. As an exercise,prove this statement.

For example, Koch [1974] presented a computer simulationshowing that two predator species can coexist on a singleprey species in a spatially homogeneous and temporallyinvariant environment. This result was then interpreted inthe context of Levin's extended exclusion principle [Levin1970]. Armstrong and McGehee (1976] provided a differentinterpretation of the relation of Koch's work to Levin's.

(v) Second-order autocatalysis:In order to demonstrate the enormous richness of the

dynamics of higher order catalytic systems, we now study thetrimolecular reaction coupled to degradation and recyclingin analogy to Egs.(7.5-3):

A + 2X ,--` 3X (7.5-12a)X d B (7.5-12b)

B r - A. (7.5-12c)The corresponding differential equations are:

da/dt = rb + c'x3 - cax2, (7.5-13a)

db/dt = dx - (d' + r)b, (7.5-13b)

dx/dt = cax' + d'b - c'x3 - dx, (7.5-13c)

and together with the conservation equation:a + b + x = co = constant.

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For simplicity, we only consider the irreversibledegradation case, i.e., d' = 0, nonetheless, the resultsobtained are also valid for d' > 0. Note that the fixedpoint Po(a = co, b = x = 0) is always stable. The two otherfixed points P and Q are positioned at:(p, q) = {coc ± [co' c2 - 4d(c + c' + cd/r) ]W')/2 (c + c'+ cd/r).Note that p > q, as p grows and q shrinks with increasingco. It is easy to show that Q is always a saddle point, andP either a sink or a source. So, for co < csad = [4d(c + c' +

cd/r) ]y'/c, PO is the only fixed point, and as co crosses csadfby a saddle-node bifurcation, P and Q emerge into theinterior of S.

To determine the stability of P, we evaluate the traceof the Jacobian, A, at P:

trA(x=p) = d - r - p2(c + c'), (7.5-14)

which is a decreasing function of co. If at Co - Csad'Eq.(7.5- 14) is negative, then P is stable for all co > csad'This corresponds to a direct transition from region I toregion V and it always happens if d <- r.

Co

1

r

Fig. 7.5.3.

Analogously to the earlier discussion, now usingx"2

as theDulac function, and using the Poincare-Bendixson theorem toexclude the existence of periodic orbits, one concludes thatall solutions have to converge to one of the fixed pointsPo, P, or Q. If the numerical value of trA of Eq. (7.5-14) atCo = csad is positive, the fixed point P is an unstable node.

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With increasing co, the eigenvalues of A become complex. Butsince trace A is decreasing, the eigenvalues have to crossthe imaginary axis at co = cH where a Hopf bifurcation takesplace. The value of cH can be obtained from Eq.(7.5-14) bysetting trace A(x = p) = 0. Marsden and McCracken [1976]showed that this Hopf bifurcation is always subcritical.This means that the bifurcating periodic orbit is alwaysunstable and occurs for co > cH, at such value of co, P isstable. A further increase of ca leads to growth of theperiodic orbit. At co = cBS, the periodic orbit includes thesaddle point Q. Thus, the periodic orbit changes to ahomoclinic orbit and disappears for co > cBS. This phenomenonis called "blue sky bifurcation" [Abraham and Marsden 1978].

When co > cBS, orbits starting from the boundary of Shave a chance to converge to P. At values close to cBS, butlarger than cBS, the admissible range of initialconcentration x(0) for final survival of the autocatalyst isvery small. Indeed, if x(0) is too small or too large , theorbit will converge to P0 and the autocatalyst will die out.For much larger values of c0, the basin of attraction of Pbecomes almost the entire S because the saddle Q tendstoward P0 for co -.

Recently, Kay and Scott [1988] have studied the behaviorof a third order autocatalytic reaction diffusion system.They have found that for indefinitely stable catalysts, themodel exhibits ignition, extinction and hysteresis, and therange of conditions over which multiple stationary statesare found to decrease as the concentration of theautocatalyst in the reservoir increases. They have alsofound that the final ignition and extinction points merge ina cusp catastrophe with the consequent loss of multiplicity.On the other hand, with a finite catalyst lifetime, thedependence of the stationary state composition on thediffusion rate or the size of the reaction zone shows morecomplex patterns. The stationary state profile for thedistribution of the autocatalytic species now allowsmultiple internal extrema, that is the onset of dissipative

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structures. And the cubic autocatalytic system also providesthe simplest, yet chemically consistent, example of temporaland spatial oscillations in a reaction-diffusion system.

We would like to mention that the existence of twostable solutions on S is a common and well-known feature ofhigher order autocatalyic reactions. It is also striking tonote that higher order autocatalytic reaction networks arevery sensitive on initial conditions. In the followingparagraphs, we shall briefly discuss competition betweenhigher order autocatalytic reactions. As in Eq.(7.5-9), wehave

A + 2X1 cy,,"C 3Xi, Ii d;- d; B, B r-+ A, 1 E [1,n].For simplicity, let us just discuss the case withcompetitors, i.e., n = 2, and irreversible degradation (di'=0). Then the dynamical equations are:

db/dt = d1x1 + d2x2 - rb, (7.5-15a)dx1/dt = x1(c1ax1 - c1'x12 - d1) , (7.5-15b)dx2/dt = x2 (c2ax2 - c2' x22 - d2) , (7.5-15c)

with a=c0-x, -x2-b.Again, the fixed points Po(x1 = xz = b = 0) is stable.

Furthermore, any xi which is very small is characterized bydx1/dt < 0, thus will tend toward zero. Contrary to thecompetition between linear autocatalytic system Eq.(7.5-10),all invariant surfaces of S are attracting. On both planes,i.e., x1 = 0 and x2 = 0, we observe all the behaviordiscussed earlier. For instance, we obtain two stable fixedpoints P1(x, > 0, xz = 0) and P2("I = 0, xZ > 0). Increasingco further for ci'> 0, we observe the creation of two pairsof fixed points in the interior of S. If co is large enough,one of the fixed point P12(5C1 > 0, xz > 0) will be stable.Nonetheless, the various bifurcations occurring withincreasing co are diffcult to analyze and one may expectchaotic behavior near the Hopf and homoclinic bifurcations.In the case of ci' = 0, only one pair of fixed pointsemerges in the interior of S. These are always a source anda saddle point. There is no stable fixed point in S even forlarge co. This indicates that in the irreversible case, all

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orbits converge to one of the planes x, = 0 or x2 = 0. Thus,at least one of the competitors has to die out, just as wehave found in the linear case, there is no region ofcoexistence of the two competitors when the autocatalyticprocesses are irreversible.

For further discussion on autocatalysis, hypercycles,and permanence, see for instance, the papers in Schuster[1984], Freedman and Waltman [1977], Epstein [1979], Eigenet al [1980]. For more quantitative results concerningcompetitions between three species, see the classical paperby May and Leonard [1975].

Even though the predator-prey model of interactingpopulations in terms of Lotka-Volterra equations and theautocatalysis equations in terms of reaction-diffusionequations are mainly for the study of population ecology andbiochemical reactions, it should be pointed out that thelaser equations for a two-level system have the exactly sameset of differential equations as the original Lotka-Volterraequations. Likewise, some problems in semiconductor physicshave very similar differential equations as theLotka-Volterra equations or the autocatalysis equations weshall see in the next section.

7.6 Examples in semiconductor physics and semiconductorlasers

By looking at the dynamical equations of thepredator-prey problem with crowding and autocatalysis, onecan easily recognize their formal similarities orequivalence to some of the problems in semiconductor physicsor semiconductor laser physics. In this section, we want toshow that the formalism and results obtained in Section 5can readily be applied to some situations in semiconductorphysics and semiconductor lasers.

(i) Limit cycle for excitons:There has been some work on the possibility of

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spontaneous oscillations of the electron concentration andtemperature due to nonlinear dependence of the Augercoefficient on temperature [Degtyarenko, Elesin and Furmenov1974]. Nonetheless, this model does not contain anautocatalytic process, here we consider such processfollowing Landsberg and Pimpale [1976] and Pimpale et al[1981] in a model with the following three steps:stimulated production of excitons: e + h + X `- 2x,radioactivedecay of excitons at recombination center: X d-r, photogeneration of carriers: r s- e + h. It is very clearthat these processes are irreversible first orderautocatalytic processes as in Eq.(7.5-3) where (e + h) = A,r = B, c'= d'= 0. The irreversibility is because thesemiconductor is assumed to be far from equilibrium. Here weassume that the semiconductor to be nondegenerate and underhigh excitation so that the electron and hole concentrationsare taken to be equal. Then the differential equationsdescribing the dynamical system are (one can obtain fromEq.(7.5-4) too):

dn/dt = gb - Cn'x, (7.6-1a)

dx/dt = Cn'x - dx. (7.6-1b)

The conservation law of total concentration and the otherdynamic equation (reduced form of Eq.(7.5-4b)) are takeninto account. Note that, the exciton decay process hasinteresting nonlinear features: the decay rate isproportional to the number of recombination centersavailable where there is a large number of excitonsavailable for decay, i.e., for x >> 1/q, dx/dt -1/q where1/q is concentration of recombination centers. But when thenumber of excitons is much lower than the number ofrecombination centers, the exciton decay rate isproportional to the exciton concentration, i.e., for x <<1/q, dx/dt = - x. Since the Michaelis-Menten S-shaped lawseems to be a good interpolation law for the exciton decayrate, we then replace Eq.(7.6-1b) by

dx/dt = cn'x - dx/(1 + qx). (7.6-lb')The exciton decay rate on recombination centers, given by

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the second term of Eq.(7.6-lb'), is shown qualitatively inFig.7.6.1.

------------------------- d/q

X

Fig. 7.6.1

For small x, (xq <<1), one has the usual exponential decay.But for large x, there is a departure from the usualexponential decay. Experimentally, such a departure has beenobserved in qualitative agreement with Fig. 7.6.2(Klingenstein and Schmod 1979]. A generalization to an n-thorder reaction is also possible (Ibanez and Velarde 1977],but we shall not discuss it any further. Note, here we havealso ignored electron-hole, electron-impurity recombinationsand other excitonic processes.

(0,0)

Fig. 7.6.2

G

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We can rescale the dimensions in such a way, (as anexercise) such that Eq.(7.6-la,b') becomesdn/dt = G - cn2x, dx/dt = cnx2 - x/(l + x). (7.6-2)

For physically meaningful solutions, we have the followingconstraints:

x >- 0, n >- Of G > Of c > 0.As in section 7.5-(ii), there are two fixed points:

Po(x = 0, n = -) and P, (x = xo, n = n0)where xo = G/(1 - G), no=- J(1 - G)/cIK and 0 < G < 1.Clearly, Po is unstable, by noting that changing variable y= 1/n, Po(x = 0, y = 0) is unstable with respect to evenlinear perturbation. To get a feel of the stability of P1,let

x(t) = xo + u(t), n(t) = no + v(t),then Eq.(7.6-2) becomes

d/dt lul = A [u] ' (7.6-3)

where G(1 - G) 2G(c/(1 - G))Y' 1

A = L-(1 - G) -2G(c/(l - G))"Note that det A = 2G (1 - G) 3i2cv. > 0 for physicallymeaningful values of G and c. And, the Poincare index of thefixed points is +1. Thus the stability of Pi is determinedby the value of Tr A, where Tr A = G(l - G) - 2[c/(l -G)]Y'. The bifurcation values of G and c are given by Tr A =0, i.e., (1 - G)3 = 4c. Since (0,0) is unstable, it appearsthat the fixed point is stable for Tr A < 0 and unstable forTr A > 0. The unstable steady state leads to oscillation.See Fig.7.6.2.

Another interesting phenomenon is the Gunn effect, orGunn instability [see for instance, Seeger 1982] in asemiconductor having a two-valley conduction band, such asGaAs, is caused by the negative resistance arising from thetransfer of electrons in high fields from a lower light-massvalley to an upper heavy-mass valley. It has been determinedthat the Gunn instability gives rise to anomalously largelow-frequency current noise over a wide range of fieldsbeyond the threshold of the instability. Recently, Nakamura

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(1988) has performed carrier dynamics simulation toinvestigate the low-frequency current noise mechanism in atwo-valley semiconductor. It has been noted that the currentnoise is caused by the chaotic transfer of carriers in theaccumulation layer of a high-field domain.

7.7 Control systems with delayed feedbackRoughly speaking, the purpose of control is to

manipulate the available inputs of a dynamical system tocause the system to behave in a more desirable manner thanthe one without control. Almost every movement our bodymakes in our daily life is under control. In fact, most ofthem involve feedback of various means. This can also easilygeneralize to our physical world.

There are two major types of control. In open-loopcontrol, the input is generated by some processes externalto the system itself, and then is applied to the system. Aspecific input may be generated by analysis such asdeveloping the yearly production plan for a company, or maybe generated repeatedly by a physical device when directinga physical parocess or machine such as programmable switchon and off of a heating system. At any rate, the underlyingfeature of open-loop control is that the input function isdetermined completely by an external process. In closed-loopcontrol, the input is determined on a continuing basis bythe behavior of the system itself as expressed by thebehavior of the outputs. This is also called feedbackcontrol because the outputs are fed back to the input. Thereare many reasons why feedback control is often preferable toopen-loop control. One is that a feedback rule is oftensimpler then a comparable open-loop scheme because it mayrequire a fair amount of computation and compleximplementation. It is very important that feedback canautomatically adjust to unforeseen system changes or tounanticipated input disturbances. One of the other importantreasons is that much feedback can rapidly adjust to changes

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leads to the fact that feedback can increase the stabilityof the system. In this respect, mathematically, a basicfeature of feedback is that it can influence thecharacteristic polynomial of the system.

Let us consider the effect of feedback applied to a verysimple system, dx(t)/dt = u(t). One can easily show thatthis system is marginally stable. Now suppose the input iscontrolled by feedback and suppose u(t) = o(xo - x(t)). Thissystem certainly yields x(t) - x0. Note that by defining y =x - x0, the feedback system is governed by the simpleequation dy(t)/dt = -ay(t), and the variable y(t) is theerror. Thus, feedback has converted the original marginallystable linear system into an asymptotically stable system.As stated before, an important objective of feedback is tomake the system more stable than it would otherwise be.Therefore, it is natural to ask how much influence feedbackcan have on the eigenvalues of a system. An important resultis the eigenvalue placement theorem. This theorem statesthat if a system is completely controllable and if all statevariables are available as outputs, then by suitable directfeedback it is possible for the feedback system to have anydesired characteristic polynomial.

Before we continue on nonlinear feedback controlsystems, we would like to introduce the result of Sontag[1982] which deals with the necessary and sufficientconditions for asymptotic controllability of a controllablesystem. The system under consideration is of the followingtype:

dx(t)/dt = f(x(t),u(t)), (7.7-1)

where the states x(t) a R", control parameter u(t) is in ametric space X, and the map f is assumed locally Liptchitzto ensure the existence and uniqueness of solutions forsmall time intervals. Furthermore, admissible controlfunctions u(t) are all measurable and locally boundedfunctions, and X is topologized by the weak topology. Thesystem Eq.(7.7-1) is asymptotically controllable if thefollowing conditions are satisfied: (i) for each state x

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there exists a control for which the corresponding solutionO(x,u,t) is defined for all t, and converges to zero; (ii)if for a given e > 0, there is a 8 > 0 such that for any xof norm less than 6, there is a u as in (i) with the ensuingorbit having UUx(t)UU < e for all t; (iii) there is aneighborhood U of the zero state, and a compact subset K ofX such that if x(0) is in U, there exists an input as in(ii) with values in K almost everywhere. Then we have thefollowing theorem:

Theorem 7.7.1 The system, Eq.(7.7-1), is asymptoticallycontrollable iff there is a Liapunov-like function for it.

Here a Liapunov-like function for Eq.(7.7-1) is a realfunction V defined on R" such that:(i) V is continuous;(ii) V(x) > 0 for x + 0, V(0) = 0;(iii) (xl V(x) < a) is bounded for all a;(iv) for each x + 0 there is a relaxed control u withV'(x,u) < 0;(v) there is a neighborhood L of the zero state and acompact subset K of U such that, for states x in L, thecontrol u can be chosen with values in K (more precisely,the measures u(t) are supported in K almost everywhere).The derivative along the chosen trajectory is defined as:VI(x,u) = limt_0 inf[(V(0(t,x,u)) - V(x))/t7.

In practice, the value of u is determined after ameasurement on the state x(t) at t was made. If thismeasurement is in error, say x(t) + e(t) is measured ratherthan x(t), the governing equation of motion will have theform

dx(t)/dt = X(x(t) + e(t)). (7.7-2)

It is this concept which leads to the definition ofstability with respect to measurement. Intuitively, X isstable with respect to measurement if any solutions ofEq.(7.7-2) and dx(t)/dt = X(x(t)), satisfying the sameinitial conditions, remain arbitrarily close over any finitepositive time interval whenever the supremum of Ie(t)I overthis time interval is restricted to be sufficiently small.

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In general, the initial value problem for such a vectorfield X may not have a solution. Nonetheless, if there is anabsolutely continuous function 0(t) which satisfies theinitial condition and do(t)/dt = X(O(t)) almost everywhere,then 0(t) is called a classical solution.

Consider a control system of the formdx/dt = g (x, u(x) ) , x = (XI , X2, ... , X") , u = (uj, . . . , U")

(7.7-3)

with values u(x) to be chosen from a control set U. Let thetarget manifold S be contained in (0,-)xR". If g is boundedand Lipschitzian and u is a given Lipschitzian control, thenan initial value problem of Eq.(7.7-3) with data x(O) = x°has a unique solution, with value at time t denoted by0(t,0,x°). Assume that 0(t,0,x°) e S. Then the question is:If S has dimension less than n in R"'', is it possible that,for each x in some neighborhood N(x°) c R" of x°, thereexists a value t(x), and 0 <_ t(x) < °°, such that 0(t(x), 0,

X) E S?Before we address this question, we shall briefly

discuss the generalized concept of solutions for suchequations as given by Filippov [1964]. Let X be a measurablefunction defined almost everywhere in a domain Q c R" withvalues in a bounded set in R. With X(x) associate theconvex set

K(X(x)) = n6,o nu(N)=o closed convex hull{X(U(x,6) - N),where U(x,d) is a closed 6 neighborhood of x, N an arbitraryset in R" and µ is n dimensional Lebesgue measure. Then anabsolutely continuous vector valued function ¢, defined on[0,T], is called a Filippov solution of dx/dt = X(x) if foralmost all t, dO(t)/dt c K{X(O(t))). In Filippov [1964], ithas been shown that such solutions will always exist, andmany of their properties are discussed. In particular, if Xis continuous, then K(X(x)) = X(x).

Let U(x,d) denote a compact, spherical neighborhood ofradius 6, about the point x in R", and coA denotes theconvex hull of a set A. A vector field X, for which aclassical solution 0 of dx/dt = X(x) with arbitrary initial

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data x° exists, is said to be stable with respect tomeasurement if given e > 0 and finite T > 0, there exists a6 > 0 such that whenever e is a measurable function on [0,T]with values in R" and norm less than 6 for which acorresponding solution µ of dx/dt = X(x(t) + e(t)), x(0) _x°, exists on [0,T], then 110 - µ1l < E. Hermes [1967] hasestablished the following theorem:

Theorem 7.7.2 If X is stable with respect tomeasurement, then every classical solution is a Filippovsolution.

If X is stable with respect to measurement, solutionsfor t >_ 0 of the initial value problem for the correspondingdifferential equation are unique, and such a solution, whenevaluated at a fixed positive time, varies continuously withthe initial data. Thus, with increasing time, solutions mayjoin but not branch. Thus, it is felt that feedback controlswhich are meaningful from the standpoint of applicationsshould lead to vector fields which are stable with respectto measurement. Nonetheless, to characterize such vectorfields directly is not a easy task. In the next section, weshall illustrate such a system in feedback control ofsemiconductor lasers and phased arrays.

For linear difference-differential equations of retardeddelay feedback, Infante [1982] has outlined acharacterization of a broad class of such difference-differential systems which have the property that theirasymptotic stability and hyperbolicity characteristics arenot affected by the value of the delays involved.

Recently, an der Heiden and Walther [1983] and an derHeiden [1983] have shown that under centain conditions, thedifference-differential equation

dx(t)/dt = f(x(t-1)) - ax(t), (7.7-4)

describing delayed feedback control systems, admits chaoticsolutions.

Recurrent synaptic feedback is an important mechanismfor regulating synchronous discharges of a population ofneurons. It can be found in the brain, the the spinal chord

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and in the sensory systems. Recurrent feedback occurs whenactivity in a population of neurons excites, via axoncollaterals, a second population, which in turn excites orinhibits the first. Models that include recurrent feedbackfall into two main categories. The first consists of modelsof a specific part of the nervous system where recurrentfeedback is important, while the models in the secondcategory are developed specifically to describe resurrentfeedback without reference to a particular part of nervesystem. Nonetheless, both types include time delays due tothe conduction times and synaptic delays in the feedbackcircuit. The model of Mackey and an der Heiden [1984]incorporates the stoichiometry of synaptic transmitter-receptor interactions and time delays and leads to anonlinear term for recurrent feedback. An even simpler modelfor recurrent feedback is that of Plant [1981], which is amodification of the FitzHugh-Nagumo equations for a nervecell [FitzHugh 1961] emphasizes the behavior of anindividual cell membrance. Recently, Castelfranco and Stech[1987] provide a more comprehensive understanding of thestructure of periodic orbits in Plant's model and viewed thesystem as a two-parameter Hopf bifurcation problemassociated with delayed feedback in the FitzHugh model.

Ikeda et al [1982] have studied delayed feedback systemsin detail and they have found that higher-harmonicoscillating states apear successively in the transition tochaos. It has been pointed out that the first ordertransitions between these states account for the frequency-locked anomaly observed by Hopf et al [1982] in a hybridoptical bistable device.

An inverse Liapunov problem was studied by Liu and Leake[1967]. They studied the controllable dynamical system ofEq.(7.7-1) and established means to determine the systemfunction f or the control function u so that Eq.(7.7-1)behaves in an "acceptable" manner such as uniform asymptoticstability in the whole (u.a.s.w.), which is equivalent toasymptotic stability in the whole if the function f is

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autonomous or f is periodic in t, and equivalent toasymptotic stability if, f is also linear. Indeed, suchstability is probably the most desirable type of stabilityfor practical systems. They have shown the followingtheorem:

Theorem 7.7.3 Let f e CO (local Lipschitz condition)and let x = 0 be the equilibrium is u.a.s.w. iff thereexists a scalar function V(x,t) such that:(i) V(x,t) is of class C', positive definite, decrescent,and radially unbounded; and(ii) whose Eulerian derivative dV(x,t)/dt is negativedefinite. Here the Eulerian derivative is defined by dV/dt =<f, grad V> + Vi, where <> denotes the inner product, Vt =aV/at, and grad V the gradient of V.

Unfortunately, there are no general means availablewhereby one can actually find V. The inverse problem isthat, given a pair of scalar functions V(x,t) anddV(x,t)/dt, one must find the function f (or u) so that theEulerian derivative is satisfied. The solution of theinverse problem together with the known results ofLiapunov's direct method can be applied directly to thesynthesis (or design) of nonlinear control systems. Theyalso have to find the necessary and sufficient conditionsfor multi- loop feedback control systems and their completeasymptotic controllability conditions.

For the feedback system of dx/dt = Ax + bf(cx), a veryinteresting stability criterion for the asymptotic stabilityin the large is the Popov criterion [Popov 1962].

In a different vein, Westcott [1986] gave a brief reviewof the application of feedback control theory tomacro-economic models.

7.8 Semiconductor laser linewidth reduction by feedbackcontrol and phased arrays

A narrow-linewidth, frequency stablized laser is a veryuseful device in many aspects of pure and applied research

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such as spectroscopy, laser communication, lidar and remotesensing, heterodyne detection, etc. Unlike conventionallasers, the field power spectra of semiconductor lasers arenot dominated by mechanical cavity fluctuations, butinstead, external noise sources such as ambient temperaturefluctuations, injection current fluctuations due to thenoises from a power supply, and so on, in addition to thespontaneous emission, contribute to the linewidth.Electrical or optical feedback techniques have been verysuccessful in reducing linewidth significantly. Simpleelectrooptic feedback systems have been utilized for diodelasers to correct for amplitude and/or frequencyfluctuations due to variations in drive current and ambienttemperature. For a recent review of frequency stablesemiconductor lasers, see, e.g., Leeand Chen [1990]. Simultaneous intensity and frequencystabilization can be achieved by detecting the fluctuationsin these characteristics and to control both the diodeoperating temperature and drive current [Yamaguchi andSuzuki 1983].

One may have noticed that the huge difference, over nineorders of magnitude between the limiting linewidth and thecavity linewidth for conventional gas or semiconductorlasers, is mainly due to the very short photon lifetime, tc,which is proportional to euK1 in the resonantors. Clearly,to increase tc, i.e., decrease the linewidth, in particularfor semiconductor lasers, is to increase the cavity length Lby placing the laser in an external resonator and by usinghigh reflectance mirrors. Indeed, semiconductor laserlinewidths in kHz regime in an external cavity have beendemonstrated [Shay 1987].

There are several papers which provide simplifiedanalytical discussions of external cavities with somerestrictions [Fleming and Mooradian 1981, Henry, 1986].These analyses can provide guidelines as to the amount oflinewidth reduction which can be realized with externalcavity. It should be pointed out that rigorous analyses of

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the external cavity of a diode laser are quite complex andnot thoroughly understood. It is known that the spectralbehavior is strongly dependent on the level of opticalfeedback [Patzak et al 1983a,b, Lenstra et al 1985, Tkachand Chraplyvy 1985, Henry and Kazarinov 1986, Henry 1986]and the phase relationship between the diode field and thefeedback field [Chen 1984]. As in injection locking, ingeneral, the higher the feedback amount is, the greater thelinewidth reduction can be achieved. Nonetheless, it shouldbe cautioned that at a varying feedback level at a certainhigh level of feedback, the linewidth reduction tends tosaturate, and is then followed by an abrupt increase inlinewidth by several orders of magnitude. This effect wasobserved for instance by Goldberg et al (1982] and Favre etal [1982]. It has been pointed out that both the saturationphenomenon and the sudden increase in linewidth areconsequences of the nonlinear dynamics, for instance,[Olesen et al 1986, Otsuka and Kawaguchi 1984]. As has beenpointed out by Lenstra et al [1985], the state of increasedlinewidth has the distinct signature of a chaotic attractor.Indeed, the coherence collapse is due to optical delayfeedback [Dente et al 1989]. Due to their high gains,semiconductor lasers are very sensitive to feedbacks.Feedback not only induces phase (i.e., frequency)fluctuations, but also induces amplitude fluctuations, andmay eventually lead to chaos.

Very recently, Li and Abraham [1989] have shown thatthe semiconductor laser with optical feedback functions as alaser with an optical servoloop can be considered aself-injection locking laser, and they also determine theeffect of various parameters on the noise reduction and onthe stability of single- mode operation of the laser.

In order to achieve high power, semiconductor lasershave to be phase locked together to form a phased array.Such phase locked semiconductor phased arrays have beendemonstrated [Teneya et al 1985, Hohimer 1986, Leger et al1988, Carlson et al, 1988]. Earlier studies of semiconductor

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phased arrays are mainly focused on their spatial modestructures under steady state operating conditions. Thesespatial mode patterns in terms of supermodes are observedand well understood [Scifres et al 1979, Botez 1985, Butletet al 1984, Peoli et al 1984]. Nonetheless, little has beensaid about the temporal or time evolutionary behavior ofsuch phase locked semiconductor arrays. Only very recently,there have been a few papers concerning the dynamicalbehavior of the phased arrays [Elliott et al 1985, Wang andWinful 1988]. What we would like to know are: (a) under whatoperating conditions will the semiconductor laser array bephase locked, (b) how long will it take for locking to takeplace, (c) will the locking be stable, i.e., be persistentfor a very long time, (d) if it can only be locked for a"short" time, then what happens after they become unlocked,(e) what are the details and routes to instabilities, (f)

can we classify the various routes to instabilities from thevariations of various operating parameters, (g) and aboveall, with reasonable operating conditions, can one prove ageneral statement concerning the existence or non-existenceof a stable region for a semiconductor laser array.

There are many ways to initiate the phase locking of thesemiconductor phased array. One way is the injectionlocking, which has been demonstrated for gain guidedsemiconductor laser array [Hohimer et al 1986]. But forindex guided laser array, some phase shifters may benecessary for phase locking. Another method for phaselocking is using a common external cavity for the laserarray, whereby the phase locking is accomplished by mutualfeedback, i.e., self-organization. Of course, in order toincrease the amount of mutual feedback, the self-imagingtechnique can be employed [Leger et al 1988]. Here we shallnot debate the relative merit of either method, or othergeneric methods, it suffices to say that in the followingformulations and discussion, the general conclusion will bethe same. Even though we shall only discuss the formulationof an semiconductor laser array undergoing mutual feedback,

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the case of injection locking can be modified trivially aswe shall indicate.

For a M element diode-laser array oscillating in acommon cavity with mutual feedback, the system of thedynamical equations is the following:

dNi- Ji(t) Ni M1/2-G(N1, Ii) {Ii ( t) +E ij Ii (t-Sji) cos ((Jjtji+4ij) )dt ed 8i j-1

2 3+AN4 , (7.8-1)+BNN

dtI[G(Ni,Ii)-]Ii(t)cos((bi(t)-4oi(t-T)+wiT+(0oi)Ti

M

+E{[Ii(t)tijlj(t-tji)]1/2xcos [ (A(aij) t+4i ( t) (t-sji) +wjTji+(P.1),

d41=

a 1 ] - [w1 (Ni) -n]Ii (t-t) ] 1/2

dt 2 Ii (t)xsin [Wi (t) (t-t) +wit+`00i]

(7.8-2)

(7.8-3)

Here i e [1,M], rs. is the semiconductor carrier diffusiontime, B is the carrier spontaneous recombinationcoefficient, A is the Augur coefficient, Eij is the couplingcoefficient and ri, is the roundtrip time between the i-thand the j-th elements, r, is the photon lifetime, r = rii isthe self-feedback delay time, a is the absorptioncoefficient, and n is the outcoupling frequency of thecoupled phased-array. For "identical" lasers, we assume thatrs. = rs. The first square bracket on the right hand side ofEq.(7.8-3) depicts the hole burning of homogeneously

389

the laser frequencies o1 (N,) depend on the carrier density

by

G(Ni) =G(Nch) +( LG) (Ni-Nth)aN,

c* 1 (Ni) i (Nh) a(0i) (Ni-N,,) .

(7.8-4)

(7.8-5)

where Nth is the threshold carrier density, Nth - Ntr Herethe coupling coefficients t,i consist only the externalmirror or grating reflectivity, transmission and diffractionlosses, and any intra-cavity loss. The product of Eii withthe appropriate phase terms correspond to the couplingcoefficients in Lang and Yariv [1986, 1987]. Furthermore,the injected current J. can be an explicit fuction of time,this will be of particular interest for applicationsutilizing FM. The mechanical analogue is a system of damped,nonlinearly coupled oscillators with time varying appliedforce. To simplify the problem further and to avoid delayfeedback instability, such as Ikeda instability [1979], andallow us to use the slow varying envelope approximation(SVEA), one can assume a short external cavity. Inparticular, the self-image cavity can satisfy this

requirement.For a constant current injection, the system is an

autonomous one. It is well-known [Lefschetz 1962, Arnold1973] that for an autonomous nonlinear dynamical systemdx/dt = f(x), where x is a M-vector, the stability criteriaof the system is equivalent to the corresponding linearizedsystem dx/dt = AX + g(x), where A is the coefficient matrixand g(X) is the nonlinear part, provided ag/axe approachzero uniformly in t as the equilibrium solution 1lxiiapproaches zero. This is always the case if each component

390

of g(x) is a polymonial in xi which begins with terms oforder 2 or higher. Put it differently, if each component ofg(x) is differentiable, it will also satisfy the condition.It is also well-established that the linearized system isstable if all the eigenvalues of the system is negative, orthe real part of the eigenvalues be negative. From theseestablished results, we have the following algorithm fordetermining the stability region of the coupledsemiconductor phased array. We first transform the coupleddynamical system into standard form, i.e., first find theequilibrium (steady state) solutions and transform theorigin to the equilibrium points by setting z = x - x0. Wethen linearize the system and write f(xo + z) in the form Az+ g(z) where A is the linear part of f(xo + z). We thencompute the eigenvalues of the linearized system, i.e., A.By varying all the operating parameters of the system, andfinding the values where all the eigenvalues are negative,one can then "map" out the stability region of the system.

Clearly, the system, Egs.(7.8-1-3), is very complex andnearly intractable even numerically. Nonetheless, we shalldiscuss its qualitative properties in a series of fulllength papers eleswhere. The most simple case which stillpreserves both the self-feedback and cross-feedback is thecoupling of two semiconductor lasers in an external cavity.As we have just mentioned, in order to avoid delay feedbackinstability, and allow us to use SVEA, we assume a shortexternal cavity, such that r < rs. The typical carrierdiffusion time is of the order of 1 nsec, then the externalcavity length is assumed to be 2 cm. Thus we have r12 = 721 =711 = 722 = r. From SVEA, we have Ii(t - r) = Ii(t).Furthermore, from the symmetry of t11 we have, E12 = E21 = x.We than have the following set of six first ordernonlinearly coupled ordinary differential equations. Noticethe symmetry between 1 and 2, in later discussions thissymmetry should be kept in mind.

dN1/dt = J/ed - N1/rs - G[I1(t) + xY. I2(t - r)391

Cos (o2r + 002) ] + BN12 + AN13,

dN2/dt = J/ed - N2/rs - G112(t) + x/ I1(t -Cos (01 r + 0o1) ] + BN22 + AN23

dI1/dt = [G - 1/rt]I1(t) + Kh[I,(t)I2(tCos[01(t) - 02(t - r) + 4)27' +

d12/dt

d¢1/dt

[G - 1/r1]I2(t) + KY.(12(t)I1(tCos[02(t) - 01(t - r) + oIr +

r)

- r) ]"'002],

(7.8-6a)

(7.8-6b)

(7.8-6c)

(7.8-6d)

a[G - 1/r]/2 - [ut,(N1) - w] - KY,(I1(t - r)/il(t)]l,sin[O,(t) - 01(t - r) + fd1r + 0.11, (7.8-6e)

d02/dt = a[G - 1/r)/2 - [e2(N2) - o] - is %( [ 12(t - r)/I2(t) ]"sin[O2(t) - 02(t - r) + 002], (7.8-6e)

As we have prescribed earlier, we can transform the"origin" of the system to (NJ, N21 Il, 121 o1, 02) , thenlinearize the system and obtain the matrix A. Theeigenvalues of A are then computed for each set of operatingparameters. We shall not go into any detail in doing so [Lee1988]. It suffices to say that with a large size computer,one can find out within what range of operating parametersthe system of two semiconductor lasers coupled by a shortexternal cavity will become phase locked and remain lockedindefinitely. Of course, one can also find the stabilityboundary of the operating parameters. For applications, Lee[1988] has found that the stability of these two coupledsemiconductor lasers is sensitive to the couplingcoefficient (or feedback amount), but most sensitive to thefrequency detuning between the two lasers and their relativeinitial phases.

It is more interesting to know when those operating(control) parameters cross the stability boundary, in otherwords, what are the routes of chaos will this system follow?

392

There have been many experimental attempts to map out thedynamics of this "simple" system, yet none have beensuccessful. The reason is that the detectors andoscilloscopes are too slow (usually of the order of onenanosecond) while the coupling of two diodes is taking placeat about 30 - 50 psec. Thus, the oscilloscope displays a bigblob of hundreds of return maps. Certainly, this situationis not satisfactory to say the least. Another approach is touse a streak camera [Elliott 1985]. But that is notsatisfactory either due to the fact that it can not takesufficient samples for a short time span, i.e., to map outthe time evolution of the dynamics. Consequently, one has tofind some way to overcome the difficulty posed by the fastcoupling of diode lasers.

We would like to point out from Egs.(7.8-6) that if thetwo coupled diode lasers are operated at small signal gainregime, i.e., G(N,) = aN, + b, where a and b are constants,the only nonlinear coupling terms between Ni and E1 (here i= 1,2) are of the form of direct product. In this case, thephase space of this system is a product space. In otherwords, the phase space of this system can be decomposed intotwo "orthogonal" subspaces, the recombination rate subspace,Ni and N2, a two-dimensional subspace, and the subspace ofamplitudes and phases of the two lasers, a four-dimensionalsubspace. Due to the product nature of the phase space, onecan determine the dynamics of the system both experimentallyand theoretically by studying its two orthogonal subspaces.Simply put, it is analogous to the understanding of thehelical motion of an electron under the influence of aconstant magnetic field. It is the product of the circularmotion of an electron under a constant magnetic field, andthe motion of the electron along z-axis. The compositemotion is the product action.

The trajectories "projected" on the (N1,N2)-subspace maybe complicated, nonetheless tractable. Furthermore, one cansolve the dynamics of Ni and N2 rate equations fairly easilyby keeping the rest constants. Moreover, from the time

393

scale, the carrier recombination rate is about onenanosecond, longer than the field couplings, which are ofthe order of 30 - 50 psec, or the round trip time of theexternal cavity. Nonetheless, in order to study the dynamicsof the two semiconductor lasers, without mixing theinstabilities and chaos induced by the nonlinear delayedfeedback [Ikeda 1979], the external cavity has to be veryshort. Indeed, in order to aviod the complexity of thenonlinear delayed feedback instability, the external cavityround trip time should not be greater than 0.1 nsec. So, themain effort of the study should be on the coupling dynamicsof the field amplitudes and phases, that is, the study ofthe four-dimensional subspace of the field amplitudes andphases.

Theoretical study of the coupled six equations,Eqs.(7.8-6), is straight forward in the sense that withappropriate numerical schemes and large enough computer, onecan have detailed numerical simulations and beautifulgraphics for the dynamics of the system. And indeed, we cantheoretically answer the questions we have posed earlier.What we are interested in, concerned with, and proposing todo, instead, is to find an experimental approach to studythe system, which will be more satisfying. As we havepointed out earlier, direct measurements are hopeless due tothe extremely fast coupling of the system. This is true evenwith the help of the very recent silicon photodetectorsinvented by Seigmann [1989], which have detector responsetime of a few psec. This is because the detectors can nothave a sampling rate of 100 GHz or higher. Indeed, it isunlikely such a device will be available in the near future.It is then clear that some means of direct detection with amuch slower sampling rate is necessary.

In the following, we will propose a set of experimentsto be performed, which can unambiguously map out thecoupling dynamics of the four-dimensional subspace of thefields and phases of two lasers experimentally. The idea isto use a laser medium which has a long radiative decay rate,

394

such as those solid-state lasers, e.g., Nd:YAG laser, whichhas decay rate of 0.23 msec, or gas lasers, such as CO2laser, which has decay rate of 0.4 msec. If the externalcavity round trip time is about 1 or 2 nsec, then thecoupling time can only be longer than this, which can beresolved by the state of art detectors with time resolutionof 0.1 nsec. In fact, we may not need such fast detectors.This is because even though the external cavity round triptime is about 2 nsec, due to relatively "low" gain of theselaser media, effective coupling may take several roundtrips, thus the coupling time is on the order of tens ofnsec. At any rate, this is not very important to ourdiscussion, we only want to point out that relatively fastyet inexpensive detectors will be sufficient.

Even for these slow decay rate laser media, we stillhave a set of six coupled, nonlinear differential equationsto deal with, which included two laser rate equations. Butsince the coupling time is stretched rather long, comparedwith semiconductor lasers, to a few to tens of nsecs, toaccommodate the slow detector response time, it is still fartoo short compared with the radiative decay time of thelaser which is of the order of 0.2 to 0.4 msec. Thus, thetwo laser rate equations can be considered as steady-state.Thus, we can study the nonlinear coupling of the fields andphases of the two lasers using existing technology.

Here we shall briefly discuss the four controlparameters for the study of the dynamics of nonlinearcoupling of two (say solid-state) lasers. For the sake ofillustration, let us here consider the coupling of two smallNd:YAG lasers. Let these two Nd:YAG lasers be monolithicones, such as the one for frequency stable applicationconsatructed by Zhou et al [1985], but with somewhat largergain volume. And they are temperature controlled withelectric coolers. Let laser A be the "reference" one, thatis, except the output power, no other parameters will bevaried. Thus, this laser can be truely monolithic in design.The other laser, laser B, has PZT mirror as the back mirror,

395

thus the laser is slightly tunable. The power reflectivityof the outcoupling mirror is the same as the "reference"laser so that they all have the same amount of feedback.Outside the outcoupler of laser B, there is an electro-opticphase modulator, this can change the relative phase of thesetwo lasers. And in front of the coupling mirror, we can puta calibrated neutral density filter to attenuate and tocontrol the mutual coupling of these two lasers. Bycontrolling the pump power of laser B, we can also vary theoutput power density of laser B. This in turn will changethe relative power density of the two lasers, andconsequently, the relative feedback amount of these twolasers. Of course, there are other control parameters onecan vary, but in order to closely simulate a diode phasedarray, these four control parameters for thefour-dimensional subspace are the only relevant ones. Fordiodes, the inject current densities are the other relevantparameters.

In solving Egs.(7.8-6c,d,e,f), we can restate theseequations by subtracting these two set of equations andobtaining the set of two nonlinear differential equations ofthe difference of field amplitude, AE, the difference ofresonance frequencies, ew, and the difference of the phasesof these two lasers, A0. Together with the other controlparameter, the feedback coupling coefficient x, we have thecompletely deterministic system. We have also simplified notonly the system, but also made the experiment simplerbecause we only have to measure all the variables or controlparameters in their difference (or relative values).Nonetheless, this simplification should not overshadow thecrucial importance of "slowing down" the dynamical couplingof two lasers by using lasers of very slow radiative decayrates.

We believe only if this four-dimensional subspace iswell understood experimentally, then do we have a chance tounderstand experimentally the full system, i.e., thecoupling of two semiconductor lasers.

396

Now let us comment on the semiconductor laser phasedarray. Of course, there are many ways to phase locksemiconductor laser array, but the most esthetic as well asthe simplest from the system point of view is the mutualfeedback approach such as the self-imaging of the Talbotcavity [Jansen et al 1989, D'Amato et al 1989]. Inprinciple, if the two-dimensional array has a very largenumber of "identical" elements (in both the geometric andphysical sense), then with a very short time the array willself- adjust to the external cavity and a stable phaselocked two- dimensional array can be formed. Nonetheless,the physical characteristics of each of the elements arecritical to the phase locking process as well as thestability of such locking. As we have pointed out earlier,the locking is less sensitive to the feedback amount and thefield intensity and gain inside each semiconductor lasers,and the locking is very sensitive to the frequency detuningand the phase. But these two most sensitive variables arevery much determined by the semiconductor laser materialprocessing. Indeed, in studying this 3N-dimensional system,where N is the number of semiconductor lasers and N is verylarge, one has to resort to nonlinear stochasticdifferential equations. The solution, if it exists, can givethe statistical properties or requirements of thesemiconductor material processing. Which in turn can givethe material tolerance of the semiconductor laser phasedarray. Now we are getting into the real application, namely,the material engineering.

An interesting problem closely related to the problemdiscussed above is the random neural networks. Recently,Sompolinsky and Crisanti [1988] have studied a continuous-time dynamical model of a network of N nonlinear elementsinteracting via random asymmetric couplings. They have foundthat by using a self-consistent mean-field one can predict atransition from a stationary pahse to a chaotic phaseoccurring at a critical value of the gain parameter. Infact, for most region of the gain parameters, the only

397

stable solution is the chaotic solution.

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442

Index

1-boundary, 411-chain, 401-cycle, 41

a-limit cycle, 159a-limit set, 157a-set, 157e-equivalent, 315e-stable, 315fi-conjugacy, 165n-conjugate, 165n-decomposition, 261n-equivalence, 165n-limit cycle, 159n-limit set, 157n-set, 157

Accumulation point, 31Abelian, 26Absolutely C1-structually stable, 262Admissible, 143Almost periodic, 238Analytic, 75Analytic index of a linear differential operator P, 133,

142, 144

Analytic manifold, 48Anosov, 260Arcwise connected, 37Asymptotic stibility, 203Asymptotically controllable, 380Asymptotically stable, 192, 200, 202, 233Asymptotically stable relative to U, 236Attracted, 202Attractive, 201Attractor, 202, 233

443

Attractor relative to U, 236Autocatalysis, 362Autonomous, 170Auxiliary functions, 206Anosov, 260Axiom A, 137, 261Axiom B, 137

Banach manifold, 124Banach space, 123Base space, 83, 85Basic sets, 261Basin of attraction, 202Bifurcation point, 268Boundary, 42Bracket of X and Y, 51Brouwer Fixed Point Theorem, 59Bundle map, 105Bundle of bases, 87Bundle space, 85

Canonical transformation, 97Catastrophe, 270Center manifold theorem, 276Chain, 42Chaos, 280Ck curve, 49

Ck perturbation, 243Classifying bundle, 106Classifying map, 107Closed, 94Closed orbit, 157Closed set of X, 33Closure of A, 33Cobordant, 73Cobound, 63Codimension one bifurcations, 272

444

Collar on M, 108Compact set, 36Compact space, 36Competitive interacting populations, 359Complete stability, 220Completely unstable, 175Connected, 38Conservative, 7Continuous, 34Continuous at p, 34Converge pointwise, 33Converge to, 30Converges locally and uniformly, 130Cooperativity, 368Coset space, 74Cosets, 26Cotangent space, 93Covering of X, 36Covering space, 44Critical point, 57, 60Critical value, 57, 60Cross-section, 86, 150Cross-section to the flow, 193

Damped anharmonic oscillator, 327De Rham cohomology group, 94Diffeomorphic, 48Diffeomorphism, 46, 48, 4 9, 125Differentiable, 46, 48, 1 24

Differentiable conjugacy, 153Differentiable manifold, 47Differentiable structure, 46Dispersive, 175Dissipative, 8Dynamical systems, 146

Effective, 74

445

Elementary transversality theorem, 122Elliptic differential operator of order k, 141Equi-asymptotically stable, 203Equi-attractive, 201Equi-attractor, 233Euler characteristic of X, 67Exact, 26, 94Exotic, 261Expansion, 184Exponential map, 79, 80Exterior derivative, 94Exterior product, 93

Faithful, 79Feedback control, 379Fiber, 83Fiber bundle associated to (B,M,G) with fiber F, 88Fiber bundle over X with fiber F, 85Fiber derivative, 101Filippov solution, 382First axiom of countability, 33Flow, 55, 146Flow equivalence, 154, 156Flow map, 153Fredholm map, 142Function of class K, 207

G acts on X to the left, 74Gauss-Bonnet theorem, 68Generalized Liapunov function, 227Generated, 26Generators, 26Generic, 253Globally asymptotically stable, 203Globally unstable, 205Grassmann manifold of k-planes in n-space, 106Group, 25

446

Group of diffeomorphisms, 49

Hausdorff space, 31Hessian, 60, 191Heteroclinic orbits, 162Homeomorphic, 35Homeomorphism, 35Homoclinic cycles, 162Homoclinic orbits, 162Homogeneous, 75Homologous, 41, 42Homology group, 42Homomorphism, 25Homotopically trivial, 43Hopf bifurcation, 273Hyperbolic, 179, 188, 193Hyperbolic attractor, 250Hyperbolic linear automorphisms, 182, 184Hyperbolic structure with respect to f, 250Hyperbolic subset of X, 251Hyperbolic toral automorphism, 151

Image, 25Imbedding, 51, 125Immersion, 51, 125Indecomposable, 261Induced bundle, 105Inner product, 95Integrable, 56, 168Integral curve, 54, 168Integral flow, 168Integral manifold, 56Interior of A, 28Interior point, 28Invariant k-form, 98Invariant set, 160Involutive, 56

447

Isomorphic, 26, 185Isomorphism, 26Isotropy, 108Isotropy group, 74Isotropy subgroup, 152

Jacobi identity, 51Jacobian matrix, 51

k-jet bundle of M and N, 114k-jet extension map, 134k-jet of f at p, 113k-jet of mappings, 114Kernel, 25

Lagrange stability, 200Lagrange stable, 167Lagrange unstable, 175Left invariant, 76Left transformation group, 74Left translation by g, 76Liapunov functions, 206Liapunov-like function, 381Lie algebra, 76Lie derivative, 95Lie group, 75Lie subalgebra, 78Lie subgroup, 78Limit, 30Limit point, 31Linear approximation, 188Linear contraction, 184Linear differential operator, 129, 134Linearization of a non-linear differential operator P, 138Linearly conjugate, 179Linearly equivalent, 180Lipschitz, 172

448

Lipschitz map, 172Local Hamiltonian, 98Local integral, 168Local one-parameter group of transformations T, 55Local stable set, 246Locally Hamiltonian, 98Locally Lipschitz, 172Lorenz equations, 288

Manifold molded on V, 124Measure zero, 58Melnikov function, 256Metric space, 27Minimal set of a dynamical system, 161Monomorphism, 104Morphism, 102Morse function, 65Multijet bundle, 122

Neat imbedding, 109Neat submanifold, 109Negatively Lagrange stable, 167Negatively Lagrange unstable, 175Negatively Poisson stable, 163Negatively prolongation,, 175Neighborhood of p, 29Non-degenerate, 96Non-degenerate critical point of f, 60Non-linear differential operator of order k, 135Non-singular, 51Non-wandering, 164Non-wandering set, 162Norm, 123Normal bundle, 89, 90Nullity, 64

One-form, 93

449

One-parameter subgroup of G, 79Open chart, 46Open covering of X, 36Open set, 28Optical instabilities, 341Orbit, 55, 146Orbit space, 147Orbital stability, 200Order of a linear differential operator P, 129Orientable, 96Origin of attraction, 203Orthogonal complement, 90

p-complex, 41p-dimensional de Rham cohomology group of M, 94p-dimensional distribution, 56p-simplex, 41p-th Betti number, 42Paracompact, 45Parallelizable, 91, 177Parametric expression for the operator P, 136Partial flow, 121, 155Partial tubular neighborhood of M, 108Partition of unity, 47Path, 37Permanence, 367Permanent, 368Phase space, 146Poincard duality, 67Poincare map, 194Poisson bracket, 98Poisson stable, 163Poisson unstable, 175Positive definite, 207Positive (negative) prolongation, 175Positive (negative) prolongational limit set, 175Positively Lagrange stable, 167

450

Positively Lagrange unstable, 175Positively Poisson stable, 163Positively prolongation, 175Positively recursive, 163Principal bundle of M, 84Principal fiber bundle, 85Product set, 28Prolongations, 114Pullback, 105

Qualitative theory of diffferential equations, 145Quasi-linear differential opeartor of order k, 136Quotient, 26Quotient space bundle, 89Quotient system, 151

r-form, 93Reaction-diffusion equations, 351Recurrent, 166Recurrent synaptic feedback, 384Region of attraction, 202, 217Region of instability, 205Region of uniform attraction, 202Region of weak attraction, 201Regular, 157Regular point, 189Regular value, 57Representation of G, 79Restriction of the flow, 34Retraction, 59Reverse flow, 150Right translation by g, 76

s-fold k-jet bundle, 122Saddle point, 148Section, 178Self positively recursive, 163

451

Semi-asymptotically stable, 233Semi-attractor, 233Semi-weak attractor, 233Sheets, 44Simply-connected, 44Singularity, 157Smooth dynamical systems, 145Smooth flow, 12Smooth manifold modeled on B, 124Spectral decomposition, 261Spherical modification, 62Split, 104Stability-additive, 238Stable, 179, 198, 201, 202, 315Stable at to, 199Stable manifold, 185, 246Stable manifold of f at p, 247, 250Stable relative to U, 236Stable set, 179, 246Stable set of size (b a) of f at x, 251Stable summand, 185Stable with respect to measurement, 383Stable with respect to x, 209Strange, 261Strange attractor, 202Strange saddles, 308Strong transversality, 261Strongly stable under perturbations, 229Structural group, 85Structurally stable, 190, 245Sub-bundle, 90Subalgebra, 78Subcovering of X, 36Subgroup, 25Submanifold, 51, 125Subspace, 29Surgery, 63

452

Suspension, 149, 150Symbol of the differential operator P, 132, 138Symplectic form, 97Symplectic manifold, 97

Tangent bundle of M, 84, 88, 125Tangent vector at p, 49Tensor bundle, 89Tensor field, 89The analytic index of P, 142The differential of f, 51The restriction of f to X, 34The restriction of the flow, 156The tangent vector to the curve a at t, 54Time map, 146Topological conjugacy, 153, 194Topological dynamics, 145Topological equivalence, 154Topological group, 74Topological manifold, 45Topological property of X, 35Topological space, 27Topological vector space, 123Topologically conjugate, 153, 155Topologically equivalent, 155Topologically transitive, 261Topology induced, 29Totally stable, 226Trajectory, 55Transitive, 74Transverse to the orbit, 193Trivial, 79, 85Trivial bundle, 86Tubular neighborhood of M, 107Tychonoff Theorem, 37Typical fiber, 83

453

Ultimately bounded,2834Unfoldings, 277Uniform attractor, 202, 233, 236Uniform attractor relative to U, 236Uniform stability with respect to x, 209Uniformly approximate, 239Uniformly asymptotically stable, 203, 219, 233Uniformly attractive, 201Uniformly globally asymptotically stable, 203Uniformly globally attractive, 203Uniformly stable, 201Uninvadable, 369Universal, 278Universal bundle over G, 106Universal covering space of X, 44Unstable, 199, 201Unstable manifolds, 246Unstable set, 246, 250

van der Polls equation, 171Vector bundle of a manifold M, 84Vector bundles, 89Vector field, 50Versal, 278Votka-Volterra equations, 359

Weak attractor, 202, 233, 236Weak attractor relative to U, 236Weakly continuous, 130Weakly attracted, 202Whitney e topology, 118Whitney (or direct) sum, 89, 104

454