Applied Mathematical SciencesVolume 38

Edit orsF. John J.E . Marsden L. Sirovich

AdvisorsM. G hil J.K. Hale J. Ke llerK. Kirchgassner BJ . MatkowskyJ.T. Stuart A. Weinstein

Applied Mathematical Sciences

1. John : Partial Differential Equations, 4th ed.2. Sirovich : Techniques of Asymptotic Analysis.3. Hale : Theory of Functional Differential Equations, 2nd ed.4. Percus : Combinatorial Methods.5. von Mises/Friedrichs: Fluid Dynamics.6. Fre iberger/Grenander: A Short Course in Computational Probability and Statistics.7. Pipkin : Lectures on Viscoelasticity Theory.8. Giacoglia: Perturbation Methods in Non-linear Systems.9. Friedrichs : Spectral Theory of Operators in Hilbert Space.

10. Stroud : Numerical Quadrature and Solution of Ordinary Differential Equations.11. Wolovich: Linear Multivariable Systems.12. Berkovitz: Optimal Control Theory.13. Bluman/Cole: Similarity Methods for Differential Equations.14. Yoshizawa: Stability Theory and the Existence of Periodic Solution and Almost Periodic Solutions .15. Braun : Differential Equations and Their Applications, 3rd ed,16. Lefschetz: Applications of Algebraic Topology.17. Collatz /Wetterling: Optimization Problems.18. Grenander: Pattern Synthesis: Lectures in Pattern Theory, Vol. I.19. Marsden/McCracken : Hopf Bifurcation and Its Applications.20. Driver: Ordinary and Delay Differential Equations.21. Courant/Friedrichs: Supersonic Flow and Shock Waves.22. Rouchelliabetsll.aloy: Stability Theory by Liapunov 's Direct Method.23. Lamperti: Stochastic Processes: A Survey of the Mathematical Theory.24 . Grenander: Pattern Analysis: Lectures in Pattern Theory, Vol. II.25. Dav ies: Integral Transforms and Their Applications, 2nd ed.26. Kushner/Clark : Stochastic Approximation Methods for Constrained and Unconstrained Systems.27. de Boor: A Practical Guide to Splines.28. Keilson: Markov Chain Models-Rarity and Exponemiality.29. de Veubeke : A Course in Elasticity.30. Shiatycki: Geometric Quantization and Quantum Mechanics.31. Reid: Sturmian Theory for Ordinary Differential Equations.32. Meis/Markowitz: Numerical Solution of Partial Differential Equations.33. Grenander: Regular Structures: Lectures in Pattern Theory, Vol. III.34. Kevorkian/Cole: Perturbation Methods in Applied Mathematics.35. Carr: Applications of Centre Manifold Theory.36. Bengtsson/Ghil /Klillen: Dynamic Meteorology: Data Assimilation Methods.37. Saperstone: Semidynamical Systems in Infinite Dimensional Spaces.38. Lichtenberg/Lieberman : Regular and Chaotic Dynamics, 2nd ed.39. Piccini /Stampacch iaiVidossi ch: Ordinary Differential Equations in R' .40. Naylor/Sell : Linear Operator Theory in Engineering and Science.41. Sparrow: The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors .42. Guckenheimerlliolmes: Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields.43. Ockend on/Taylor: Inviscid Fluid Flows.44. Pazy: Semigroups of Linear Operators and Applications to Partial Differential Equations.45. Glashoff/Gustafson: Linear Operations and Approximation: An Introduction to the Theoretical Analysis

and Numerical Treatment of Semi-Infinite Programs.46. Wilcox: Scattering Theory for Diffraction Gratings.47 . Hale et al: An Introduction to Infinite Dimensional Dynamical Systems-Geometric Theory.48. Murray : Asymptotic Analysis.49. Ladyzhenskaya: The Boundary-Value Problems of Mathematical Physics.50. Wilcox: Sound Propagation in Stratified Fluids.51. Golub itskyischaeffer: Bifurcation and Groups in Bifurcation Theory, Vol. I.

(continued following index)

A.J. LichtenbergM.A. Lieberman

Regular andChaotic Dynamics

Second Edition

With 225 Figures

Springer Science+Business Media, LLC

A.J. LichtenbergM.A. LiebermanDepartment of Electrical Engineering

and Computer SciencesUniversity of CaliforniaBerkeley, CA 94720USA

Editors

F.JohnCourant Institute of

Mathematical SciencesNew York UniversityNew York, NY 10012USA

J.E. MarsdenDepa rtment of

MathematicsUniversity of CaliforniaBerkeley, CA 94720USA

L. Sirov ichDivision ofApplied Mathematics

Brown UniversityProvidence, RI 02912USA

Mathematics Subject Classification (1991): 70Kxx; 60Hxx; 34Cxx; 35Bxx

Library of Congress Cataloging in Publication DataLichtenberg, Allan J.

Regular and chaotic dynamics/AJ. Lichtenberg, M.A. Lieberman .­2nd ed.

p. cm.-(Applied mathematical sciences; v. 38)Includes bibliographical reference s and index .

1. Nonlinear oscillations. 2. Stochastic processes.3. Hamiltonian systems . 1. Lieberman, M.A. (Michael A.)II . Title . III. Series: Applied mathematical sciences (Springer­Verlag New York Inc.) ; v. 38.QAI.A647 vol. 38 1992[QA867.5]510 s---dc20[531'.32] 91-39849

This book is the second revised and expanded edition of "Regular and Stochastic Motion,"which was published in the Applied Mathematical Sciences series Vol. 38 by Springer-Verlagin 1983.

Printed on acid-free paper.© 1983, 1992 Springer Science+Business Media New YorkOriginally published by Springer-Verlag New York, Inc in 1992.Softcover reprint of the hardcover 2nd edition 1992

All right s reserved. This work may not be translated or copied in whole or in part without thewritten permission of the publisher Springer Science+Business Media, LLCexcept for brief excerpts in connection with reviews or schola rlyanalysis. Use in connection with any form of information storage and retrieval, electronicadaptation, computer software, or by similar or dissimilar methodology now known or hereafterdeveloped is forbidden.The use of general descriptive names, trade names, trademarks, etc., in this publication, even ifthe former a re not especially identified, is not to be taken as a sign that such names, asunderstood by the Trade Marks and Merchandise Marks Act, may accordingly be used freelyby anyone.

Production managed by Francine Sikorski; manufacturing supervised by Robert Paella.Typeset by Integral Typesetting, Norfolk, UK.

9 8 7 654 3 2 I

ISBN 978-1-4419-3100-9 ISBN 978-1-4757-2184-3 (eBook)/978-1-4757-2184-3DOI 10.1007

To Elizabeth and Marlene

Preface to the Second Edition

What's in a name? The original title of our book, Regular and StochasticMotion, was chosen to emphasize Hamiltonian dynamics and the physicalmotion of bodies. The new edition is more evenhanded, with considerablymore discussion of dissipative systems and dynamics not involving physicalmotion. To reflect this partial change of emphasis, we have substituted themore general terms in our title. The common usage of the new terms clarifiesthe emphasis of the book.

The main change in the book has been to expand the sections ondissipative dynamics, including discussion of renormalization, circle maps,intermittancy, crises, transient chaos, multifractals, reconstruction, andcoupled mapping systems. These topics were either mainly in the mathemati­cal literature or essentially unstudied when our first edition was written. Thevolume of work in these areas has surpassed that in Hamiltonian dynamicswithin the past few years.

We have also made changes in the Hamiltonian sections, adding many newtopics such as more general transformation and stability theory, connectedstochasticity in two-dimensional maps , converse KAM theory, new topicsin diffusion theory, and an approach to equilibrium in many dimensions.Other sections such as mapping models have been revised to take intoaccount new perspectives. We have also corrected a number of misprintsand clarified various arguments with the help of colleagues and students,some of whom we acknowledge below. We have again chosen not to treatquantum chaos, partly due to our own lack of acquaintance with the subject.

Since writing the first edition, a number of monographs and texts haveappeared that are complementary to ours, covering material with a differentemphasis, or covering material not covered at all in our book. In writingthe sections on dissipative dynamics, we benefited from monographs byGuckenheimer and Holmes (1983) and by Schuster (1988) : The former isdetailed and mathematically oriented with somewhat limited scope; the lattercovers some of the topics we have added, at a similar level. New books on

vii

viii Preface to the Second Edition

Hamiltonian mechanics have recently appeared; one, a concise treatment butincluding quantum chaos and a detailed treatment of integrability by Tabor(1989), and another, a thorough account relating Hamiltonian chaos toplasma and wave problems by Sagdeev et al. (1988). There is also a recentgeneral text by Jackson (1989) and an account of numerical methods byChua and Parker (1989). The above list is by no means exhaustive.

There are many colleagues whose assistance was helpful in preparation ofthis new edition. We thank Per Bak, David Bruwiler , Jim Crutchfield, CelsoGrebogi, and Jerrold Marsden for reading sections of the manuscript relevantto their work; Chris Goedde for supplying material not yet published fromhis thesis; and Jim Howard, Chris Silva, and M. Nambu for carefully readingportions of the original edition and pointing out a number of misprints andminor errors. Particular thanks go to Boris Chirikov and associates whotranslated the original edition into Russian . Besides bringing misprints to ourattention, more substantive points were addressed. Once again, we gratefullyacknowledge our research support in the dynamics area from the NationalScience Foundation, the Office of Naval Research, and the Department ofEnergy. This continuing support has been essent ial for our developing manyof the topics that we treat in the new edition.

Preface to the First Edition

This book treats stochastic motion in nonlinear oscillator systems. Itdescribes a rapidly growing field of nonlinear mechanics with applicationsto a number of areas in science and engineering, including astronomy, plasmaphysics, statistical mechanics, and hydrodynamics. The main emphasis is onintrinsic stochasticity in Hamiltonian systems, where the stochastic motionis generated by the dynamics itself and not by external noise. However, theeffects of noise in modifying the intrinsic motion are also considered. Athorough introduction to chaotic motion in dissipative systems is given inthe final chapter.

Although the roots of the field are old, dating back to the last century whenPoincare and others attempted to formulate a theory for nonlinear perturba­tions of planetary orbits, it was new mathematical results obtained in the1960s, together with computational results obtained using high-speed com­puters, that facilitated our new treatment of the subject. Since the newmethods partly originated in mathematical advances, there have been twoor three mathematical monographs exposing these developments. However,these monographs employ methods and language that are not readilyaccessible to scientists and engineers, and also do not give explicit techniquesfor making practical calculations.

In our treatment of the material, we emphasize physical insight rather thanmathematical rigor. We present practical methods for describing the motion,for determining the transition from regular to stoch astic behavior, and forcharacterizing the stochasticity. We rely heavily on numerical computationsto illustrate the methods and to validate them.

The book is intended to be a self-contained text for physical scientists andengineers who wish to enter the field, and a reference for those researchersalready familiar with the methods. It may also be used as an advancedgraduate textbook in mechanics . We assume that the reader has the usualundergraduate mathematics and physics background, including a mechanicscourse at the junior or senior level in which the basic elements of Hamiltoniantheory have been covered . Some familiarity with Hamiltonian mechanics at

ix

x Preface to the First Edition

the graduate level is desirable, but not necessary. An extensive review of therequired background material is given in Sections 1.2 and 1.3.

The core ideas of the book, concerning intrinsic stochasticity in Hamil­tonian systems, are introduced in Section 1.4. Our subsequent exposition inChapters 2-6 proceeds from the regular to the stochastic. To guide the readerhere , we have "starred" (*) the sections in which the basic material appears.These "starred" sections form the core of our treatment. Section 2.4a onsecular perturbation theory is of central importance. The core material hasbeen successfully presented as a 30-lecture-hour graduate course at Berkeley.

In addition to the core material, other major topics are treated. The effectsof external noise in modifying the intrinsic motion are presented in Section5.5 (using the results of Section 5.4d) for two degrees of freedom and inSection 6.3 for more than two degrees of freedom, and an application isgiven in Section 6.4. Our description of dissipative systems in Chapter 7 canbe read more-or-less independently of our treatment of Hamiltonian systems.For studying the material of Chapter 7, the introduction in Section 1.5 andthe material on surfaces of section in Section 1.2b and on Liapunovexponents in Sections 5.2b and 5.3 should be consulted. The topic of per ioddoubling bifurcations is presented in Sections 7.2b, 7.3a, and Appendix B (seealso Section 3.4d). Other specialized topics such as Lie perturbation methods(Section 2.5), superconvergent perturbation methods (Section 2.6), aspectsof renormalization theory (Sections 4.3, 4.5), noncanonical methods (Section2.3d), global removal of resonances (Section 2.4d and part of 2.4c), variationalmethods (Sections 2.6b and 4.6), and modulational diffusion (Section 6.2d)can generally be deferred until after the reader has obtained some familiaritywith the core material. 1

This book has been three and a half years in the writing. We have receivedencouragement from many friends and colleagues. We wish to acknowledgehere those who reviewed major sections of the manuscript. The final drafthas been greatly improved by their comments. Thanks go to RD.I.Abarbanel, J.R . Cary, B.V. Chirikov, R.H. Cohen, D.F. Escande, J. Ford,J. Greene, R.H.G. HeIleman, P.J. Holmes, J.E. Howard, O.E. Lanford,D.B. Lichtenberg, R. Littlejohn, B. McNamara, H. Motz, e. Sparrow,J.L. Tennyson, and A. Weinstein. Useful comments have also been receivedfrom G. Casati, A.N. Kaufman, l .C, Percival, and G.R. Smith. We are alsopleased to acknowledge the considerable influence of the many publishedworks in the field by B.V. Chirikov. Many of the ideas expressed herein weredeveloped by the authors while working on grants and contracts supportedby the National Science Foundation, the Department of Energy, and theOffice of Naval Research. One of the authors (A.J.L.) acknowledges thehospitality of St. Catherine's College, Oxford, and one of the authors(M.A.L.) acknowledges the hospitality of Imperial College, London, wheremuch of the manuscript was developed.

1 The section numbers refer to the first edition . For the second edition, the reader should consultthe Contents.

Contents

Preface to the Second EditionPreface to the First EditionList of Symbols

CHAPTER I

Overview and Basic Concepts

1.1. An Introductory Note*1.2. Transformation Theory of Mechanics

*1.2a. Canonical Transformations*l.2b. Motion in Phase Space*1.2c. Action-Angle Variables

1.3. Integrable Systems*l.3a. One Degree of Freedom

1.3b. Linear Differential Equations1.3c. More than One Degree of Freedom1.3d. Noncanonical Variables and Reduction

*1.4. Near-Integrable Systems*1.4a. Two Degrees of Freedom*1.4b. More than Two Degrees of Freedom

1.5. Dissipative Systems1.5a. Strange Attractors1.5b. The Lorenz System

CHAPTER 2

Canonical Perturbation Theory

2.1. Introduction2.la. Power Series2.1b. Asymptotic Series and Small Denominators2.1c. The Effect of Resonances

*2.2. Classical Perturbation Theory*2.2a. One Degree of Freedom*2.2b. Two or More Degrees of Freedom

viiix

xvii

I77

12212425293244484860626365

70

70737577787883

xi

Xli

2.3. Adiabatic Invariance*2.3a. Introduction and Basic Concepts*2.3b. Canonical Adiabatic Theory*2.3c. Slowly Varying Harmonic Oscillator2.3d. Noncanonical Methods

2.4. Secular Perturbation Theory*2.4a. Removal of Resonances*2.4b. Higher-Order Resonances*2.4c. Resonant Wave-Particle Interaction2.4d. Global Removal of Resonances

2.5. Lie Transformation Methods2.5a. General Theory2.5b. Deprit Perturbation Series2.5c. Adiabatic Invariants

2.6. Superconvergent Methods2.6a. Kolmogorov's Technique2.6b. Singly Periodic Orbits

CHAPTER 3Mappings and Linear Stability

*3.1. Ham iltonian Systems as Canonical Mappings*3.1a. Integrable Systems*3.1b. Near-Integrable Systems*3.1c. Hamiltonian Forms and Mappings

*3.2. Generic Behavior of Canonical Mappings*3.2a. Irrational Winding Numbers and KAM Stability*3.2b. Rational Wind ing Numbers and Their Structure*3.2c. Complete Description of a Nonlinear Mapping*3.2d. Numerical Examples

3.3. Linearized Motion*3.3a. Eigenvalues and Eigenvectors of a Matrix*3.3b. Two-Dimensional Mappings3.3c. Linear Stability in Higher Dimensions3.3d. General Stability Concepts

*3.4. Mapping Models*3.4a. Physical Problems and Models*3.4b. Numerical Results*3.4c. Fixed Points and Linear Stability*3.4d. Bifurcation Phenomena*3.4e. Hamiltonian Formulation

*3.5. The Separatrix Motion*3.5a. Driven One-Dimensional Pendulum*3.5b. The Separatrix Mapping

3.6. Mappings from Perturbed Linear Oscillators3.6a. Tiling of the Pha se Space3.6b. Phase Space Structure

Contents

939396

100102109110117122128133135136141149152154

162

164164167169174174183188191195196200208210215216220223226229231231236237238241

Contents

CHAPTER 4

Transition to Global Stochasticity

*4.1. Introduction*4.1a. Qualitative Description of Criteria*4.1b. The Standard Mapping

*4.2. Resonance Overlap*4.2a. Rationale for Criteria*4.2b. Calculation of Overlap Criteria

4.3. Growth of Second-Order Islands4.3a. Elliptic Fixed Points4.3b. The Separatrix

*4.4. Stability of High-Order Fixed Points*4.4a. The Basic Elements of Greene's Method*4.4b. Numerical Evaluation

4.5. Renormalization for Two Resonances4.6. Nonexistence of Invariant Curves

*4.7. Summary and Conclusions

CHAPTER 5

Stochastic Motion and Diffusion

5.1. Introduction5.2. Definitions and Basic Concepts

*5.2a. Ergodicity*5.2b. Liapunov Characteristic Exponents5.2c. Concepts of Stochasticity5.2d. Randomness and Numerical Errors

5.3. Determination of Liapunov Exponents and KS Entropy5.3a. Analytical Estimates

*5.3b. Numerical Methods*5.4. Diffusion in Action Space

*5.4a. The Fokker-Planck Equation*5.4b. Transport Coefficients*5.4c. Steady-State and Transient Solutions

5.5. The Effects of Correlations5.5a. Higher-Order Transport Corrections5.5b. Diffusion near Critical K5.5c. Long-Time Correlations and Averaging5.5d. Quasiaccelerator Modes

5.6. The Effect of Extrinsic Stochastic Forces5.6a. Introduction5.6b. Diffusion in the Presence of Resonances5.6c. Diffusion on Two Space and Time Scales

5.7. Slow Separatrix Crossings

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293294294296302309312314315320321324326328328339342345349349353357363

xiv

CH A PT ER 6

Three or More Degrees of Freedom

*6.1. Resonance in Multidimensional Oscillations*6.1a. Geometric Relations*6.1b. Examples of Arnold Diffusion

6.2. Diffusion Rates Along Resonances*6.2a. Stocha stic Pump Diffusion Calculation6.2b. Coupling Resonance Diffusion6.2c. Man y Resonance Diffusion6.2d. Modulational Diffusion

6.3. Extrin sic Diffusion6.3a. Resonance Streaming6.3b. Diffusion of a Parameter

6.4. Diffusion in Toroidal Magnetic Fields6.4a. Magnetic Islands6.4b. Drift Surfaces and Diffusion in Static Fields6.4c. Time-Varying Fields6.4d. The Self-Consistent Problem

6.5. Many Degrees of Freedom6.5a. Strongly Nonlinear Coupled Systems6.5b. The Ferrni-Pasta- Ularn Model and the

Korteveg-de Vries Equation6.5c. Discretized Sine-Gordon Equation

CHAPTER 7

Bifurcation Phenomena and Transitionto Chaos in Dissipative Systems

Contents

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373374380386387393397399406406413418418425428436437440443

450

457

7.1. Simple and Strange Attractors 4577.l a. Basic Properties 4587.1 b. Simple Attractors and Their Bifurcation s 4617.1c. Examples of Strange Attractors 4687.ld. Geometric Properties of Strange Attractors 474

7.2. One-Dimensional Noninvert ible Maps 4787.2a. Basic Properties 4787.2b. Periodic Behavior 4827.2c. Chaotic Motion 497

7.3. Bifurcation Phenomena in Two-Dimensional Maps 5087.3a. Dissipative Maps 5097.3b. Area-Preserving Maps 5127.3c. Transition Between Hamiltonian and Dissipative Chao s 519

7.4. Circle Maps 5257.4a. Basic Formalism 5267.4b. Crit ical Behavior and Universality 5297.4c. Supercritical Behavior 5327.4d. Physical Examples 534

7.5. Intermittency 5377.5a. Intermittency Mechanisms 5387.5b. Renormal izat ion-Group Analysis 542

Contents

7.5c. Power Spectra and Experimental Results7.6. Crises and Basin Boundaries

7.6a. Crises7.6b. Fractal Basin Boundaries

7.7. Determination of Homoclinic Intersection

CHAPTER 8

Chaotic Motion in Dissipative Systems

8.1. Transient Chaos8.1a. Weak Dissipation8.1b. Transient Chaos Near a Crisis

8.2. Invariant Distributions on Strange Attractors8.2a. Reduction to a One-Dimensional Map8.2b. Direct Iteration of an Initial Distribution

8.3. Multifractals and Their Properties8.3a. Generalized Dimensions8.3b. Scaling Index Spectrum

8.4. Reconstructing Dynamics8Aa. Delay Coordinates and Embedding Dimension8.4b. Statistical Quantities8Ac. Short-Time Prediction

8.5. Spatio-Temporal Dynamics8.5a. Maps on a Lattice8.5b. Self-Organized Criticality

8.6. The Fluid Limit8.6a. Fourier Mode Expansions8.6b. The Transition to Turbulence8.6c. Hamiltonian Chaos in Fluids

APPENDIX A

Applications

A.1. Planetary MotionA.2. Accelerators and BeamsA.3. Charged Particle ConfinementAA . Charged Particle HeatingA.5. Chemical DynamicsA.6. Quantum Systems

BibliographyAuthor IndexSubject Index

xv

545548549557560

570

570571579580581583588588592597598602604613613620628630634638

644

644646648650651653

655671679