Lorenz, Chaos

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Nonlinear Dynamical Economics and Chaotic Motion Second Edition by Hans-Walter Lorenz Volkswirtschaftliches Seminar Georg-August-Universit¨ at Platz der G¨ ottinger Sieben 3 W-3400 G¨ ottingen, Germany

description

Chaos theory

Transcript of Lorenz, Chaos

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Nonlinear Dynamical Economics

andChaotic Motion

Second Edition

by

Hans-Walter Lorenz

Volkswirtschaftliches SeminarGeorg-August-Universitat

Platz der Gottinger Sieben 3W-3400 Gottingen, Germany

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V

To My Parents

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VII

...only nonlinear differential equations have interesting dynamics.

M. Hirsch (1984)

Unfortunately, many of the mostimportant processes in nature are inherently nonlinear.

R.L. Devaney (1992)

There are no true fractals in nature.( There are no true straight lines or circles either!)

K. Falconer (1990)

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Prefaces

Preface to the Second Edition

Usually, the first edition of a book still contains a multiplicity of typographic, con-ceptional, and computational errors even if one believes the opposite at the timeof publication. As this book did not represent a counterexample to this rule, thecurrent second edition offers a chance to remove at least the known shortcomings.

The book has been partly re-organized. The previously rather long Chapter 4has been split into two separate chapters dealing with discrete-time and continuous-time approaches to nonlinear economic dynamics. The short summary of basicproperties of linear dynamical systems has been banned to an appendix becausethe line of thought in the chapter seems to have been unnecessarily interruptedby these technical details and because the book concentrates on nonlinear systems.This appendix, which mainly deals with special formal properties of dynamical sys-tems, also contains some new material on invariant subspaces and center-manifoldreductions. A brief introduction into the theory of lags and operators is followedby a few remarks on the relation between the ‘true’ properties of dynamical systemsand their behavior observable in numerical experiments. Additional changes in themain part of the book include a re-consideration of Popper’s determinism vs. inde-terminism discussion in the light of chaotic properties of deterministic, nonlinearsystems in Chapter 1. An investigation of a simultaneous price-quantity adjustmentprocess, a more detailed inquiry into the uniqueness property of limit cycles, anda short presentation of relaxation oscillations are included in Chapter 2. Chapter3 now starts with an extended discussion of different structural stability concepts.While the material on chaotic dynamics in Chapters 4 and 5 still concentrates on themotion on attractors, the importance of complex transient motion is emphasizedin the current edition.

The literature on chaotic dynamics in economics is rapidly growing. It is there-fore difficult if not impossible to keep track of all the advances made in the last

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years. As this book concentrates on methodological aspects and usually discussesonly simple economic examples, not all economically relevant contributions in theliterature could be presented in detail. The papers known to the author are how-ever listed in the appropriate sections.

Most numerical calculations and associated plots in this edition were performedwith the help of the Dynamical Software package and the Dynamics program.This is not mentioned because the responsibility for the correctness of the numer-ical results should be shifted to other sources. It should only prevent the readerinterested in performing his own calculations from re-inventing the wheel and turnhis attention to the existing elaborated packages. All other illustrations were pro-duced with a standard CAD program or commercial plotting routines; the manu-script was again typeset in TEX.

It is a pleasure for me to thank all those friends and colleagues who commentedon improving the text. Particular thanks go to C. Chiarella, P. Flaschel, D. Furth,L. Nicelli, and B. Woeckner who all provided more or less extensive error lists. G.Konigsberg copy-edited several new parts of the text. The assistance of B.K.P. Hornof Y&Y in the management of diverse PostScript fonts is greatly appreciated.

Gottingen, February 1993 Hans-Walter Lorenz

Preface to the First Edition

The plan to publish the present book arose while I was preparing a joint workwith Gunter Gabisch (Gabisch, G./Lorenz, H.-W.: Business Cycle Theory. Berlin-Heidelberg-New York: Springer). It turned out that a lot of interesting materialcould only be sketched in a business cycle text, either because the relevance forbusiness cycle theory was not evident or because the material required an interestin dynamical economics which laid beyond the scope of a survey text for advancedundergraduates. While much of the material enclosed in this book can be foundin condensed and sometimes more or less identical form in that business cycletext, the present monograph attempts to present nonlinear dynamical economicsin a broader context with economic examples from other fields than business cycletheory.

It is a pleasure for me to acknowledge the critical comments, extremely detailedremarks, or suggestions by many friends and colleagues. The responses to earlierversions of the manuscript by W.A. Barnett, M. Boldrin, W.A. Brock, C. Chiarella, C.Dale, G. Feichtinger, P. Flaschel, D.K. Foley, R.M. Goodwin, D. Kelsey, M. Lines, A.Medio, L. Montrucchio, P. Read, C. Sayers, A. Schmutzler, H. Schnabl, G. Silverberg,H.-W. Sinn, J. Sterman, and R. Tscherning not only encouraged me to publishthe book in its present form but helped to remove numerous errors (not onlytypographic ones) and conceptual misunderstandings and flaws. Particular thanks

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go to G. Gabisch who initiated my interest in nonlinear dynamics and encouragedthe writing of this text. A. Johnson and R. Phillips copy-edited parts of the textand helped to remove many misleading formulations and stylistic shortcomings. Itseems to be unnecessary to stress that all remaining errors will debit my personalaccount.

Large parts of the manuscript were written while I was visiting the Universityof Southern California. Without the inspiring environment of the Modelling Re-search Group and the extraordinary help of the staff the book would not have beencompleted in due time.

The work was partly supported by the Deutsche Forschungsgemeinschaft. Thefinal manuscript was typeset in PCTEX.

Gottingen, March 1989 Hans-Walter Lorenz

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Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1. Economic Dynamics, Linearities, and the Classical MechanisticWorldview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5

1.1. Some Reflections on the Origin of Economic Dynamics . . . . . . . . . . . . . . . . . 61.2. The Deterministic Worldview and Deterministic Theories . . . . . . . . . . . . . . 131.3. The Dominance of Linear Dynamical Systems in Economics . . . . . . . . . . . 19

2. Nonlinearities and Economic Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.1. Preliminary Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.2. The Poincare-Bendixson Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.2.1. The Existence of Limit Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .402.2.2. The Kaldor Model as a Prototype Model in Nonlinear

Economic Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .432.2.3. A Classical Cross-Dual Adjustment Process . . . . . . . . . . . . . . . . . . . . . . 47

2.3. The Uniqueness of Limit Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512.3.1. The Lienard Equation and Related Tools . . . . . . . . . . . . . . . . . . . . . . . 512.3.2. The Symmetric Case: Unique Cycles in a Modified Phillips

Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542.3.3. The Asymmetric Case: Unique Cycles in a Kaldor Model . . . . . . . . 57

2.4. Predator-Prey Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612.4.1. The Dynamics of Conservative Dynamical Systems . . . . . . . . . . . . . . .612.4.2. Goodwin’s Predator-Prey Model of the Class Struggle . . . . . . . . . . . .67

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2.4.3. Other Examples and Predator-Prey Structures inDissipative Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

2.5. Relaxation Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .732.6. Irreversibility and Determinism in Dynamical Systems . . . . . . . . . . . . . . . . . .77

3. Bifurcation Theory and Economic Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 803.1. Preliminaries and Different Concepts of Structural Stability . . . . . . . . . . . . 813.2. Local Bifurcations in Continuous-Time Dynamical Systems . . . . . . . . . . . . .87

3.2.1. Fold, Transcritical, and Pitchfork Bifurcations . . . . . . . . . . . . . . . . . . .873.2.2. The Hopf Bifurcation in Continuous-Time Dynamical Systems . . .95

3.2.2.1. The Hopf Bifurcation in Business-Cycle Theory . . . . . . . . 1013.2.2.2. Closed Orbits in Optimal Economic Growth . . . . . . . . . . . 107

3.3. Local Bifurcations in Discrete-Time Dynamical Systems . . . . . . . . . . . . . . . 1103.3.1. Fold, Transcritical, Pitchfork, and Flip Bifurcations . . . . . . . . . . . . .1103.3.2. The Hopf Bifurcation in Discrete-Time Dynamical Systems . . . . . 115

4. Chaotic Dynamics in Discrete-Time Economic Models . . . . . . . . . . . . . . . . . . . . . 1194.1. Chaos in One-Dimensional, Discrete-Time Dynamical Systems . . . . . . . . 121

4.1.1. Basic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1224.1.2. Chaos in Descriptive Growth Theory . . . . . . . . . . . . . . . . . . . . . . . . . . .1384.1.3. Chaos in Discrete-Time Models of Optimal Economic Growth . . 1434.1.4. Other Economic Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

4.2. Chaos in Higher-Dimensional Discrete-Time Systems . . . . . . . . . . . . . . . . . 1494.2.1. Some Basic Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1494.2.2. An Economic Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

4.3. Complex Transients in Discrete-Time Dynamical Systems . . . . . . . . . . . . . 1574.3.1. Complex Transient Behavior in One-Dimensional Systems . . . . . .1584.3.2. Horseshoes, Homoclinic Orbits, and Complicated

Invariant Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

5. Chaotic Dynamics in Continuous-Time Economic Models . . . . . . . . . . . . . . . . . .1675.1. Basic Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1675.2. The Coupling of Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .174

5.2.1. Toroidal Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1745.2.2. International Trade as the Coupling of Oscillators . . . . . . . . . . . . . 180

5.3. The Forced Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1825.3.1. Forced Oscillator Systems and Chaotic Motion . . . . . . . . . . . . . . . . . 1835.3.2. Goodwins’s Nonlinear Accelerator as a Forced Oscillator . . . . . . . 186

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5.3.3. Keynesian Demand Policy as the Source of Chaotic Motion . . . . .1875.3.4. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

5.4. Homoclinic Orbits and Spiral-Type Attractors . . . . . . . . . . . . . . . . . . . . . . . . 1925.4.1. The Shil’nikov Scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1935.4.2. Spiral-Type Chaos in a Business Cycle Model with Inventories . . 195

6. Numerical Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2016.1. Spectral Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2026.2. Dimension, Entropy, and Lyapunov Exponents . . . . . . . . . . . . . . . . . . . . . . . 205

6.2.1. Phase Space Embedding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2056.2.2. Fractal Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2086.2.3. Correlation Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2116.2.4. Lyapunov Exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2136.2.5. Kolmogorov Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2186.2.6. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

6.3. Are Economic Time Series Chaotic? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2226.4. Predictability in the Face of Chaotic Dynamics . . . . . . . . . . . . . . . . . . . . . . . .228

7. Catastrophe Theory and Economic Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2337.1. Basic Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2347.2. The Kaldor Model in the Light of Catastrophe Theory . . . . . . . . . . . . . . . .2397.3. A Catastrophe-Theoretical Approach to Stagflation . . . . . . . . . . . . . . . . . . . 241

8. Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248A.1. Basic Properties of Linear Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 249A.2. Center Manifolds and the Reduction of (Effective) Dimensions . . . . . . . . . .264A.3. A Brief Introduction to the Theory of Lags and Operators . . . . . . . . . . . . . . .270A.4. Numerical Simulations and Chaotic Dynamics in Theoretical

Economics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

Name Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309

Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315

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Introduction

The history of economic science abounds in examples of the emergence anddecline of fashionable trends in economic thought. Basic and paradigmatic

attitudes toward the conceptual understanding of an economy, concentrations onspecific classes of economic models which are believed to be an optimal reflectionof economic reality, or the usage of formal or verbal techniques whose applicationsare believed to provide new insights into existing paradigms have rarely gainedlasting serious attention over the decades. It is this transitoriness which allows toassign many textbooks and monographs to a certain era.

In order for a discipline to be considered a serious scientific field, a standardcollection of ideas, methods and concepts has to emerge over the decades which isaccepted by the majority of scientists in that field and which is not easily vulnerableto the challenge of fashionable and short-lived trends. These scientific fundamentalsof economics are characterized by two essential properties:

• The foundation of modern economics dates back to the 18th century and hasnot undergone a drastic restructuring in the subsequent years. Unlike otherdisciplines in which the emergence of a new set of ideas has had revolution-ary effects on the development of the field (e.g., consider the changes arisingin biology with evolution theory, or quantum mechanics’ revolutionary effectin physics), scientific progress in economics seems to consist mainly in refine-ments and/or modifications (as sophisticated they may be) of accepted centraltheories.

• The formal apparatus of mainstream economics is borrowed from mathematicsand the natural sciences, especially from physics. Abstracting from the tightconnections between mathematical statistics and econometrics, economics hasonly rarely contributed to the advances of formal science and has adapted itselfto existing formalisms.

Modern economic theory not only has its heritage in but also continues to em-ploy the ideas of classical and neoclassical economists of the 18th and 19th cen-

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tury. Classical and neoclassical economics emerged in a scientific environment thatwas dominated by the grandiose inventions of classical physics and tremendousadvances brought through the application of classical mechanics to engineeringproblems. The mechanistic weltanschauung that characterized scientific thought inmany different disciplines at least until the turn of the century postulates a deter-ministic framework in which empirically observable phenomena follow strict andwell-defined laws comparable to Newton’s famous basic laws of gravitation. If theinvolved laws are precisely known, predictions on the outcome of a process can bemade with the same precision. The task of the scientist therefore remains to uncoverthese immanent rules. The writings of Walras, Marshall, Jevons, or Pareto aredominated by the attempt to find these immanent rules in economic activities andto formalize them in the way of classical mechanics. A major part of microeco-nomic theory and welfare economics, whose invention is usually attributed to theseauthors, is characterized by the attempt to explain human behavior deterministi-cally from assumed preference orderings and associated optimization procedureswhich resemble methods of mechanical or engineering problem solving. This ba-sic attitude toward an understanding of economic life has obviously survived untiltoday and will probably persist as the mainstream paradigm of economic thoughtfor years to come.

This characterization of economic theory is not an attempt to classify economicsas a dependance of other more advanced sciences. Due to its character as a dis-cipline which has to rely more than other practical sciences on abstract thoughtexperiments, and in which measurement procedures depend more than in otherfields on theoretical reflections, economics obviously has not experienced incen-tives strong enough to necessitate any drastic modifications of its formal apparatusand conceptual framework. Furthermore, it may be argued that advances made inseveral natural sciences such as biology, physics, and chemistry simply have had norelevance to economic theory.

During the last two or three decades several of the natural sciences have expe-rienced increasing efforts to diverge from their immanent heritage in the mecha-nistic weltanschauung, which continues to prevail in many other disciplines. Whilequalitative advances made in physics like the development of quantum mechanics,relativity theory, and thermodynamics already suggested a basic failure of classicalmechanics as early as around the beginning of this century, a formal phenomenonseems to initiate a divergence from the mechanistic attitude in other disciplinesas well. The mathematical discovery of chaotic or irregular dynamical systems hasinitiated a renewed interest in nonlinear dynamics, which do not simply constitutesome kind of a generalization of known linear systems, but which indeed concernthe very conceptual framework of an understanding of actual phenomena. As it willbe demonstrated at some length below, the mechanistic worldview can be referredto as the linear worldview, and the concept of nonlinearities can have dramatic ef-fects on the capability to predict the behavior of even simply structured dynamicalmodels.

With unusual immediacy, new results on the effects of nonlinear dynamicalsystems in experimental mathematics, physics, chemistry, and biology have beenpromptly applied to economic dynamics, though these early works were surely out-

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side mainstream theorizing. Meanwhile, an impressive list of publications now ex-ists, indicating that nonlinear systems with chaotic properties are not untypical ineconomics. This book attempts to introduce the basic concepts of chaotic dynami-cal systems and to familiarize the reader with the existing literature. Furthermore,the aim of the book consists in activating interest in the consequences of the pres-ence of nonlinearities for economic theory’s conceptualization of reality.

As was mentioned above, theories and concepts come and go in scientific life,especially in fields of applied philosophy like sociology or economics. Whetherthe chaos property of some dynamical systems will indeed revise the mainstreamparadigm, or whether it will be shown that it is only a marginal curiosity in economicdynamics can be determined only by scientific progress. In any case, chaotic dynam-ics constitute an exciting example of how complicated some dynamical systems maybe, although they may at first seem to be qualitatively identical with well-knownregular systems.

Naturally, this book is not an essay on the purely mathematical aspects of non-linear dynamical systems. It is designed as a survey of recent developments in dy-namical systems theory and its economic applications. It is the aim of the bookto familiarize economists with the existing literature in dynamical systems theory,and not to provide a satisfactory overview from a mathematical point of view. Thus,the interested reader will be referred to the genuine mathematical literature for allproofs of the mentioned theorems and for a deeper mathematical understandingas often as possible.

The book is organized as follows: Chapter 1 attempts to demonstrate that thegeneral attitude of dynamical economics toward reality is an inheritance from themechanistic worldview of the 18th and 19th century. The philosophically more ed-ucated reader who is also familiar with the history of science is cordially requestedto excuse the excursion into a basically distinctive field which nevertheless is en-lightening with respect to several of the topics enclosed in this book. Such a discus-sion seems to be mandatory when an attempt is made to evaluate the influence ofcomplex dynamical systems on the determinism/indeterminism controversy dom-inating the science-theoretic literature during the first half of the 20th century.Chapter 1 also attempts to illustrate this worldview by a short survey of assump-tions and methods in standard economic dynamics which generally can be coinedlinear dynamics. The basic tools for analyzing nonlinear dynamical systems are in-troduced in Chapter 2. It includes topics like the Poincare-Bendixson theorem,the uniqueness of limit cycles, and – as an example of a conservative dynamicalsystem – Goodwin’s predator-prey model of the class struggle, which can be trans-formed into a dissipative dynamical system under additional assumptions. Chapter3 is devoted to a subject which is becoming more and more important in economicdynamics, namely bifurcation theory. In addition to the renowned Hopf bifurca-tion, economic examples of other bifurcation types like the transcritical, fold, or flipbifurcation are presented for discrete-time and continuous-time systems. Chapter4 constitutes one of the two main chapters of this book. It contains an introduc-tion to discrete-time, one-dimensional, chaotic dynamics and provides examplesof these “strange” phenomena from several economic sub-disciplines. The chap-ter concludes with a short outline of the emergence of strange dynamics in two-

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and higher-dimensional, discrete-time systems and a discussion of complex tran-sient motion. The mathematically more sophisticated higher-dimensional chaos incontinuous-time models is presented in Chapter 5 which should be considered asan outline of future research. It concentrates on standard scenarios like coupledoscillator systems and forced oscillators. It also contains a discussion of spiral-typechaos which might be a very long-lasting transient phenomenon. Chapter 6 dealswith the empirically most important question of establishing chaos in observed timeseries. Chapter 7 then presents an outline of catastrophe theory whose relevanceto the advance of dynamical economics may not seem to be obvious but never-theless merits special attention. Catastrophe theory represents a particular tool tomodel the evolution of economies whose variables can be categorized as slow andfast variables. Catastrophe theory permits to model sudden jumps in the evolutionof a variable in a completely endogenous fashion. A few concluding remarks arecontained in the final Chapter 8. The book closes with an appendix that containssome material which is either mandatory for an understanding of several conceptsintroduced in the main text or which supplements some statements. It recalls basicelements in the theory of dynamical systems, including the dynamic properties oflinear one- and two-dimensional systems in discrete and continuous time, differentapproaches to the modeling of lag structures, and the use of operators in express-ing these lag structures. It also contains a few warning remarks regarding the use ofnumerical simulation techniques in investigating nonlinear differential equations.

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Chapter 1

Economic Dynamics, Linearities, and the

Classical Mechanistic Worldview

Nonlinear economic dynamics may be considered just a collection of modelswith essentially nonlinear ingredients that require the use of a particular set of

(relatively new) mathematical tools. As such, nonlinear economic dynamics has arank comparable to that of game theory, optimal control, or many other innovationsin economic theory made during the last 50 years. However, nonlinear dynamicalsystems emerging in several fields have never been evaluated only from an exclusiveformal point of view. The potential complexity and impredictability of nonlineardynamical systems have almost immediately initiated a discussion of basic science-theoretic themes. Popular treaties of the subject occasionally talk of a scientificrevolution or employ similar spectacular expressions. However, it seems as if inseveral examples of these inquiries the scientific environment which is supposed toencounter such a revolution is not always described with a sufficient accuracy. Thefollowing remarks do not (and cannot, actually) attempt to provide a completelysatisfactory account of the origin of economic theorizing and the extend to whichnonlinear dynamics might contribute to a change in the attitude toward economicdynamic processes. The sole purpose of the following notes consists in encouragingfurther reflections on the role of dynamical systems in the modeling of dynamiceconomic processes.

The first section recalls a few original quotations from the ancestors of mod-ern economic theory (with an emphasis on the dynamic aspects of economic the-ory). The overall imitation of physics’ methodology in the writings of 19th centuryeconomists is demonstrated with several quotations from those authors who obvi-ously felt obliged to justify their procedures. As the mechanistic worldview domi-

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nated the basic attitude toward life in those days, a more careful description andevaluation of this worldview and the challenge it encountered at the beginning ofthis century is presented in the second section. The chapter closes with a few reflec-tions on the resistance observable in the economics profession to a concentrationon nonlinear phenomena in economic dynamics.

1.1. Some Reflections on the Origin of Economic Dynamics

Economics in its modern form was introduced as a serious and distinguished sci-ence during the second half of the 18th century. Unlike earlier attempts to un-derstand economic phenomena (usually in the context of political economy like,e.g., mercantilism) the writings of Adam Smith or David Ricardo constitute thefirst successful approaches toward an abstract explanation of human economic be-havior. One reason why economics emerged as a science in that particular periodsurely has to do with the expansion of capitalism in the advanced societies of thatday and the increasing complexity of trade. It is not surprising that economicsas a modern science originated in Great Britain, which not only is considered thehomeland of capitalistic production but which also had been the dominant factorin international trade for more than 150 years. Much of the early economists’ inter-est was therefore devoted to the major economic subjects of the day like the effectsof international trade on the prosperity of the domestic economy.1

It cannot have been the political and economic environment of the late 18th andearly 19th century alone which stimulated an interest in focusing research on eco-nomic problems and which initiated the development of economics into its presentday form (although the development of this science is inherently connected to thesocial environment). There had been other events with similar importance to theeconomic development of a political unit which had not initiated a comparable in-terest in economic affairs. Economic considerations of, e.g., the mercantilistic policyin 17th century France and other European countries were intimately connectedthrough absolutistic ideals of improving the welfare of the nation, occasionally in-carnated in the personal welfare of its emperor. Thus, the “economist” of the daywas incorporated into the national administration and was given no incentive todwell upon his own independent individualistic ideas and concepts.

This mercantilistic attitude in absolutistic nations came in conflict with the emer-gence, popularization, and final success of the enlightenment movement in the 18thcentury. The enlightenment’s concentration on individualism, which laid the foun-dations for capitalistic (and political) development in the advanced economies likeBritain, arose in an intellectual atmosphere dominated by the writings of Leibniz,Voltaire, Kant, Newton and other enlightenment philosophers. Several of theseauthors who profoundly reformed modern western thought (some of whom were

1 In many cases, inquiries into international trade represent the renowned work of clas-sical writers; for example, most economists will probably remember David Ricardomainly for his investigations of comparative cost advantages rather than for his laborvalue theory.

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probably the last generally educated and interested scholars in modern history) werenot only concerned with philosophical questions of Being but also strongly inter-ested in the natural sciences. The enlightenment period of 18th century Europehas gained favorable retrospective interest not exclusively due to its concentrationon human affairs, but also from its successes in the investigation of natural phe-nomena. Whereas scientific pioneers like Galileo, Kepler, or Descartes had torecant their ideas or seek refuge, the 18th century was characterized by an opennessto enlightening ideas, probably because of the stringency of the results of severalauthors and experimentalists in the natural sciences. The effects that the writing ofVoltaire or Newton had on the academic community of their day can probablynot be overestimated.

It was in this era of close ties of enlightenment philosophy to advances in thenatural sciences and political and economic development in which the writings ofthe now classical economists were published. As educated academics, A. Smith orlater D. Ricardo and J.S. Mill must have been familiar with at least the generalideas of enlightenment philosophy and the advances made in physics. Their workmust have been influenced, directly or indirectly, both by the political and socialimplications of that philosophy, and also through its basic approach toward anunderstanding of natural phenomena.

Abstracting from several spectacular inventions, a major reason for the strongimpact of the natural sciences on daily life and the academic community consistedin the fact that physics occurred as being a precise science in the sense that anexperiment with a careful description of the environment leads to unambiguousresults. If the environment does not change, an experiment’s outcome will remainconstant as well. The hypothetical possibility of repeating an experiment infinitelyoften with the same outcome laid the foundation for determining the physical con-stants and for deriving basic laws of motion underlying the experiment. Once thelaws of motion and the physical constants are known, it is possible to predict the out-come not only of the particular experiment from which they are derived, but alsoof related and qualitatively similar events in general surroundings. If science wouldnot have been characterized by this ability to precisely predicting the outcome ofphysical processes, the major inventions made in the 18th and 19th century wouldprobably not have been possible and physics may not have had any impact on othersciences at all.

At a relatively early stage in the development of classical mechanics the viewwas expressed that the basic physical laws of motion constitute the essential dy-namic principles of the entire cosmos. In reflecting on the predictability question,Laplace wrote the following, often quoted statement in 1776:

The present state of the system of nature is evidently a consequence of what it was inthe preceding moment, and if we conceive of an intelligence which at a given instantcomprehends all the relations of the entities of this universe, it could state the respectivepositions, motions, and general affects of all these entities at any time in the past or future.Physical astronomy, the branch of knowledge which does the greatest honor to the humanmind, gives us an idea, albeit imperfect, of what such an intelligence would be. Thesimplicity of the law by which the celestial bodies move, and the relations of their massesand distances, permit analysis to follow their motion up to a certain point; and in order to

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determine the state of the system of these great bodies in past or future centuries, it sufficesfor the mathematician that their position and their velocity be given by observation for anymoment in time. Man owes that advantage to the power of the instruments he employs,and to the small number of relations that it embraces in its calculations. But ignorance ofthe different causes involved in the production of events, as well as their complexity, takentogether with the imperfection of analysis, prevents our reaching the same certainty aboutthe vast majority of phenomena. Thus there are things that are uncertain for us, thingsmore or less probable, and we seek to compensate for the impossibility of knowing them bydetermining their different degrees of likelihood. So it is that we owe to the weakness of thehuman mind one of the most delicate and ingenious of mathematical theories, the scienceof chance or probability. 2

In principle everything therefore follows deterministic rules. Either the humanincapability or technical restrictions prevent a complete comprehension of actualempirical phenomena. Laplace’s demon represents a universal scientist who is notlimited by these technical and mental restrictions. It should be noted that thisdemon is not a divine being but that in principle every human being can attainto its capabilities. While a more detailed discussion of this attitude toward realitycan be found below, this attitude should tentatively be denoted as the mechanistic,deterministic worldview.

The tremendous success of this approach in explaining natural phenomena inmechanical, celestial, optical, etc. problems constituted a stimulant for the newlyemerging branches of philosophical thinking in the 19th century. The determinis-tic worldview and the attitude toward the predictability problem began to becomeinfluential in the social sciences as well. While, as was pointed out by Crutch-field et al. (1986), a direct application of Laplace’s statement on predictions tohuman affairs implies that no free human will exists at all, the philosophical devel-opment incorporated this idea in a somewhat hidden manner. Hegel’s philosophyof history, and later Marx’s deterministic laws of economic and social development,indicated that in the course of the 19th century a tendency to compare the overalleffects of human action with qualitatively the same kind of laws of motion, whichhad been applied to the natural sciences, emerged. The philosophical attitude ofthe early 19th century was dominated by an entity called weltgeist which constituteda surrogate for the legislation of the medieval universe: the determinism of classicalphysics, idealistic philosophy, or Marxian sociology began to replace the theologicalnotion of a divine predestination of human life.

If no truly free human will exists, it is possible to generalize individual humanbehavior and to abstract from singular phenomena based in the isolated mindsof human beings. It is therefore possible to describe the actions of an individualaccording to typical patterns of behavior, provided he/she is not characterized bypathological attitudes toward reality. This idea that individuals behave to some de-gree according to typical patterns constitutes the essential prerequisite in establish-ing economics as a scientific branch. Typical patterns of economic behavior wereintroduced to economics by means of a rather simple approach; for example, if therationale of a typical agent consists in maximizing a predetermined utility function

2 Quoted from Crutchfield et al. (1986).

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which lacks psychological or sociological considerations, the fundamental problemof explaining individual economic behavior is replaced by the assumption of indi-viduals acting rationally, i.e., maximizing utility, in a given environment. What haslater been termed the axiomatic foundation of economics is basically nothing morethan the hypothetical determination of fundamental behavioral laws from whichmost results in economics follow tautologically, though usually not obviously.

This determination of fundamental behavioral patterns by hypotheses circum-vented the basic problem of studying individuals acting within an economy andcleared path toward a precise economic science, which resembles a strong similar-ity with classical physics as the most advanced science in the 18th and 19th centuries.While this similarity of emerging economics to physics was probably only vaguelyevident to classical writers until the mid-19th century, the beginning of the mathe-matical formalization of economics in the second half of that century let economicsappear either as a transformation of physical methods to problems of human life oras an application of mathematics, with a status equal to physics. The predecessorsof modern mathematical economics, e.g., L. Walras, W.S. Jevons, and V. Pareto,were not only aware of the similarity, but propagated the use of the methods ofphysics in economics.3 It seems as if the representatives of the Lausanne Schoolconsidered physics as a scientific idol among applied sciences, which is supportedby the fact that some of them were not educated economists, but had their aca-demic origin in mathematics or in the engineering sciences.4 Walras repeatedlymentioned his aim to structure economics in a manner similar to physics5 andclaimed that the classical and pre-classical writers were already implicitly guided bythe same idea:

...the theory of price determination of economic goods or the pure economic theory appears(to have) the character of a real, namely physico-mathematical science. ... Isn’t it truethat all those English economists from Ricardo to J.S. Mill have treated pure economicslike real mathematics? Their sole error ... was that they attempted to develop this branchof mathematics by means of common everyday-language and that they could handle ittherefore only with difficulties and without complete success. ... I...have been concernedwith the development of pure economics as a physico-mathematical science for severalyears. 6

3 Standard references for questions concerning the relation between physics and eco-nomics include, for example, Georgescu-Roegen (1971) and Mirowski (1988).

4 V. Pareto had a doctoral degree in railroad engineering and, like his predecessor L.Walras in Lausanne, had not published much on economic theory when he got his firstacademic appointment. However, Debreu’s (1986) statement that Walras and Paretohad published only novels and other belletristic literature before their first appoint-ments is misleading.

5 In a rather enthusiastic fashion, Walras’ German translator, L.v. Winterfeld, comparedWalras with the astronomer J. Kepler: “. . .Walras appears to me as the Kepler of economics,who incontestably and for all time proves the laws which once were suspected and expressed by (the)German scholar . . .H.H. Gossen in the style of a Kopernikus.” Own translation (H.-W.L.)from the German preface to Walras (1876)

6 Walras (1874), p. 7. Own translation (H.-W.L.) from Walras (1876).

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In an even more pointed style, Jevons (1871) associated the survival of economicswith its use of mathematics:

It is clear that Economics, if it is to be a science at all, must be a mathematical science.7

I. Fisher wrote in his 1891 doctoral thesis:

Scarcely a writer on economics omits to make some comparison between economics andmechanics. One speaks of a “rough correspondence” between the play of “economic forces”and mechanical equilibrium. ... In fact the economist borrows much of his vocabularyfrom mechanics. Instances are: Equilibrium, stability, elasticity, expansion, inflation,contraction, flow, efflux, force, pressure, resistance, reaction, distribution (price), levels,movement, friction. 8

Walras, like Pareto, Cournot, and other early mathematical economists, at-tempted to develop a logically consistent edifice of thought. They clearly saw therestrictions of abstract thought experiments and therefore insisted on a separationof the categories of pure and applied economics. It is certainly inappropriate toclaim that they considered real economies as systems which behave completely anal-ogously to a physical system. Concerning the modeling of economic systems in pureeconomics, however, physics did not only serve as a paragon of the useful exploita-tion of mathematics as an instrument in developing a logically consistent theory. Inaddition to the adoption of its formal methodological approach, physics providedthe basic qualitative foundations of scientific economics. As was mentioned above,the deterministic, mechanistic worldview of physics in the 19th century dominatednot only the internal scientific community but also had a widespread influenceon other disciplines and also on the public weltanschauung. Much in the spirit ofLaplace’s statement, economic systems were therefore interpreted as systems whosedevelopment could be calculated with preciseness if an appropriate degree of in-formation about the structure, the parameters, and the initial values of the systemswere provided to the economist.

This favoring of a methodological approach derived from physics is most clearlyevident in a statement made by J.S. Mill, originally published in 1843, which showsthat physics was not only favored by mathematically educated scholars like Walrasor Pareto:

The phenomena with which this science is conversant being the thoughts, feelings, andactions of human beings, it would have attained the ideal perfection of a science if itenabled us to foretell how an individual would think, feel, or act, throughout life, with thesame certainty with which astronomy enables us to predict the places and the occultationof the heavenly bodies. It need scarcely be stated that nothing approaching to this canbe done. ... This is not, however, because every person’s modes of thinking, feeling, andacting, do not depend on causes; ... (T)he impressions and actions of human beings are

7 Jevons (1871), p. 3.8 Cf. Fisher (1961), p. 25. Fisher himself attempted to develop a consistent value theory

analogous to the theory of equilibrating water cisterns. He even constructed mechanicaldevices to illustrate his ideas.

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... the joint result of (the) circumstances and of the characters of the individuals: andthe agencies which determine human character are so numerous and diversified, ... , thatin the aggregate they are never in any two cases exactly similar. ... Inasmuch, however,as many of those effects ... are determined, ... it is evidently possible to make predictionswhich will almost always be verified. ... For the purposes of political and social sciencethis is sufficient. 9

It must be stressed, however, that the orientation of economics to the paradigms andmethods of physics was already being questioned by economists who are nowadayscategorized as the founders of deterministic (neo-) classical economics.10 A. Mar-shall repeatedly drew attention to the idea that the appropriate fellow-disciplinein the natural sciences which is most closely analogous to economics (as far as thesubject of the field is concerned) is not physics but biology

... the forces of which economics has to take account are more numerous, less definite,less well known, and more diverse in character than those of mechanics. ... economics,like biology, deals with a matter, of which the inner nature and constitution, as wellas the outer form, are constantly changing. ... If however we look at the history of suchstrictly economic relations as those of business credit and banking, of trade unionism or co-operation, we see that modes of working, that have been generally successful at some timesand places, have uniformly failed at others. The difference may sometimes be explainedsimply as the result of variations in general enlightment, or of moral strength of characterand habits of mutual trust. But often the explanation is more difficult. 11

Other authors searched for analogies with even other disciplines. Menger (1871)described his marginal utility concept “just as a difficult as yet untreated topic inpsychology”.12 Edgeworth (1881) called one of his master pieces “Mathematical Psy-chics”. It should be noted that these influences have not always been uni-directional:the American psychological school adopted several ideas developed by Edgeworth.

This construction of analogies between different sciences is usually called reduc-tionism in the science-theoretic literature. A scientific procedure is called reduction-istic if basic properties of a particular science’s study object are derived with the helpof another science’s methodology and existing knowledge. When the statementsof classical mechanics are indeed generally valid and if the deterministic worldviewprevails, all scientific questions (in all fields) can consequently be treated with thehelp of the principles of physics. The standard, hierarchical reduction scheme inTable 1.1 which covers only a few interesting sciences is due to Medawar (1969) and

9 Mill (1973), pp. 847f., emphases in original. For the purpose of this little excursioninto the history of science, Mill’s Logic can be considered the gap filling contributionbetween enlightenment philosophy, the methodology of the subsequent developmentof classical mechanics, and the methodology of economics and other social sciences.

10 Compare, e.g., Blaug (1978), p. 311, for the resistance to the emerging mathematicalmethods among well-reputed economists.

11 Marshall (1938), p. 772. Compare also Hodgson (1993) for a recent discussion ofMarshall’s attitude toward biology.

12 Menger (1871), p. 94. Own translation (H.-W.L.).

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describes economics as a science which can be treated with biological principles.Biology itself is nothing else than a particular investigation of the working of basicchemical processes and, finally, physics describes the essential relations inherent inall observable phenomena.

4. Economics↑

3. Biology↑

2. Chemistry↑

1. Physics

The Reduction Scheme in theDeterministic Worldview

Source: Medawar (1969), pp. 15ff.Table 1.1

A consequent application of this re-duction scheme implies that the fields2.-4. do not possess a real right to ex-ist as self-sustained sciences. If the con-stituting properties of a particular fieldcan be described with the help of themethods and qualitative results of sci-ences situated in front of it, this field ob-viously does not possess those essentialproperties which could justify the dif-ferentiation as a separate science. Thesubject of reducible sciences appears asa specific application of the more gen-eral science in the scheme.

It might be suspected that the scientific reductionism was a historic phenome-non that does not have a serious recent relevance. However, modern fields in thenatural sciences like molecular biology reduce biological phenomena to chemicalprocesses. Parts of evolutionary economics emphasize the biological principles ofnatural selection.13 Finally, if the above mentioned discussion of the analogy be-tween the methodology of physics and economics appears just as a historic anecdotedating back to the last century the reader should compare Jojima (1985), Sebba(1953), or Thoben (1982) for an indication that the discussion is still going on(although this happens to take place somewhere at the outskirts of mainstreameconomics).

Aside from this general recalling of reductionism as a procedure which is nottoo uncommon in the history of sciences and the few critical remarks that em-phasize analogies to other sciences than physics, Marshall, Menger, Jevons andmost contemporaries nevertheless considered physics as the science which can serveas a paragon in respect to both the formal apparatus and the involved worldviewin economic theorizing. Marshall’s general understanding of economics as asub-discipline of natural philosophy and especially Walras’ concentration on themathematical methodology have, in the scientific spirit of the last century, survivedin mainstream economics until today. A majority of the topics covered by modernmathematical economics, especially in the general equilibrium framework, still dealwith the same problems which interested classical economists like Walras, and it isthis tradition inherited from the classical writers, which still allows one to assign theterm “mechanistic worldview” to most economic approaches. Though this term isoften quoted (mainly among critics of neoclassical economics) it nevertheless seemsuseful to investigate it more carefully. It will turn out that the common association

13 Compare also the standard discussion on reductionism in modern evolutionary biologyitself. Cf. Dawkins (1987).

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of this term with “determinism” in a most general sense cannot cover all facettes ofthe relevant, basic science-theoretic discussion. Without a more elaborated discus-sion of the relevant terms it seems to be impossible to evaluate the above mentionedstatements that nonlinear dynamics tackles basic science-theoretic foundations ofseveral disciplines.

1.2. The Deterministic Worldview and Deterministic Theories

In the light of the discussion in the previous section classical physics has played aprominent role in the emergence of the deterministic worldview. It is thus useful torecall the standard paradigm governing research in physics and most other naturalsciences roughly until the end of the 19th century.14

• A physical phenomenon can be isolated from the environment. The study of theisolated (or de-coupled) physical systems and processes (for example in labora-tory experiments) can provide a precise understanding of the problem’s nature.The abstraction from noisy surroundings during this isolation may reveal thepure properties of a physical system.

• Laboratory experiments can be repeated as often as desired. In these experi-ments constants can be derived which permit the formulation of laws of nature.These laws have an arithmomorphic character, i.e., they can be formulated in math-ematical terms and follow the standard mathematical rules.15

• The interaction of different isolated phenomena occurs in an additive manner,i.e., it is dominated by the principle of superposition. This implies that “the most gen-eral motion of a complicated system of particles is nothing more than a linear superpositionof the motions of the constituent elements.”16

• If it is not possible to properly analyze all constituent elements of a given system,perturbing an existing linear model (which was constructed by superposition)can always explain the originally disregarded phenomena.

The paradigmatic attitude toward the study object expressed by the above list canbe called the mechanistic worldview. Georgescu-Roegen (1971) summarizes thisparadigm as follows:

...a science is mechanistic if, first, it assumes only a finite number of qualitatively differentelements, and if, second, it assumes a finite number of fundamental laws relating theseelements to everything else in the same phenomenal domain. 17

Other terms can be used in characterizations of the paradigm. West (1985) iden-tifies the procedure expressed in the aforementioned list with a linear science:

14 Cf. West (1985), for a longer discussion.15 Cf. Georgescu-Roegen (1971), p. 44, for an intensive discussion of the term.16 Cf. West (1985), p. 70 .17 Georgescu-Roegen (1971), p. 115.

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Physical reality could therefore be segmented: understood piece-wise and superposed backagain to form a new representation of the original system. ... Thus the philosophy was tosolve the linear problem first, then treat the remaining interaction (that was not treatedquite properly) as a perturbation on the linear solution, assuming throughout that theperturbation is not going to modify things significantly. 18

It can in fact be shown that the majority of the most important theoretical discov-eries in classical physics followed this procedure. The investigations of sound as awave phenomenon by Newton, Lagrange, and Laplace, or the findings on thevibration of strings by D. Bernoulli, Lagrange, and Euler are good exampleshow a complex phenomenon was separated into single elements which could beanalyzed by means of simple techniques.19

Classical economic writers incorporated this procedure in analyses of economicbehavior. The following items appear as the most important properties of (neo-)classical economic analyses in the present context:

• Individual behavior (or the behavior of individuals in an economic unit like amarket) can be isolated from the economy as a whole.

• Human behavior can be described in terms of general behavioral patterns, suchthat the analysis may indeed abstract from individual behavior.

• Individual human behavior is comparable to the physical laws of motion, it isboth regular and predictable. If the environment is known with precision, indi-vidual behavior within that environment is deterministic.

• The behavior of a society consists of the additive actions of its members. Theprinciple of superposition implies that the behavior of a society as a whole doesnot differ from the sum of the individual actions.

This economic worldview implies that an economy can be described by linear (orquasi-linear) functional relations. It abstracts from the presence of unpredictable(irrational) individual behavior, from restrictions in the environment, from non-additive interdependence between different individuals and/or actions, etc. A lotof progress has been made since the days of the classical (neoclassical) writers withrespect to the above mentioned and other limitations, but the dynamic aspects ofthe theory are still more or less characterized by the same concentration on linearrelations as was the case during the first formalizations of the development of aneconomy over time.

At the turn of the century physics began to experience a basic revolution (theuse of the term seems to be undisputable for a description of that event). Quantummechanics and later relativity theory constituted a challenge to the dominatingclassical mechanistic paradigm. It was demonstrated in the subsequent years thatclassical mechanics was only an approximation to those phenomena that happen

18 West (1985), p. 7019 Cf. West (1985), pp. 68 ff., for a short survey. It is remarkable that Euler personally

rejected the superposition principle though he actually proved its correctness in thecase of the wave equation.

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to take place on a scale immediately observable by human beings. Heisenberg’squantum mechanics with the unsharpness relation and Schrodinger’s equations havedemonstrated that the best that can be done in many cases is to provide a stochasticdescription of possible phenomena.20 The conflict between this newly emergingparadigm and Laplace’s determinism is obvious.

The basic indeterminism (in the classical sense) of phenomena on the molec-ular as well as the cosmic layer initiated a long discussion in the science-theoreticliterature on the relevance of the mechanistic worldview. When phenomena areessentially indeterministic it can obviously be doubted whether the formulationof deterministic, arithmomorphic laws of nature makes any sense at all. Stochasticdescriptions of reality may be considered the only meaningful approach to explana-tions of physical phenomena. Alternatively, it might be supposed that deterministiclaws of nature represent good approximations of reality on that scale immediatelyobservable by human beings. It turns out that a rough distinction between deter-minism and indeterminism is not extremely well-suited for a discussion of the rel-evance of the deterministic worldview. Popper (1982) thus distinguished betweenscientific determinism and a deterministic theory.21

The doctrine of ‘scientific’ determinism is the doctrine that the state of any closed physicalsystem at any given future instant of time can be predicted, even from within the system,with any specified degree of precision, by deducing the prediction from theories, in conjunc-tion with initial conditions whose required degree of precision can always be calculated(in accordance with the principle of accountability) if the prediction task is given. 22

The scientific determinism can therefore be interpreted as a deterministic worldviewwhich is exclusively based on empirical knowledge collected in the form of scientificactivity. In contrast, Popper defines a deterministic theory (a “prima facie deterministictheory”) as follows:

A physical theory is prima facie deterministic if and only if it allows us to deduce, froma mathematical exact description of the initial state of a closed physical system whichis described in terms of the theory, the description, with any stipulated finite degree ofprecision, of the state of the system at any given future instant of time. 23

The distinction of the two terms is useful for two reasons. The scientific deter-minism (i.e., the mechanistic worldview in Georgescu-Roegen’s term and Laplace’ssense) does not necessarily result in the construction of deterministic theories. It

20 Even the motion in (usually) simple devices like mechanical clocks (which occasionallyappear as the incarnation of the mechanical approach) might be called indeterministicwhen the molecular layer is considered instead.

21 The reference originally constituted a postscript to the English translation of his Logicof Scientific Discovery, first published in German in 1934, which has had a major impacton the methodology of all social sciences, including economics.

22 Popper (1982), p. 36. The principle of accountability states that it is possible to determinethe required precision of initial points for a desired precision in the predictions.

23 Popper (1982), p. 31.

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will become obvious from the material presented in subsequent chapters that dy-namical systems which reflect the deterministic worldview, i.e., equations withoutany stochastic elements, might not constitute deterministic theories. An example

Scientific Determinism Deterministic Theory

For each phenomenon: For each mathematicallydescribable phenomenon:

• initial values with • initial values withinfinite precision finite precision

• prediction with • prediction with giveninfinite precision mathematical precision

Popper’s Distinction between the Deterministic Worldview and a Deterministic TheoryTable 1.2

is the famous three-body problem: although the basic laws of motion are preciselygiven, the motion of three or more bodies with special mass constellations in spacecannot be calculated with an arbitrary precision. Thus, Newtonian mechanics can-not be called a deterministic theory in the sense of Popper’s definition while itcertainly reflects a deterministic worldview.24 Without a distinction between thetwo terms, Newtonian mechanics should be called indeterministic; obviously, thiswould contradict the self-assessment of classical authors. Alternatively, even if thedeterministic worldview prevails theoretical investigations of particular phenomenado not necessarily have to make use of deterministic theories. The influence of hu-man incapabilities have already become obvious from Laplace’s statement; onlyhis demon is able to comprehend the actually infinite numbers of freedom. Thus,stochastic descriptions of reality may constitute sufficiently accurate approaches tophysical phenomena.

A second justification for the distinction can be seen in the idea that an inde-terministic worldview is not necessarily irreconcilable with the use of deterministictheories. First, an indeterministic worldview does not exclude the possibility that de-terministic niches exist which can be treated with the help of deterministic theories.Second, a deterministic theory can serve as an approximation of those phenomenawhich actually should be considered indeterministic. The success of Newtonianmechanics in calculating the motion of known celestial bodies is not diminished bythe fact that on the molecular layer the motion is indeterminate. This seems to beparticularly relevant for economics when economic models are considered thoughtexperiments instead of precise pictures as close to reality as possible.

Popper’s (1982) distinction between scientific determinism and a deterministictheory is useful when classical mechanics and all reduced sciences are consideredthe standard references for deterministic theories. Popper’s main contribution

24 Popper himself has another view on this last statement, cf. Popper (1982), p. 31.

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in this context probably consists in his provision of incentives to concentrate onstochastic approaches in the social sciences. It does not seem to be quite clearwhat kind of appearance such stochastic descriptions of economic phenomenashould have. However, it seems as if descriptions of, e.g., consumer behavior en-tirely in terms of well-defined preferences and deterministic environments withoutany stochastic influences contradict Popper’s anticipation of an indeterministic ap-proach to economic phenomena.

The critique of the deterministic worldview in economics refers to those parts ofmainstream economics whose scientific origins date back to the late 19th century.Indeed, while neoclassical economists were still arguing in terms of mainstreamclassical mechanics, a new way of thinking eventually emerged in the natural sci-ences which involved a drastically different attitude toward reality. Around the turnof the century, advances made in the natural sciences and mathematics raised an-other doubt to the validity of the mechanistic worldview. While the development ofrelativity theory or quantum mechanics with its unsharpness relation constituted achallenge to the deterministic worldview, the discovery of mathematical propertiesof several dynamical systems represented a challenge to the deterministic theory itself.It was shown that problems can emerge in predicting the evolution of dynamicalsystems which are completely deterministic in the sense that no stochastic elementsare involved in the definition of the system. It should be noted that the relevanceof new developments in mathematics and physics either went unrecognized by themajority in the physics profession or was considered to be only marginally signifi-cant to mainstream science. Thus, with mainstream science still elaborating on theclassical mechanistic paradigm, classical economists should not be discredited fortheir attempts to adapt the methodology of emerging formal economic theory towell-accepted paradigms.

Despite the fact that physics was still dominated by the classical paradigm atthe turn of the century, this does not mean that the public was not open to newideas. In fact, mathematicians like H. Poincare had attained a reputation over thedecades which initiated an uncountable number of honorary lectures, not only forthe mathematical community, but also for a broader public audience. For example,as early as 1908 Poincare stated in front of a general audience:

A very small cause which escapes our notice determines a considerable effect that we cannotfail to see, and then we say that the effect is due to chance. If we knew exactly the laws ofnature and the situation of the universe at the initial moment, we could predict exactlythe situation of that same universe at a succeeding moment. But even if it were the casethat the natural laws had no longer any secret for us, we could still only know the initialsituation approximately. If that enabled us to predict the succeeding situation with thesame approximation, that is all we require, and we should say that the phenomenon hadbeen predicted, that it is governed by laws. But it is not always so; it may happen thatsmall differences in the initial conditions produce very great ones in the final phenomenon.A small error in the former will produce an enormous error in the latter. Prediction becomesimpossible, and we have the fortuitous phenomenon. 25

25 Poincare (1952), p. 76. Originally published in Poincare (1908), p. 68. I am gratefulto D. Farmer for providing this reference to me.

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18 Chapter 1

The very essence of Poincare’s statement was not immediately realized in the math-ematics community, though his work not only initiated research in several dynamicphenomena but even still constitutes a challenge to recent mathematicians. Fatou(1919) and Julia (1918) made important observations about the behavior of iter-ated complex maps, but it took nearly fifty years before some of the basic results ofPoincare’s work were exploited in a pioneering, but generally unnoticed work ofE.N. Lorenz (1963). His inspection of a dynamical system in the context of a me-teorological phenomenon impressively demonstrated the conceptual impossibilityof precisely predicting a dynamical system’s future development without an abso-lutely precise knowledge of the system parameters and the initial values of the statevariables. It was left to the currently renowned work by Ruelle/Takens (1971)and Li/Yorke (1975) to encourage a wide interest in nonlinear dynamics, whichsometimes even appears to be a fashionable scientific trend.26

An immediate consequence of the results obtained in studying nonlinear dy-namical systems consists in the need for a revision of Popper’s distinction betweenscientific determinism and deterministic theories. Popper’s concept of a determin-istic theory is based on the mathematical properties of basically linear dynamicalsystems. When nonlinear dynamical systems do not possess the predictability prop-erty known from established linear deterministic systems, deterministic theories(i.e., arithmomorphic theories without any stochastic components) have to be dis-tinguished according to their possible output. Deterministic theories can behave

Scientific DeterministicDeterminism Theories

For each For linear and For chaotic non-phenomenon quasi-linear systems linear systems

• initial values with • initial values with • initial values withinfinite precision finite precision finite precision

• prediction with • prediction with • prediction onlyinfinite precision given mathe- for short time

tical precision intervals

A Revision of Popper’s Distinction in the Light of Nonlinear Dynamical SystemsTable 1.3

in the way described in the Popperian scheme. When their behavior does not es-sentially differ from the behavior of linear systems they will occasionally be called

26 Of course, this does not mean that in the course of the century there was no mathemat-ical progress in the theory of nonlinear dynamical systems. Indeed, relaxation oscilla-tions, for example, were intensively discussed in the 1920s. The work of Cartwright,Levinson, and Littlewood in the late 1940s actually laid the foundations for the recentanalysis of chaotic dynamical systems.

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1.3. The Dominance of Linear Dynamical Economics 19

quasi-linear dynamical systems. However, it cannot be excluded that determinis-tic theories behave in an indeterministic fashion: although the functional form ofthe systems is completely deterministic (i.e., without any stochastic components),the finite precision in the initial values is responsible for the fact that predictionover more than a very short time interval is impossible. These complex nonlineardynamical systems thus require a revision of Popper’s scheme (cf. Table 1.3).

The theoretical and empirical research in several disciplines, mainly in the nat-ural sciences, has concentrated on the investigation of nonlinear systems duringthe last two decades. While nonlinear approaches have occasionally been enthusi-astically adopted in some disciplines, economics (at least mainstream economics)seems to be characterized by a general hesitancy in exploiting the (mathematically)new ideas. This may be explained by the fact that linear dynamical systems are theappropriate environment for those economic ideas whose origin dates back to thewritings of the 19th century classical economists.

On the other hand, nonlinearities have been investigated for a long time inthe minds of those economists who have developed a more critical attitude towardthe functioning of a market economy. For the reader who is interested in recentdevelopments in the theory of nonlinear dynamical economics, it will probably besurprising which topics had been selected and solved by economic writers like R.Goodwin many years before the scientific community became aware of the impor-tance of those nonlinearities.27 It is noteworthy, however, that the impetus for thestudy of nonlinear dynamical systems originated once again in the natural sciences,this time with the sometimes spectacular advances made in the analysis of practicalphysical or biological phenomena.

The following section attempts to explain why mainstream economics still con-centrates on linear models though the foundations of nonlinear economics werelaid more than forty years ago.

1.3. The Dominance of Linear Dynamical Systems in Economics

A continuous-time dynamical system is called a linear system when it can be writtenin the form

x(t) = Ax(t) + c, x, c ∈ Rn, t ∈ R, (1.1)

with x(t) as the n-dimensional vector of state variables at the point in time, t, x(t) =dx(t)/dt as the vector of time derivatives of the state variables, A as an n× n matrixof constant coefficients, and c as an n-dimensional column vector of constants.Analogously, a discrete-time dynamical system is called a linear system when it canbe written in the form

xt+1 = Axt + c, x, c ∈ Rn, t ∈ Z, (1.2)

27 Cf. Harcourt (1984) or Velupillai/Ricci (1988) for honory lectures on Goodwin’swork.

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20 Chapter 1

with xt as the vector of state variables in the discrete period t, and A and c definedas above. The solution of these dynamical systems for low values of n and a briefsurvey of a few stability criteria are contained in the Appendix A.1. The basicdynamic properties of these systems relevant for this section are the following:

• Abstracting from a few exceptional examples, linear dynamical systems possessonly single fixed points (equilibria). It follows that global and local analyses ofthe dynamic behavior coincide.

• The fixed points of linear dynamical systems belong to one of the followingcategories: 1) stable/unstable nodes (monotonic convergence/divergence to/from the fixed point), 2) stable/unstable foci (monotonic oscillations toward/away from the fixed point, 3) centers, or 4) saddle points.

Center dynamics occur only for particular, numerically precise parameter constella-tions and usually become relevant only in descriptions of limit cases. Unstable nodesand foci imply the eventual motion of the state variables toward infinity. Thus, if aneconomist’s task is to model the bounded motion of an economy with the help oflinear dynamical system, stable nodes and foci represent the appropriate types offixed points. Several economic applications (including perfect foresight models,for example) concentrate on saddle points because the motion along the stablemanifold (cf. the horizontal dashed lines in Figure A.1.f) represents the uniquepath toward the fixed point.

In the face of the various types of dynamic behavior outlined in the rest of thisbook linear dynamical systems are thus able to describe only a small number of dy-namic phenomena. As linear systems dominate dynamical economics, the questionarises what the reasons for this concentration on a limited set of hypothetically pos-sible dynamic phenomena are. Basically, two possible answers to this fundamentalquestion can be distinguished.28

i) Compared with some branches of the natural sciences, economics has laggedbehind in the technical as well as the methodological aspects of scientific work.

ii) Economics is characterized by a paradigmatically motivated concentration onfixed points (equilibria), to the point that other dynamic phenomena than thestability of these fixed points are ignored though they are at least known to existin the formal mathematical literature.

In the following, both complexes will shortly be discussed in separate sub-sectionsthough they are actually immanently identical.

ad i)

West (1985) distinguishes five stages of scientific progress:29

Stage 1: Verbal description of the subject and the immanent logic of the problem.

28 Of course, other pragmatic reasons cannot be excluded. Compare, for example, theenlightening introduction in Boldrin/Woodford (1990).

29 Cf. West (1985), pp. 3-10.

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1.3. The Dominance of Linear Dynamical Economics 21

Stage 2: Formal identification of the problem and quantification of the mathe-matical relations.

Stage 3: Consideration of the dynamic aspects of the mathematical model in theform of linear dynamical systems.

Stage 4: Re-consideration of the basic scientific principles and testing whethermodels in stage 3 can represent all mentally possible phenomena. Estab-lishment of the need to include nonlinear aspects in dynamical models.

Stage 5: Development of complete nonlinear models, which are indeed able toexplain the phenomena outlined in the general reflections in stage 1.

It is possible to assign distinguished economists to the different stages of scientificprogress according to this rough classification: the classical writers like Smith,Ricardo, Malthus, etc. dominate the first stage. Stage 2 is occupied by neoclassicaleconomists like Marshall, Walras, Pareto and others, who also bridged the gapto the third stage of scientific progress. The mathematically sophisticated literatureon the existence of equilibria and its stability in a general equilibrium framework,dominated by the work of Arrow and Debreu, has to be ascribed to this stage. Stage4 in the above list already leads to the frontiers of recent research in economics.The recent theoretical attempts to demonstrate the possibly drastic divergence inthe behavior of nonlinear models from linear ones raise questions concerning thegeneral validity of the standard linear and mainstream thought experiments.

A general, nonlinear, dynamic, economic theory representing stage 5 is there-fore obviously not in sight. While other sciences are also still far away from acomplete realization of the programme, it seems as if nonlinear phenomena havealready been incorporated into other disciplines with more acceptance than in eco-nomics. It may be argued that it is simply a matter of time until economics adoptsthose new techniques which become more and more important in other disciplinesbecause economics has always reacted sluggishly to new formal developments. How-ever, in contrast to the situation at the end of the last century, economists usuallydo not lack a profound mathematical background anymore. While this excuse forinvestigating mainly linear systems is thus not acceptable anymore, it may be worth-while to elaborate a little bit more on the second justification of the use of lineardynamical systems.

ad ii)

The concentration on linear dynamical systems in economics is usually justified (ifat all) with an excuse. The phenomenon under consideration is actually thoughtof as being nonlinear just because no convincing argument can be delivered whycomplex structures like economic systems should be characterized by highly stylizedand simple relations in the form of linear equations. However, as linear models canbe analyzed much easier than nonlinear ones (at least in low-dimensional cases),actual phenomena are approximated by stylized linear structures.30

30 Usually, this simplification goes hand in hand with the prospect of future research inwhich the influence of nonlinearities should be investigated. Compare also Baumol(1987), p. 105, for a discussion of this procedure.

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22 Chapter 1

This simplification can certainly be justified in many cases, especially when thetrue dynamical structure does not diverge essentially from the assumed linear form.If, e.g., the number of equilibria in a dynamical system is identical in linear and non-linear formulations, if the nonlinear functions diverge only minimally from linearforms, and if the interplay of the different nonlinear functions does not imply phe-nomena which are unobservable in linear systems, the usage of linear functionsmay indeed lead to a (qualitatively) sufficiently good approximation of the system’strue behavior. However, in higher-dimensional systems it will become increasinglydifficult to discriminate between good and bad approximations. Indeed, it may be-come impossible to evaluate the effects of neglecting a special (maybe numericallysmall) nonlinear term which perhaps can drastically change the dynamic behaviorof the system.

It may be that some economists are not aware of the potential qualitative differ-ences between linear and nonlinear dynamical systems. One major reason for theconcentration on linear systems may, however, have its origin in paradigmatic idealsof the functioning of an economy. If one is (explicitly or implicitly) guided by theclassical mechanistic weltanschauung outlined in Sections 1.1 and 1.2, then there isindeed no need to consider anything other than linear systems. The fact that lin-ear dynamical systems behave in a very regular fashion and that the most complexdynamic behavior, namely steady regular oscillations, can be modeled only by as-suming a numerically exact parameter constellation support the basic idea that aneconomy’s equilibrium is asymptotically stable. In addition, the dynamic behavioris predictable. A model which demonstrates the impossibility of predictions canbe considered to be part of a negative theory in economics: when the provisionof predictions is regarded a justification for the mere existence of economics sev-eral nonlinear dynamic models will certainly be treated very skeptically (and theirdestructive effects will be emphasized).

Consequently, as can be expected, the different scientific economic schools havedeveloped a different attitude toward nonlinearities in economic models. Linearmodels have been employed especially by neoclassical and “new” classical writerswho, after the (neo-) Keynesian disequilibrium interlude, have concentrated onthe investigation of equilibrium economics once again. The assumption of lineardynamical systems in these classical models is often justified by technical reasons:

The predominant technical requirement of econometric work which imposes rational expec-tations is the ability to write down analytical expressions giving agent’s decision rules asfunctions of the parameters of their objective functions and as functions of the parametersgoverning the exogenous random process they face. Dynamic stochastic maximum prob-lems with quadratic objectives, which produce linear decision rules, do meet this essentialrequirement ... Computer technology in the foreseeable future seems to require workingwith such a class of functions, and the class of linear decision rules has just seemed mostconvenient for most purposes. ... It is an open question whether for explaining the centralfeatures of the business cycle there will be a big reward to fitting nonlinear models. 31

This opinion will probably not be shared by every Rational Expectations theoristbut it implicitly uncovers the ignorance of the importance of nonlinear phenom-

31 Cf. Lucas/Sargent (1978), p. 314.

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1.3. The Dominance of Linear Dynamical Economics 23

ena. In fact, former schools in equilibrium economics were not all characterizedby this limited methodological point of view. The literature that concentrated onthe properties of tatonnement processes in a general equilibrium context in the1950s and 1960s also dealt with nonlinear systems. However, the focus of researchdid not consist of investigations into the effects of different kinds of possibly in-volved nonlinearities. Nearly all contributions concentrated on the question whichassumptions are necessary and/or sufficient to ensure the stability of a general equi-librium. This literature therefore excluded all those effects of nonlinearities whichconstitute an essential deviation from the qualitative behavior of linear systems. Or,in other words, only those nonlinearities were considered whose implied behavioris sufficiently close to that of linear systems.

Concentration on linear dynamical structures implies a conceptual problemwhich becomes evident in attempts to describe and explain actual time series. Theseseries are obviously not characterized by the regular kind of behavior which is typi-cal in deterministic linear systems; instead, several irregularities in the form of, e.g.,different types of noise, different frequencies in oscillating series, etc., seem to beinvolved. The New Classical Macroeconomics overcomes this problem by introduc-ing stochastic exogenous disturbances in basically linear dynamical structures.32 Aneconomy isolated from its surrounding is believed to behave in a regular fashion,i.e., if it is displaced from its fixed point it returns toward this state in the form ofa monotonic or regularly oscillating motion. The observed irregularity in actualeconomies’ time series can then be explained by the influence of random terms,which do not necessarily have any purely economic meaning.33 When superim-posed stochastic disturbances take place in every period in discrete-time systems(or at each point in time in continuous-time systems) an interesting phenomenoncan be observed in these systems. When the deterministic part of the model impliesoscillations with monotonic decreases in the amplitude, i.e., when the fixed pointis a stable focus, the introduction of the disturbances generates persistent fluctua-tions. Figure 1.1 attempts to illustrate this phenomenon. A linear, discrete-time,dynamical systems generates the continuously drawn time series of one of the twostate variables. The oscillation is dampened and x1 converges toward its fixed-pointvalue. The dotted line represents the time series generated by the same determin-istic system but with superimposed, normally distributed, stochastic disturbances.The time series displays persistent fluctuations. Remarkably, the system generatesup- and downswings of the time series which prevail for several periods; the intro-duction of permanent stochastic disturbances therefore does not simply imply apositive or negative offset of the deterministic time series with the magnitude of thestochastic term.34

32 Compare also Brock (1991) for a discussion of the standard macroeconomic (macroe-conometric) approach in the New Classical tradition.

33 The observation that the influence of additive random terms in linear business cyclemodels indeed implies theoretically generated time series which closely resemble actualseries dates back to Frisch (1933), Slutzky (1937), and Kalecki (1954).

34 This effect has actually been known for a long time in the economic dynamics literature,cf. Samuelson (1947), pp. 335ff.

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24 Chapter 1

x1

Time

Persistent Oscillations in a Stochastic Linear Dynamical SystemFigure 1.1

It can be argued that assuming these linear structures with superimposed sto-chastic influences is justified when actual time series do not suggest a falsificationof this hypothesis. However, it was demonstrated by Blatt (1978, 1980, 1983)that statistical procedures may be misleading in discriminating between linear andnonlinear structures. Suppose that a time series is generated by a linear dynamicmodel with stochastic influences. It can be expected that a linear regression willfit the data extremely well. On the contrary, suppose that a time series is gener-ated by a deterministic nonlinear model. It is not immediately clear that a linearregression will reject the hypothesis of a linear structure with stochastic influences.Blatt (1983) performed the following experiment:35 consider the discrete-time,multiplier-accelerator model of Hicks (1950) with ceiling and floor.36 The modelis nonlinear because the ceiling (the maximal growth path) and the floor (the min-imal growth path determined by autonomous investment) constitute upper andlower bounds to the endogenous fluctuations. It is crucial to the nonlinear ver-sion of the Hicks model that the endogenous (linear) fluctuations are exploding.Blatt assumed the following parameter specifications in the endogenous part ofthe Hicks model, i.e., the second-order difference equation

Yt = C0 + I0 + (c+ β)Yt−1 − βYt−2

= 25.0 + (0.75 + 1.5)Yt−1 − 1.5Yt−2,(1.3)

with c as the marginal propensity to consume and β as the accelerator. The values ofthe parameters in (1.3) imply exploding oscillations. A time series generated by thedeterministic nonlinear model, i.e., equation (1.3) with upper and lower bounds,was investigated by postulating the linear stochastic equation

Yt = A+ (c+ β)Yt−1 + βYt−2 + ut. (1.4)

35 Compare also Brock (1988b) for a discussion of Blatt’s results.36 Cf. Gabisch/Lorenz (1989), pp. 49ff., for a description of the model.

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1.3. The Dominance of Linear Dynamical Economics 25

Surprisingly, this linear lagged model fits the data of the nonlinear model suffi-ciently well. Standard statistics for the model are

Yt = 28.5 + (0.713 + 0.887)Yt−1 − .887Yt−2

(R2 = 0.92, DW = 2.17, H = 0.56)(1.5)

withH as the alternative Durbin statistics. The estimated value of the accelerator β islower than 1, indicating that the economy is inherently stable though the underlyingdynamical system (1.3) is unstable. An inspection of the statistics does not leavemuch room for rejecting the linear structure of the time series, even when theprinciples of critical rationalism are kept in mind. It can be suspected that similareconometric investigations of actual time series suggested the presence of linearstochastic structures in a multitude of cases and that the presence of nonlinearitieshas probably been rejected too many times.

Nonlinear approaches to economic dynamics have been investigated mainly byeconomists who felt uncomfortable with the classical paradigm of equilibrium eco-nomics. Most contributions to nonlinear economic dynamics in the postwar era aretherefore credited to authors usually assigned to post-Keynesianism, neo-Keynes-ianism, neo-Ricardianism, etc., though these contributions did not always make useof the mathematical advances in dynamical systems theory of the day. However,it would be misleading to attribute research in nonlinear economic dynamics ex-clusively to these schools. Nonlinearities have played a particular role in severalfields dominated by neoclassical writers. For example, oscillating control trajecto-ries were known to exist in nonlinear optimal control theory long before the profes-sion became aware of the potential relevance of nonlinearities in other fields. Mostinterestingly, recent work on the effects of nonlinearities in the standard domainof mathematical economics, namely the general equilibrium analysis, is becomingmore and more important.

Oscillatory motion of economic variables has almost always been identified withfluctuations observable in capitalist, market-oriented, Western economies. The ideathat phenomena like investment cycles could have been an empirical fact in theformer socialist East European countries usually did not come to mind. However,recent work by Brody/Farkas (1987) and Simonovits (1991a,b) indicates thatsuch cycles were at least theoretically possible in socialist economies. It seems asif the emergence of oscillatory behavior in dynamic economic systems cannot beexcluded per se in models of various economic schools.

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Chapter 2

Nonlinearities and Economic Dynamics

I f the world is not linear (and there is no qualitative reason to assume the con-trary), it should be natural to model dynamic economic phenomena in the form

of nonlinear dynamical systems. However, there will not always exist an advan-tage in such a modelling. It depends crucially on the kind of nonlinearity in amodel and sometimes on the subject of the investigation whether techniques ap-propriate to nonlinear systems provide new insights into the dynamic behavior ofan economic system. Nonlinearities may be so weak that linear approximations donot constitute an essential error in answering qualitative questions about the sys-tem, e.g., whether or not the system converges to an equilibrium state. While thisis certainly true for many low-dimensional systems, the effects of nonlinearities inhigher-dimensional systems cannot always be anticipated with preciseness, implyingthat linear approximations should be treated with skepticism especially when thenonlinearities obviously diverge from linear structures.

Unfortunately, the techniques for analyzing nonlinear dynamical systems are farless developed than for linear models. In detail, it is usually not possible to solvea nonlinear dynamical system anymore, i.e., to provide an explicit expression thatdelivers the value of a variable at a specific point in time when an initial value isgiven.1 What is left to an analysis of nonlinear systems is the description of the qual-itative behavior in the sense that it is occasionally possible to determine under whatconditions a dynamical system exhibits a closed orbit or displays related dynamicphenomena. Occasionally it is also possible to exclude the occurrence of somephenomena typical for nonlinear dynamical systems. In these cases linear approx-

1 It should be noted that the same is actually true for high-dimensional linear systemswhere computational difficulties usually preclude the determination of a solution.

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2.1. Preliminary Concepts 27

imations can provide a sufficiently precise description of the dynamical propertiesof a given nonlinear system.

This chapter deals mainly with the concept of closed orbits in a dynamical sys-tem. Although it is exclusively defined for the two-dimensional case, the Poincare-Bendixson theorem has become one of the most popular tools in analyzing non-linear dynamical systems. The power of this tool will be demonstrated through thepresentation of economic examples from standard Keynesian business-cycle theoryand simultaneous price-quantity adjustment processes. As the Poincare-Bendixsontheorem does not exclude the existence of multiple closed orbits, a separate sec-tion is devoted to the question of the uniqueness of these cycles. An exampleof a so-called conservative dynamical system is provided through a presentation ofGoodwin’s (1967) model of the class struggle, which in terms of dynamical systemstheory is an example of a so-called predator-prey system. The chapter closes with ashort section on relaxation oscillations, i.e., a particular type of dynamic behavior thatemerges when the adjustment speed of one of the state variables is very large, and afew remarks on the irreversibility phenomenon observable in dissipative dynamicalsystems.

2.1. Preliminary Concepts

In this chapter, only continuous-time dynamical systems will be discussed.2 Most ofthe following concepts can be transformed to the case of discrete-time dynamicalsystems. Consider the n-dimensional, ordinary differential equation system defin-ing the motion of the state variables xi, i = 1, . . . , n 3

x1 = f1(x1, . . . , xn),...

xn = fn(x1, . . . , xn),

(2.1.1)

or, in vector notation,

x = f(x), x ∈ W ⊂ Rn, (2.1.2)

with W as an open subset of Rn and a dot over a variable denoting the operatord/dt. The functions fi, i = 1, . . . , n are usually assumed to be C∞. Differentialequation systems like (2.1.1) describe vector fields in W, i.e., for each x ∈ W thedynamical system unambiguously determines the direction and the speed of changeof that point. Figures 2.1.a and 2.1.b depict two examples of vector fields in theplane. A solution curve, trajectory, or orbit is defined as Φt

(x(0)

), i.e., when a certain

2 A discussion of the advantages and disadvantages of different time concepts will beavoided in this book. Cf. Gandolfo/Martinengo/Padoan (1981) for a discussion.

3 The dependence of the variables on t will be ignored for notational convenience.

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28 Chapter 2

2.1.a 2.1.bStylized Vector Fields in R

2

Figure 2.1

2.2.a 2.2.bA Solution Curve and the Flow of a Dynamical System

Figure 2.2

x(0) is given, Φt

(x(0)

)provides the values of x at t (cf. Figure 2.2.a). The flow

Φt(x): Rn → Rn of system (2.1.2) describes the future development of all x(0) ∈ W(cf. Figure 2.2.b with t0, t1, and t2 as distinct points in time, t ∈ R).

Central in the discussion of nonlinear dynamical systems is the notion of an at-tractor. There exist subtle differences between different definitions of an attractorin the literature; the following definition should be understood as a working def-inition. An attractor is an example of an invariant set with specific properties. Aset D ⊂ Rn is invariant for the flow Φt(x) of a system like (2.1.1) if Φt(x) ∈ D forx ∈ D ∀ t ∈ R.4

4 A set is called a positive invariant set if it is invariant for t ≥ 0. If it is invariant for t < 0,it is said to be negatively invariant. Cf. Wiggins (1990), p. 14.

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2.1. Preliminary Concepts 29

Definition 2.1:5 A closed invariant set A ⊂ W is called an attractingset if there is some neighborhood U of A such that Φt(x) ∈ U ∀ t ≥ 0and Φt(x) → A when t→ ∞ for all x ∈ U.

A repelling set is defined by letting t → −∞ in Definition 2.1. An attracting set istherefore a set to which trajectories starting at initial points in a neighborhood ofthe set will eventually converge. The set of all initial points which are attracted byA is called the basin of attraction of A.

Definition 2.2: Let U be a neighborhood of an attracting set A. Thebasin of attraction B(A) is the stable manifold of A, i.e., B(A) =⋃

t≤0 Φt(U).

The shaded areas in Figures 2.3.a and 2.3.b depict basins of attraction for the casesin which the attractor is a single point (cf. 2.3.a) and in which the attractor is aclosed curve (cf. 2.3.b). The basin of attraction is delimited by its basin boundary.

2.3.a 2.3.bBasins of Attraction of Attracting Sets A

Figure 2.3

In most parts of this book the term “attracting set” will be identified with an“attractor”. However, it should be noted that the expressions can be distinguished.An attractor can be defined as a topologically transitive attracting set.6

Attracting sets can often be detected by locating a trapping region (cf. Wiggins(1990), p. 43):

5 Cf. Guckenheimer/Holmes (1983), p. 34, or Wiggins (1990), p. 43.6 A closed invariant set is said to be topologically transitive if, for any two open sets U,

V ⊂ A there exist t0 > 0 such that Φt(U) ∩ V �= ∅ ∀ t > t0, cf. Wiggins (1990), p. 45,and Ruelle (1989), p. 151f. Examples showing that an attractor may be only a subsetof an attracting set can be found in Eckmann/Ruelle (1985), p. 623. Compare alsoWiggins (1990), pp. 44f.

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30 Chapter 2

Definition 2.3: A closed, connected set D is a trapping region ifΦt(D) ⊂ D ∀ t ≥ 0 or, equivalently, if the vector field on the boundaryof D is pointing toward the interior of D.

The intersection⋂

t>0 Φt(D) of all trajectories in D is then an attracting set. Trap-ping regions can be identified with the help of Lyapunov functions which will beintroduced later in this section.

Consider an initial point that does not belong to an attractor, i.e., x(0) �∈ A, andsuppose that the trajectory starting at x(0) eventually approaches the attractor. Thepart of the trajectory Φt(x(0)) which is not yet on the attractor is called a transient.Transients may exhibit wild behavior in the initial phase of the convergence towardan attractor.

2.4.a 2.4.bWandering and Non-Wandering Points

Figure 2.4

The motion on transients and on attractors can also be distinguished by intro-ducing the notion of wandering and non-wandering sets.7 A point x(0) is non-wan-dering under the flow Φt(x) if for any neighborhood U

(x(0)

), there exists a t0 ≥ 0

such that Φt(U)∩U �= ∅ for t > t0, i.e., a trajectory starting in an arbitrary neighbor-hood of x(0) eventually returns to this neighborhood. The set of non-wanderingpoints is called the non-wandering set. The wandering set is the complement of thenon-wandering set. Examples of non-wandering sets are asymptotically stable fixedpoints and stable limit cycles (to be introduced below). Points on transients and ontrajectories diverging from repellers are examples of wandering sets. Figure 2.4.ashows an example of a wandering point x(0). In order to be a non-wandering point,any neighborhood U of the initial point must fulfill the above mentioned require-ment in the definition of such a point. However, it is trivial to find a neighborhood(the shaded area) with the property that a trajectory leaving the neighborhoodnever returns to this set. Thus, x(0) in Figure 2.4.a is a wandering point. Figure

7 Cf. Guckenheimer/Holmes (1983), p. 236.

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2.1. Preliminary Concepts 31

2.4.b shows an example of a non-wandering point. The motion takes place on theclosed curve in a continuous fashion, i.e., all points on the curve are subsequentlyvisited by the trajectory starting at x(0). Thus, when the trajectory on the closedcurve leaves the neighborhood U (shaded area) it will eventually return to thisneighborhood. Obviously, this is independent of the seize of U. Thus, the pointx(0) in Figure 2.4.b (and any other point in the closed curve as well) is an exampleof a non-wandering point.

————–

Two types of regular attractors have found particular attention, namely fixed-pointattractors and closed orbits. They will briefly be described in the following.

1. Fixed Point Attractors. For a long time, economics has concentrated on a specialkind of attractor, namely fixed-point (or equilibrium-point)8 attractors. A surveyof some techniques to establish the stability of fixed points in linear dynamicalsystems is contained in Appendix A.1. In considering nonlinear systems, the localand global stability properties of a fixed point must be distinguished.9

Local Stability of Fixed Points

Let x∗ = (x∗1 , . . . , x∗n) be a fixed point of (2.1.1) with x = 0 = f(x∗). The followingtwo local stability concepts are the most relevant local concepts for economics:

• The fixed point is locally stable (locally Lyapunov stable) if for every ε > 0 thereexists a δ > 0 such that for all |x(0) − x∗| ≤ δ one has |Φt

(x(0)

)− x∗| ≤ ε ∀t.• The fixed point is asymptotically stable if it is stable and if there exists a δ > 0 such

that for |x(0) − x∗| ≤ δ one has limt→∞ |Φt(x(0) − x∗| = 0.

The two stability concepts are illustrated in Figure 2.5. In order to be locally Lya-punov stable a trajectory starting in a neighborhood of the fixed point x∗ (deter-mined by δ and indicated by the light grey-shaded area in Figure 2.5.a) is requiredto stay in a neighborhood determined by ε (dark grey-shaded area). As the ε –neighborhood can be larger than the δ – neighborhood a trajectory is allowed tomove away from x∗ for a while but must not leave the ε – neighborhood. The local-asymptotic-stability concept is depicted in Figure 2.5.b. A trajectory starting in a δ

8 The term “equilibrium” as a description of a fixed point of a dynamical system will beavoided as often as possible in the course of the book. The term is mostly used in itseconomic meaning and indicates the congruence of supply and demand in a marketand/or the identity of planned and actual individual actions.

9 Compare also Hahn (1984), pp. 748ff., and Takayama (1974), p. 356, for a detaileddiscussion of various stability concepts.

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32 Chapter 2

2.5.a 2.5.bLyapunov Stability and Asymptotic Stability

Figure 2.5

– neighborhood of x∗ converges toward the fixed point; the Euclidian distance be-tween points Φt

(x(0)

)on a trajectory and the fixed point decreases for increasing

values of t.

2.6.a 2.6.b 2.6.cLinear Invariant Subspaces and Nonlinear Manifolds

Figure 2.6

The local behavior of a nonlinear dynamical system near a fixed point can beinvestigated by inspecting the behavior of the linear part of the system. This is truefor the following reason. Consider the nonlinear system (2.1.2) and assume that itcan be decomposed into a linear part and a nonlinear part, i.e.,

x = Ax + g(x), (2.1.3)

where A is an n×n – matrix of constant coefficients and g(x) is a nonlinear vector-valued function. It is demonstrated in the appendix that it is possible to determineinvariant eigenspaces for the linear part of (2.1.3), i.e., x = Ax. Assume that the stableand unstable eigenspaces are described by the linear curvesEs andEu, respectively,in Figure 2.6.a and 2.6.b.

In nonlinear systems, the analogs of the linear invariant subspacesEs andEu willbe called the nonlinear, local (with respect to a fixed point x∗), invariant manifolds

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2.1. Preliminary Concepts 33

W s andWu, respectively. Let U ⊂ Rn be a neighborhood of a fixed point x∗. Thenthe nonlinear manifolds are defined as: 10

W s = {x ∈ U | Φt(x) → x∗ as t→ ∞ and Φt(x) ∈ U ∀ t ≥ 0}Wu = {x ∈ U | Φt(x) → x∗ as t→ −∞ and Φt(x) ∈ U ∀ t ≤ 0}. (2.1.4)

An important property of these nonlinear local manifolds is depicted in Figure2.6.a. The manifold W s is tangent to the subspace Es at the fixed point x∗, andthe manifold Wu is tangent to the subspace Eu at this fixed point. Furthermore,it follows from the Hartman-Grobman theorem11 that the flow of the nonlinear system(2.1.2) is equivalent with the flow of the linear system x = Ax in a neighborhood ofa fixed point x∗. The meaning of “equivalence” will be discussed in greater detailin Chapter 3. For the moment it suffices to say that two systems are equivalent if thetrajectories of the linear and nonlinear systems have the same orientation and moveinto the same direction at analogous initial points. Figures 2.6.b and 2.6.c attemptto illustrate this equivalence. Invariant subspaces Es and Eu are shown in Figure2.6.b; the local manifolds W s and Wu are depicted in 2.6.c. In a neighborhoodU (indicated by the shaded area) the trajectories move into similar directions. Itis impossible that a trajectory in Figure 2.6.c moves into a completely differentdirection than the analogous trajectory in Figure 2.6.b.

As it is possible to analyze the local behavior of a nonlinear dynamical system inthe neighborhood of a fixed point x∗ with the help of the linear part in (2.1.3), it isdesirable to isolate this linear part. The Taylor expansion of a Cm function f : R → R

at a point x∗ is defined as

f(x) = f(x∗) +11!df(x∗)dx

(x− x∗) +12!d2f(x∗)dx2 (x− x∗)2

+13!d3f(x∗)dx3 (x− x∗)3 + . . .+

1m!

dmf(x∗)dxm

(x− x∗)m.(2.1.5)

In a linear Taylor expansion only the first two terms are considered and all remainingterms are dropped. The linear Taylor expansion of a differential equation system(2.1.1) (or (2.1.2)) yields

x = f(x∗) + J|x=x∗ (x − x∗), (2.1.6)

with J|x=x∗ as the Jacobian matrix of partial derivatives evaluated at x∗. When x∗ isa fixed point, f(x∗) is, of course, equal to zero.

10 Cf. Guckenheimer/Holmes (1983), pp. 13f., for details. The negative time directionhas been chosen because otherwise it would not have been possible to express the originof a diverging trajectory.

11 Cf. Guckenheimer/Holmes (1983), p. 13, for details.

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34 Chapter 2

The properties of the Jacobian matrix, defined as

J|x=x∗ =

∂f1

∂x1

∂f1

∂x2. . .

∂f1

∂xn

∂f2

∂x1

∂f2

∂x2. . .

∂f2

∂xn...

.... . .

...∂fn∂x1

∂fn∂x2

. . .∂fn∂xn

, (2.1.7)

and its eigenvalues are analogous to those of the coefficient matrix A in lineardynamical systems (cf. (A.1.3) in Appendix A.1). For example, if the real parts ofthe eigenvalues of (2.1.7) are negative, then the fixed point is locally asymptoticallystable.

When one (or more) of the eigenvalues of the Jacobian matrix J, evaluated atthe fixed point x∗, equals zero or has zero real parts (i.e., when one of the linearinvariant subspaces is a center subspace (cf. Appendix A.1.3)) the above mentionedanalogies between the eigenspaces and the local nonlinear manifolds do not holdanymore. When there are such zero roots it is not possible anymore to analyzethe local behavior of a nonlinear dynamical system by inspecting the behavior ofits linear part. In such cases it is necessary to calculate the center manifold and toinvestigate the dynamic behavior restricted to this manifold. An introduction tocenter manifold theory is contained in the Appendix A.2.

Global Stability of a Fixed Point

The distinction between the local and global stability of a fixed point is a necessityin studying nonlinear dynamical systems. While local stability in a linear systemalso implies global stability, nonlinear dynamical systems can be characterized bymultiple fixed points which (in continuous-time systems) are alternatively locallyasymptotically stable and unstable. The concept of global asymptotic stability isdefined in analogy to the local asymptotic stability with the modification that δ canbe arbitrarily large:• The fixed point is globally asymptotically stable if it is stable and limt→∞ |Φt

(x(0)

)−x∗| = 0 for every x(0) in the domain of definition of (2.1.1).

A useful tool in investigating the global stability of a fixed point is the concept of aLyapunov function.12

Theorem 2.1 (Lyapunov (1949)): Let x∗ be a fixed point of a dif-ferential equation system and let V : U → R be a differentiable functiondefined on some neighborhood U ⊂ W ⊂ Rn of x∗ such that:

12 Cf. Hirsch/Smale (1974), pp. 192ff., and Guckenheimer/Holmes (1983), pp. 4f. Ex-tensive treatments of the usage of Lyapunov functions can be found in Hahn (1967)and Lasalle/Lefschetz (1961).

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2.1. Preliminary Concepts 35

(i) V (x∗) = 0 and V (x) > 0 if x �= x∗, and

(ii) V (x) ≤ 0 in U − {x∗}.

Then x∗ is stable. Moreover, if

(iii) V (x) < 0 in U − {x∗},

then x∗ is asymptotically stable.

Note that the neighborhood U ⊂ W can be chosen arbitrarily large. Thus, a fixedpoint is globally asymptotically stable if the conditions (i) - (iii) are fulfilled for theentire domain of definition of a system like (2.1.2).

The choice of the appropriate Lyapunov function in economic applications isnot always obvious. Good candidates in different fields are welfare functions, thenational product, or arbitrary constructions which resemble the notion of potentialsin physics.13

2. Cyclical Attractors. The present monograph does not focus on the question of(global or local) stability of a fixed point but on dynamic phenomena other thanthe (possibly complicated) convergence to a fixed-point attractor. The followingdiscussion concentrates on attractors in the form of closed orbits. A point x is saidto be in a closed orbit if there exists a t �= 0 such that Φt(x) = x. If a closed orbit isan attractor it will be called a limit cycle in the following.

Definition 2.4: A closed orbit Γ is called a limit cycle if there is a tubu-lar neighborhood U(Γ ) such that for all x ∈ U(Γ ), any flow Φt(x) ap-proaches the closed orbit.

A Limit CycleFigure 2.7

13 Cf. Chapter 7 for the role of potentials in catastrophe theory.

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36 Chapter 2

An example of a limit cycle in R2 is depicted in Figure 2.7. Trajectories starting atinitial points in the closed orbit will stay on the cycle forever. Trajectories starting atinitial points in the neighborhood U(Γ ) of the closed orbit will approach the cyclein a spiraling fashion.

————–

In higher-dimensional systems (n ≥ 3) more complicated attractors are possible.Several complicated attractors will be introduced in Chapter 4 and 5.

In nonlinear dynamical systems more than a single fixed point can exist. In fact,the existence of a multiplicity of fixed points can be viewed as the origin of variouskinds of complicated dynamic behavior. It is thus important to know the numberof fixed points when a nonlinear dynamical system with potentially complicatedbehavior is to be investigated. The Poincare index is a helpful tool for this purpose.14

The Determination of the Poincare Index of a ContourFigure 2.8

Suppose that a two-dimensional, continuous-time dynamical system generatesa vector field as in Figure 2.8.15 The fixed point in the center of the vector fieldis obviously unstable. The Poincare index of the contour D (i.e., the closed curveencircling the fixed point) is determined in the following way: Mark the pointsof intersection of D with the vector field and note the orientation of each singlevector. Start somewhere on D, e.g., at the intersection point # 1, and move alongD in a counterclockwise manner, i.e., with positive orientation. Obviously, duringthe journey on D the orientations of the vectors change. After a full 2π motion,the vector orientation is again the same as that at the beginning but it may havechanged by 3600 or by −3600 (i.e., in a counterclockwise or clockwise manner)

14 Cf. Andronov/Chaikin (1949) and Milnor (1965) for detailed treatments of indextheory. Economically motivated discussions can be found in Dierker (1974) and Varian(1981).

15 The higher-dimensional (n ≥ 3) analog of index theory is degree theory. Cf. Chow/Hale(1982) for details.

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2.1. Preliminary Concepts 37

during the wandering on the contour. In Figure 2.8 a counterclockwise change inthe orientation of the vector field can be observed but there exist dynamical systemswith a clockwise change.

When a single, complete, counterclockwise rotation of the vector field can be ob-served during the counterclockwise motion along D the Poincare index of the con-tour is defined to be IP = 1. Each additional rotation of the vector field increasesthe index by 1. The Poincare index is the number of complete counterclockwise ro-tations of the vector field during a single counterclockwise motion alongD. When aclockwise rotation can be observed it contributes a value of −1 to the count, i.e., fora single clockwise rotation during a counterclockwise motion along D the Poincareindex is IP = −1.16

2.9.a: IP = 1 2.9.b: IP = 1

2.9.c: IP = −1 2.9.d: IP = 1

Poincare Indices of Different Dynamical SystemsFigure 2.9

16 When the vector field rotated by 2πk with k as an integer during a single counterclock-wise motion along D, the Poincare index is therefore identical with k.

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38 Chapter 2

In Figure 2.8 the contour D encloses a fixed point. It is, of course, possible todetermine the Poincare index of a contour without a fixed point in its interior butthe index is typically calculated for the purpose of investigating the number of fixedpoints in a specific dynamical system. The Poincare index IP (x∗) of a fixed point isobtained by calculating the index of a contour D that encircles a single, isolatedfixed point x∗. Figures 2.9.a – 2.9.c illustrate the determination of the indices fordifferent kinds of fixed points. Instead of drawing the entire vector field it sufficesin most cases to consider a few trajectories and their (tangential) vectors at thepoints of intersection of the trajectory and the contour D. The stable node (2.9.a)and the stable focus (2.9.b) have indices IP = +1, while the saddle point in 2.9.chas an index IP = −1.17 Figure 2.9.d illustrates the determination of the index ofa closed orbit. Applying the same technique as above uncovers that a closed orbithas a Poincare index of IP = +1.18

The importance of Poincare indices becomes obvious by the fact that the indexof a contour D is equal to the sum of the indices of the objects encircled by thecurve.19 Figures 2.8 and 2.9.a-c depict the case where the contour D and the en-circled fixed points have the index IP = +1. Similarly, a contour without fixedpoints in its interior has index IP = 0. Assume that it is known that a dynamicalsystem generates a closed orbit. Take the orbit itself as the contour D. As its indexis IP = +1 it follows that the orbit must encircle at least one fixed point. Whenthe index of a known fixed point (calculated by drawing a contour in a sufficientlysmall neighborhood of the point) is, e.g., IP = −1 then it follows that there mustbe additional fixed points encircled by the closed orbit.20 For example, the saddleloop in Figure 2.10 has an index of IP = +1 and the index of the saddle fixed pointB is IP = −1. Thus, there are additional fixed points (points A and C in Figure2.10 with indices IP (A) = 1 and IP (C) = 1). However, the argument cannot beapplied in the reverse way: when the sum of the known fixed points encircled by aclosed orbit equals +1, one cannot be sure that all fixed points are indeed known.

17 The Poincare index of some fixed points can also be determined analytically. Let J bethe Jacobian of a dynamical system evaluated at the fixed point x∗. Then the index is

IP (x∗) =

{+1 if det(−J) > 0;−1 if det(−J) < 0.

If det(−J) = 0, the index has to be calculated by the method described above, cf. Varian(1981). Compare also the formula provided in Wiggins (1990), p. 35.

18 The reader may verify that this is true independent of the orientation of the motion onthe closed curve.

19 This follows from the Poincare-Hopf theorem; cf. Guillemin/Pollack (1974), pp.132ff., or Varian (1981), p. 100.

20 When all fixed points are hyperbolic (cf. Section 3.1) then the number of fixed pointsis odd. When this number is 2n + 1, n fixed points are saddles and n + 1 are eithersinks or sources.

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2.2. The Poincare-Bendixson Theorem 39

2.2. The Poincare-Bendixson Theorem

In most economic applications, especially when dealing with nonlinear dynamicalsystems, it is desirable to establish results on the global behavior of dynamical sys-tems. Unfortunately, the global character of the results must be compensated bycompromising in respect to the dimension of the dynamical system; this restric-tion arises because it is possible to completely categorize the global behavior of adynamical system only in the two-dimensional case.

The Poincare-Bendixson theorem employs the notion of a limit set which has notbeen mentioned yet.

Definition 2.5:21 An ω – limit set of a point x ∈ W is the set of allpoints ∈ W with the property that there exists a sequence ti → ∞ suchthat limi→∞Φti

(x) = . The α – limit set is defined in the same way butwith a sequence ti → −∞.

A Saddle LoopFigure 2.10

In R2, three different types of limit sets can be distinguished:22

• Fixed point attractors.• Limit Cycles.• Saddle loops, i.e., fixed points and the trajectories connecting them.

The first two types of limit sets have already been discussed. An example of a saddleloop (or homoclinic orbit) is depicted in Figure 2.10 (it is also possible to have only a

21 The letters α and ω constitute an asses’ bridge: the letters are the first and last letters ofthe Greek alphabet, respectively. The α – limit set represents the set of points where themotion starts; the ω – limit set contains all points where the motion ends. Cf. Wiggins(1990), pp. 41f., for an explanation why the sequence {ti}, i → ∞, is considered insteadof t → ∞.

22 Cf. Guckenheimer/Holmes (1983), p. 45.

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40 Chapter 2

single loop). The associated dynamical system has two unstable fixed points (A andC) and a saddle as a third fixed point (B). The ω - limit set in this example consistsof the union of the two loops (i.e., the trajectories that leave the saddle and returnto it23) and the saddle point. Saddle loops can occur in a variety of constellationsof multiple fixed points and can enclose closed orbits.24

The subject of the Poincare-Bendixson theorem is to provide sufficient conditionsfor the existence of limit cycles in particular sub-areas of the plane.25

2.2.1. The Existence of Limit Cycles

Consider the two-dimensional differential equation system

x1 = f(x1, x2),

x2 = g(x1, x2),(2.2.1)

and assume that an initial point x(0) =(x1(0), x2(0)

)is located in an invariant set

D ⊂ R2.

A Limit Cycle in a Compact Set DFigure 2.11

When the set contains limit sets, basically all three types of limit sets mentionedabove are possible. The Poincare-Bendixson theorem discriminates between thesedifferent types:

23 These specific trajectories are also known as separatrices.24 Cf. Guckenheimer/Holmes (1983), p. 46.25 A complete discussion of the theorem can be found in Hirsch/Smale (1974), Chapter

11, to which the interested reader is strongly referred. Further presentations can befound in Arrowsmith/Place (1982), pp. 109ff., Boyce/DiPrima (1977), Chapter 9,and Coddington/Levinson (1955), Chapter 16. A concise overview is contained inVarian (1981).

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2.2.1. The Existence of Limit Cycles 41

Theorem 2.2 (Poincare-Bendixson): A non-empty compact limit setof aC1 dynamical system in R2, which contains no fixed point, is a closedorbit.

The proof of the theorem can be outlined by a simple geometrical consideration.Figure 2.11 depicts an example of an invariant set D in the plane. On the boundaryof D, the vector field points inwards the set, implying that a trajectory will stay in itfor all t once it has entered the set. The question then arises how a trajectory mightwander when it has entered the set. When the fixed point does not belong to thementioned limit set, i.e., when it is unstable, trajectories starting in a neighborhoodof the fixed point will be repelled from it (cf. Figure 2.11). As trajectories of acontinuous-time dynamical system cannot intersect, the only possible limit sets inD in the case of unstable fixed points are closed orbits and saddle loops. As saddleloops imply the existence of at least one additional fixed point (in the form of asaddle), this possibility is excluded by Theorem 2.2. It follows that if the fixed pointin Figure 2.11 does not belong to the ω – limit set of the points in D then all initialpoints in D must converge toward a closed curve for t→ ∞.

While the fixed point has been excluded from the limit set in D, a closed orbitin R2 always encloses a fixed point:26

Theorem 2.3: A closed trajectory of a continuously differentiable dy-namical system in R2 must necessarily enclose a fixed point with x1 =x2 = 0.

The proof follows immediately from the Poincare index theory outlined in theprevious section.

Summarizing, the following procedure is appropriate in applying the Poincare-Bendixson theorem to a specific dynamical system in R2.

• Locate a fixed point of the dynamical system and examine its stability properties.• If the fixed point is unstable, search for an invariant set D enclosing the fixed

point. When a closed orbit does not coincide with the boundary of D, the vectorfield described by the function f and g must point into the interior of D.

Actually, the set D must not necessarily have the form of the set described in Figure2.11, i.e., a simply connected set.27 Assume that D is described by the tubular,

26 Cf. Boyce/DiPrima (1977), p. 445, and Hirsch/Smale (1974), p. 252.27 A “simply connected” set is a set that consists of one piece (or two or more touching

pieces) and which does not contain any holes in it. The first two sets outlined below are

simply connected sets while the third set is an example of a connected but not simplyconnected set, cf. Arrowsmith/Place (1982), p. 111, and Debreu (1959), p. 15.

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42 Chapter 2

shaded area in Figure 2.7. The boundary of the invariant set is described by theoutermost and innermost closed ellipses. Theorem 2.2 implies that the tubular areacontains at least one closed orbit. However, when the innermost ellipse is repellingin both directions, the set of points enclosed by this curve can again be consideredthe boundary of another invariant set which might contain additional closed orbits.This procedure can be continued, and eventually the above mentioned stability/instability criterion of the fixed point again becomes relevant. It follows that theinstability of the enclosed fixed point is a prerequisite when a simply connectedarea is considered. However, this instability does not exclude that a variety of closedorbits exist in appropriate tubular invariant sets.

The search for the set D constitutes the essential difficulty in applying thePoincare-Bendixson theorem to a dynamical system. On the other hand, it is rela-tively easy to exclude the existence of closed orbits in a system like (2.2.1). Let S bea simply connected domain in W ⊆ R2.

Theorem 2.4 (Bendixson): 28 Assume the functions f and g in (2.2.1)having continuous first order derivatives in S. If the sum (∂f/∂x1 +∂g/∂x2) has the same sign throughout S, then there is no periodic solu-tion of (2.2.1) lying entirely in S.

The Poincare-Bendixson theorem thus provides sufficient conditions for the exis-tence of closed orbits in a set D but it does not say anything about the number ofthese orbits. The above consideration shows that it is possible that more than a sin-gle closed orbit exist.29 When several cycles exist, it is obviously impossible that allcycles are attracting, i.e., that they are limit cycles in the sense of Definition 2.4. Pro-vided that the fixed point is unstable, the innermost cycle in D is stable. Additionalcycles with increasing amplitude are then alternatively unstable and stable.

The most serious disadvantage of the Poincare-Bendixson theorem is the factthat it is restricted to two dimensions. Analogous theorems in higher dimensionsdo not exist. This is not due to a lack of mathematical research, but to a conceptualproblem. While in the two-dimensional case the planar set D can be divided intoan inner and outer region with the above mentioned implications, things get drasti-cally more difficult in the three-dimensional case. Suppose that a closed set D ⊂ R3

exists with the vector field pointing inwards this set and that the unique fixed pointis unstable. Nevertheless, it is possible that no closed orbits exist because a trajec-tory can arbitrarily wander in R3 without intersecting itself and without necessarilyapproaching a limit set (cf. Figure 2.12).

Despite the fact that this limitation usually restricts the application of the theo-rem to highly aggregated model-economies, it provides the theorist with a powerfultool in facing complicated two-dimensional dynamical systems which sometimescannot be described by means of graphical phase diagrams alone.

28 Cf. Andronov/Chaikin (1949), p. 227, and Boyce/DiPrima (1977), p. 446.29 Cf. Section 2.3 for sufficient conditions for the uniqueness of limit cycles.

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2.2.2. The Kaldor Model 43

The Invalidity of the Poincare-Bendixson Theorem in R3

Figure 2.12

2.2.2. The Kaldor Model as a Prototype Model in Nonlinear Economic Dynamics

As early as in 1940 N. Kaldor presented a business-cycle model which is able togenerate endogenous limit cycles, and which in the sequel has served as the pro-totype model for nonlinear dynamical systems in economics.30 Actually, Kaldor’scontribution should be considered in conjunction with the work of Kalecki (1937,1939), who investigated similar models but concentrated on different aspects ofstability.

The key to Kaldor’s model can be found in his assumptions on investmentbehavior in a one-sector model. Investment depends positively on income, but thepropensity to invest decreases if income diverges from its stationary equilibriumlevel. Furthermore, at a given level of income, investment decreases if the capitalstock increases,31 i.e., I = I(Y,K); IY > 0, IK < 0 and there exists a Y1 such thatIY Y > 0 (< 0) if Y < Y1 (Y > Y1), with Y as income, K as the capital stock, I asgross investment, and the subscripts denoting the partial derivatives with respect tothe nth argument (cf. Figure 2.13).

For the sake of simplicity, assume that savings depends linearily on income inthe usual way,32 i.e., 0 < SY < 1, and, additionally, on the capital stock with SK >

30 A more intensive discussion of the Kaldor model and its formal reconsideration byChang/Smyth (1971) can be found in Gabisch/Lorenz (1989), pp. 122ff.

31 Cf. Gabisch/Lorenz (1989), pp. 122-129, for economic justifications of these assump-tions.

32 Kaldor himself assumed a sigmoid shape of S(Y, ·). The linearity assumption does notchange the qualitative results presented below.

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44 Chapter 2

Kaldor’s Investment Function for Different K-valuesFigure 2.13

0.33 Income changes proportionally to the excess demand in the goods market.Together with a standard capital accumulation equation the Kaldor model can bewritten as

Y = α(I(Y,K) − S(Y,K)

),

K = I(Y,K) − δK,α, δ > 0, (2.2.2)

with δ as the constant depreciation rate and α as an adjustment coefficient.Consider first the local stability of the fixed point of system (2.2.2), i.e., the point

(Y ∗,K∗) for which Y = K = 0. A linear Taylor expansion of (2.2.2), evaluated atthe fixed point (Y ∗,K∗), yields the Jacobian matrix

J =

(α(IY − SY ) α(IK − SK)

IY IK − δ

), (2.2.3)

with the determinant

det J = α(IY − SY )(IK − δ) − αIY (IK − SK), (2.2.4)

and the trace

tr J = α(IY − SY ) + (IK − δ). (2.2.5)

33 This assumption is not very convincing. Chang/Smyth (1971) therefore assumed thatSK < 0, i.e., a standard wealth effect prevails. However, the different signs do notessentially effect the results when IK − SK < 0 is assumed.

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2.2.2. The Kaldor Model 45

It follows from the consideration of linear continuous-time systems in the AppendixA.1.1 that the eigenvalues are

λ1,2 =tr J ±√

(tr J)2 − 4 det J2

. (2.2.6)

The determinant must be positive in order to exclude the possibility of a saddlepoint. The fixed point is then locally asymptotically stable if the real parts of theeigenvalues, i.e., the trace of the Jacobian, are negative. Inspection of (2.2.5) showsthat this is the case if α(IY − SY ) < −(IK − δ). As the right hand side of theinequality is positive, the difference between the marginal propensity to invest andto save must therefore be smaller than a positive value.

Kaldor explicitly assumed that (IY − SY ) > 0 at the fixed point.34 Figure 2.14demonstrates the model (2.2.2) for this constellation of the slopes at the stationaryequilibrium.

Multiple Goods-Market Equilibria in the Kaldor ModelFigure 2.14

When the trace is positive, i.e., α(IY − SY ) + (IK − δ) > 0, the fixed point isunstable. The first requirement of the Poincare-Bendixson theorem is thereforefulfilled.

Second, it should be examined whether the Bendixson criterion, i.e., Theorem2.4, is fulfilled. As the slope IY decreases for Y diverging from the stationary equi-librium, the term (IY − SY ) changes its sign twice at appropriate income levels.Thus, depending on the magnitude of (IK − δ), it is possible though not neces-sary that the trace of the Jacobian changes its sign, too. The Bendixson criteriontherefore does not exclude the existence of closed orbits.

34 In fact, Kaldor intended to express the instability of the stationary equilibrium by thisassumption.

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46 Chapter 2

The Phase Portrait of the Kaldor ModelFigure 2.15

The question of whether a compact invariant set D exists such that the vec-tor field (2.2.2) points inwards that set can be answered by means of a graphicalargument. Figure 2.15 represents the phase portrait of the Kaldor model.

Consider first the set of points (Y,K) with the property that the capital stockdoes not change, i.e.,

K = 0 = I(Y,K) − δK. (2.2.7)

Total differentiation yields

dK

dY |K=0= − IY

IK − δ> 0. (2.2.8)

Thus, the locus of all points in the set {(Y,K) |K = 0} is an upward sloping curve.For all K above the curve K = 0, investment decreases because of (IK − δ) < 0,hence K < 0. In the same way, K is positive for all K below the curve for K = 0.

The set of points (Y,K) with Y = 0 is given by

Y = 0 = I(Y,K) − S(Y,K). (2.2.9)

It follows that

dK

dY |Y =0=SY − IYIK − SK

� 0. (2.2.10)

The sign of (2.2.10) depends on the values of SY and IY . The difference SY − IYis positive for low as well as for high levels of income and is negative for normallevels in the neighborhood of the fixed point. It follows that the curve for Y = 0

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2.2.3. A Classical Cross-Dual Adjustment Process 47

is negatively sloped for low and for high values of Y and is positively sloped in aneighborhood of Y ∗. Income increases (decreases) for all points below (above)the curve Y = 0.

It is relatively easy to find a set D with the desired properties in the Kaldormodel.35 The subset D = {(Y,K)| 0 ≤ Y ≤ Y1, 0 ≤ K ≤ K1}, i.e., the shadedarea in Figure 2.15, is compact, and the vector field obviously points inwards the seton the boundary. Thus, the requirements of the Poincare-Bendixson theorem arefulfilled and it has been shown that the Kaldor model exhibits limit cycles.

2.2.3. A Classical Cross-Dual Adjustment Process

One of the most intensively investigated and best-understood examples in economicdynamics is the so-called tatonnement process originally introduced (in passing, actu-ally) by L. Walras (1954). In a pure exchange economy with price-taking individ-uals economic intuition suggests that the price of a good i changes when the excessdemand for this good differs from zero, and the problem arises how to illustratethe convergence toward a simultaneous equilibrium with zero excess demand forall goods.36 For the purpose of a didactical illustration, Walras introduced theconcept of the auctioneer who subsequently visits all markets in the economy andwho is the only person being able to change prices. After gradually adjusting pricesin a single market according to the observed demand and supply quantities in theface of announced hypothetical prices, the auctioneer moves to the next marketwhere individuals take the eventually found equilibrium price in the previous mar-ket into account. In this second market, the equilibrium price is found in the samefashion, and the auctioneer moves to the next market, etc. It is worthwhile stressingthat this process does not represent a simultaneous price adjustment in all marketsand that it is only a heuristic and didactical description of such a process.

The mathematical treatments of the Walrasian tatonnement that have beenpublished since the late 1930s departed from this heuristic character of Walras’soriginal process and have assumed a simultaneous adjustment in all prices.37 Con-sider a pure exchange economy with n different goods and m price-taking, utility-maximizing individuals. The price of a single good i is denoted by pi, i = 1, . . . , n.The aggregate excess demand for good i is zi and depends on the vector p =(p1, p2, . . . , pn) ∈ Rn

+ of the prices of all goods. When the change in price pi is afunction fi: Rn → R of the excess demand zi for this good, i.e., pi = fi

(zi(p)

), the

35 In other examples the search for this set D can be difficult. Cf. Gabisch/Lorenz(1989), pp. 143ff., for a discussion of a non-Walrasian business-cycle model by Benassy(1984) with a complicated compact set D.

36 The question of whether the price of a single good converges to its partial equilibriumvalue never seemed to be a problem for Walras. Cf. Newman (1965) and Walker (1987)for these interpretations of Walras’s work.

37 The list of original mathematical treatments of the tatonnement process include Arrow(1959), Arrow/Hurwitz (1958), Samuelson (1947), and Uzawa (1961). A survey ofthe most relevant results of the tatonnement literature can be found in Hahn (1984).

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48 Chapter 2

simultaneous change in all prices is described by the system of differential equations

p1 = f1(z1) = h1(p),p2 = f2(z2) = h2(p),

...pn = fn(zn) = hn(p).

(2.2.11)

A lot of attention has been paid to the question of the local and global stabilityof the fixed point p∗ of (2.2.11) with the property p∗i = 0 ∀ i = 1, . . . , n, and sev-eral sufficient conditions at least for the local stability of the fixed point have beenprovided. Though extensive treatments of the subject exist in the literature, thesufficient conditions like gross substitutability between all goods appear to be arbi-trary and the problem seems to be far away from being finally solved. However, ofparticular importance is the answer to the question of whether the process (2.2.11)applies to economies with production as well.38

It might be argued that the excess demands zi(p) in (2.2.11) should simply bereplaced by the differences xi − yi, i = 1, . . . , n, where xi represents the aggregatedemand of households for good i and yi is the aggregate supply of price-takingfirms. This argument implies that the demand of households and the supply of firmsare always represented by points on the aggregate demand and supply functions,respectively. Such an assumption appears to be natural but it reflects the idea thatthe agents can adjust to internal disequilibria infinitely fast. Assume on the contrarythat firms, for example, need time to adjust their production plan when (at a givenproduction level) prices change so that the profit maximizing output changes aswell. When a single market is considered, aggregate output is assumed to changeaccording to the function k: R → R, i.e., y = k(yd − y), k′ > 0, with yd(p) as thedesired output (the supply function) at a price p, and y as the actual output. Withthis discrepancy between actual and desired output the price adjustment has to beassumed to depend on the actual excess demand, i.e., p = f(x− y). A partial viewof a single market thus leads to the two-dimensional differential equation system39

p = f(x(p) − y

),

y = k(yd(p) − y

).

(2.2.12)

The equation system (2.2.12) is a so-called cross-dual adjustment process, and it isoccasionally claimed that this process is suited to reflect Marshallian and Walrasian

38 Extensive discussions of adjustment processes in economies with production can befound in, e.g., Amano (1968), Davies (1963), Marschak (1941), and Takayama (1974).

39 Usually, a slightly different formulation of this process is investigated. Beckmann/Ryder (1969) and Mas-Colell (1986) incorporate the Marshallian and Walrasian ter-minology of the selling or offer price which is the inverse of yd(p). The output adjustmentequation then turns into y = k

(p − c(y)

), where c(y) is the marginal cost associated

with the production of y.

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2.2.3. A Classical Cross-Dual Adjustment Process 49

ideas on the price and quantity adjustment in an economy with production.40 How-ever, Marshall and Walras emphasized the entry and exit of firms in a marketand it is obvious that this effect cannot be satisfactorily modeled without furtherassumptions. Therefore, the system (2.2.12) will be interpreted in the sequel just asan example of sluggish adjustment on the supply side of the market with a constantnumber of firms.41

Consider the following algebraic specification of (2.2.12) with constant adjust-ment coefficients α > 0 and β > 0 in the price and quantity adjustment equations:

p = α(x(p) − y

),

y = β(yd(p) − y

).

(2.2.13)

Assume that the system possesses a unique fixed point (p∗, y∗). The Jacobian matrixof (2.2.13) is

J|(p∗,y∗) =

(αxp(p∗) −αβydp(p∗) −β

), (2.2.14)

with det J = αβ(ydp(p∗)− xp(p∗)

)and tr J = αxp(p∗)− β. As saddle points should

be excluded, assume that the determinant of J is positive. Obviously, this is alwaysthe case when ydp(p∗) > xp(p∗), i.e., when the supply function is steeper than thedemand function.

When the demand function can be derived from the utility maximization pro-cedure of a Representative Consumer, the slope of the demand function x(p) isalways negative. It follows that the trace tr J is negative for all p. Thus, the fixedpoint (p∗, y∗) is locally asymptotically stable in this case.

A negative slope of the demand function seems to be intuitively plausible andthe microeconomic textbook literature usually deals with non-negative slopes onlyin conjunction with negative income effects. However, recent work in general equi-librium analysis has made evident that the aggregation procedure can lead to avariety of different shapes of the aggregate demand function. Even if all individualagents encounter the usual convexities and if their demand functions are nega-tively sloped, it cannot be excluded without further assumptions that the aggregatedemand function is positively sloped in a certain region of the (p∗, y∗) plane.42

The dynamic effects of the presence of a demand function with a positive slopeat the fixed point of the simple cross-dual system (2.2.13) were investigated by Mas-

40 Cf. Goodwin (1953, 1970) and Morishima (1959). Discussions of stabilizing processesof this kind can be found in Flaschel (1991, 1992) and Flaschel/Semmler (1987).

41 Cf. Novshek/Sonnenschein (1986, 1987) for more appropriate models with a varyingnumber of firms. A discussion of those models with an emphasis on possibly complexbehavior is contained in Lorenz (1992a).

42 The precise result is essentially due to Debreu (1974) and Sonnenschein (1972). Com-pare also the work of Dierker (1974), Hildenbrand/Kirman (1988), Kirman (1989),and Shafer/Sonnenschein (1982). Saari (1991) discusses the implications of this re-sult for the possible emergence of complicated dynamics.

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50 Chapter 2

The Mas-Colell Scenario of a Cross-Dual ProcessFigure 2.16

Colell (1986) who assumed an S-shaped demand function as in Figure 2.16.43 Inthis scenario the fixed point becomes unstable when the trace tr J is positive at thefixed point, i.e., when xp(p∗) > β/α. It should be noted that the fixed point cantherefore always become unstable when the slope xp(p∗) is positive and when theadjustment coefficients α and β take on appropriate values.

The subset D ⊂ R2, on whose boundary the vector field points inwards the set,cannot be found in the same easy way as in the Kaldor model. Define this invariantset as D = {(y, p)| 0 ≤ y ≤ y2, 0 ≤ p ≤ p2}, i.e., the shaded area in Figure 2.16. Thedirections of change of y and p can immediately be determined. As the changes in yand p depend on the quantities x(p)−y and yd(p)−y, consider the phase diagramin the horizontal direction: p is positive (negative) to the left (right) of the demandfunction; y is positive (negative) to the left (right) of the supply function. Withtwo exceptions, the vector field points inwards the set D on its boundary. The twoexceptional regions on the boundary are the intervals [0, p1) and (y1, y2]. Formally,either y or p can become negative in these intervals on the boundary. In order toexclude these technical difficulties assume that44

limp→0

p = 0 if y ∈ (y1, y2] and limy→0

y = 0 if p ∈ [0, p1).

Under these two assumptions the vector field never points out of the set D andeventually points toward the interior of the set. As the fixed point (p∗, y∗) is unsta-ble, the Poincare-Bendixson theorem implies the existence of at least one closedorbit in D.

43 Beckmann/Ryder (1969) assumed an S-shaped marginal cost function in order to ob-tain qualitatively similar results.

44 Of course, there exist other assumptions that ensure the boundedness of the set Dwith the desired properties. For example, the critical region (y1, y2] disappears if thedemand function converges asymptotically to the y–axis.

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2.3.1. The Lienard Equation 51

2.3. The Uniqueness of Limit Cycles

As was mentioned in Section 2.2., the Poincare-Bendixson theorem does not ex-clude the possibility of multiple closed orbits which are alternatively stable andunstable. However, the question of how many cycles exist in a dynamical system isextraordinarily important, because in case of multiple cycles the initial conditionsdetermine the final motion of a system with a specific amplitude. It is important toknow, especially in business-cycle models, whether by choice of the initial conditionsthe amplitude of the cyclical motion can be decreased or not.

Multiple Limit CyclesFigure 2.17

Unfortunately, this question of how many cycles exist cannot be answered forall dynamical systems. Although the theory of two-dimensional dynamical systemsis fairly well-developed, the problem of the uniqueness of limit cycles has not beenfinally solved and research is still going on.45 One of the few nonlinear systemsfor which it is indeed possible to establish sufficient conditions for the existence ofunique cycles is the so-called generalized Lienard equation.

2.3.1. The Lienard Equation and Related Tools

This section introduces two theorems on the uniqueness of limit cycles which ap-pear to be particularly useful for economic dynamics. A thorough discussion ofseveral other theorems with a varying degree of generality can be found in Yan-Qian (1986).

45 It might be considered interesting that the number of limit cycles in two-dimensionaldynamical systems with polynomial expressions of various degree was part of Hilbert’s16th unsolved mathematical problem. Cf. Hilbert (1990), p. 317.

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52 Chapter 2

Consider the two-dimensional differential equation system46

x = y − F (x),y = −g(x),

(2.3.1)

or, written as a second-order differential equation,

x+ f(x)x+ g(x) = 0, (2.3.2)

with f(x) = dF (x)/(dx). This so-called generalized Lienard equation was originallyformulated to model the dynamics of a spring mass system with g(x) as the springforce and f(x)x as a dampening factor. Setting g(x) = x and F (x) = (x3/3 − x)in (2.3.2) yields the so-called van-der-Pol equation

x+ (x2 − 1)x+ x = 0, (2.3.3)

which can be considered a prototype equation in two-dimensional nonlinear sys-tems theory. Levinson/Smith (1942) proved the following theorem for the equa-tion (2.3.2).

Theorem 2.5 (Levinson/Smith):47 Equation (2.3.2) has a uniqueperiodic solution if the following conditions are satisfied.

a) f and g are C1.b) ∃ x1 > 0 and x2 > 0 such that for −x1 < x < x2 : f(x) < 0, and > 0

otherwise.c) xg(x) > 0 ∀ x �= 0

d)∫∞

0 f(x)dx =∫ ±∞

0 g(x)dx = ∞e) G(−x1) = G(x2) where G(x) =

∫ x

0 g(ξ)dξ.

Condition e) is fulfilled if f(x) is even and g(x) is odd.48

The theorem allows to establish the uniqueness of limit cycles in a convenientway. The symmetry assumption e) represents the only more or less severe specifica-tion in a two-dimensional system.49

A weaker theorem that does not dwell on this symmetry requirement is due toZhifen (1986). The theorem represents a very convenient tool in establishing theuniqueness of limit cycles though it appears to be extensive at first glance.

46 Cf. Hirsch/Smale (1974), p. 215, and Boyce/DiPrima (1977), pp. 447ff.47 Cf. Levinson/Smith (1942), pp. 397f.48 A function is even if f(x) = f(−x), e.g., a parabolic function with the origin as the

center. A function is odd if −g(x) = g(−x), e.g., a cubic equation.49 For example, it can easily be shown that the van der Pol equation (2.3.3) fulfills the

requirements of the Levinson/Smith theorem.

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2.3.1. The Lienard Equation 53

Theorem 2.6 (Zhifen (1986)): Consider the system of differentialequations

x = φ(y) − F (x),y = −g(x).

(2.3.4)

If the following conditions are satisfied:

1. a) g(x) fulfills the Lipschitz condition50 in any finite interval;

b) xg(x) > 0 ∀ x �= 0;

c) G(∞) = G(−∞) = ∞ with G(x) =∫ x

0 g(ξ) dξ,

2. a) f(x) = F ′(x) ∈ C0(−∞,∞);

b) F (0) = 0;

c)f(x)g(x)

is nondecreasing when x increases in (−∞, 0) and (0,∞);

d)f(x)g(x)

�= constant when 0 < |x| � 1,

3. a) φ(y) fulfills the Lipschitz condition in any finite interval;

b) yφ(y) > 0 ∀ y �= 0;

c) φ(y) is nondecreasing; φ(−∞) = −∞; φ(∞) = ∞;

d) φ(y) has right and left derivatives, φ′+(y) and φ′−(y), at y = 0;

e) φ′+(y) φ′−(y) �= 0 when f(0) = 0,

then the system (2.3.4) has at most one limit cycle, and (if it exists) isstable.

Note that this theorem does not exclude the case in which no limit cycle exists atall. The existence of the limit cycle must be proved separately. For example, thiscan be done with the help of the Poincare-Bendixson theorem. However, for theparticular case of dynamical systems of the form (2.3.4), several theorems exist thatrepresent easier ways to establish limit cycles in these systems.51

50 A function g(x), x ∈ D, fulfills the Lipschitz condition if there is a positive constant k(the Lipschitz constant) such that for every x ∈ D and x′ ∈ D

|g(x) − g(x′)| ≤ k|x− x′|.

The Lipschitz condition is fulfilled when g(x) is continuous and when the derivativeg′(x) exists and is continuous on D. Cf. Brock/Malliaris (1989), pp. 15ff., for details.

51 The best-known of these theorems is due to A. Filippov, cf. Yan-Qian (1986), p. 96. Aneconomic application of the theorem in the context of the model discussed in Section2.3.3 below can be found in Galeotti/Gori (1990).

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54 Chapter 2

Theorem 2.7:52 When the following conditions hold for (2.3.4):

1) xg(x) > 0 when x �= 0, and G(±∞) = ∞ with G(x) =∫ x

0 g(ξ) dξ,2) xF (x) < 0 when x �= 0 and |x| is sufficiently small,3) there exist constants M > 0 and K > K ′ such that

F (x) ≥ K when x > M, and F (x) ≤ K ′ when x < −M,

then system (2.3.4) has stable limit cycles.

Figure 2.18 illustrates the requirements of Theorem 2.7 for the special case f(x) =x2 − a and g(x) = bx. F (x) is then a cubic function with a negative slope at theorigin such that (3) is immediately fulfilled. Furthermore, xg(x) > 0 ∀ x �= 0 andG(±∞) = ∞.

The Case of a Cubic F (x) in Theorem 2.7Figure 2.18

Theorems 2.5 and 2.6 and related theorems that rely on the Lienard equation arenot the only tools for establishing the uniqueness of limit cycles. Averaging methodsallow for quantitative approximations of limit cycles in many cases53, implying thatthe number of cycles and their stability can directly be examined.

The two Theorems 2.5 and 2.6 will be illustrated with two economic examplesin the following two sections.

2.3.2. The Symmetric Case: Unique Cycles in a Modified Phillips Model

The Lienard-van-der-Pol equation has received relatively little attention in eco-nomic dynamics probably because of the restrictive symmetry assumption of the

52 Cf. Yan-Qian (1986), p. 92.53 Cf. Guckenheimer/Holmes (1983), pp. 166ff. Chiarella (1990) discusses several en-

dogenous business-cycle models with the help of averaging methods.

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2.3.2. Unique Cycles in a Modified Phillips Model 55

well-known Levinson/Smith theorem or because it is not always possible to reducea given dynamical system to a form (2.3.1). A remarkable exception can be found inIchimura (1955) with an examination of some traditional mathematical business-cycle models.54

In the following, a simple modification of Phillips’ (1954) continuous-time,multiplier-accelerator model will be discussed.55 Consumption, C, depends onincome in the usual way:

C(t) = cY (t), 0 < c ≤ 1, (2.3.5)

with Y as net income. The desired capital stock, Kd, depends linearily on income:

Kd(t) = vY (t), v > 0. (2.3.6)

It is assumed that firms change their capital stocks as soon as the actual stock differsfrom the desired one:

K = I(t) = β(Kd(t) −K(t)

)= β

(vY (t) −K(t)

), β > 0, (2.3.7)

with I as net investment. The coefficient β is an adjustment parameter and ex-presses the reaction speed of investment in response to a discrepancy between ac-tual and desired stock.

Assume that income changes according to the excess demand,C(t)+I(t)−Y (t),in the goods market:

Y (t) = α(C(t) + I(t) − Y (t)

), α > 0, (2.3.8)

with the coefficient α as an adjustment parameter.Differentiating (2.3.7) with respect to time,

I(t) = β(vY (t) − I(t)

), (2.3.9)

and substituting for I and I in the differentiated form of (2.3.8) yields the linearsecond-order differential equation with constant coefficients

Y (t) +(α(1 − c) + β − αβv

)Y (t) + αβ(1 − c)Y (t) = 0. (2.3.10)

Let y = Y − Y ∗, with Y ∗ as the fixed-point value of net income. Equation (2.3.10)then turns into

y(t) +(α(1 − c) + β − αβv

)y(t) + αβ(1 − c)y(t) = 0. (2.3.11)

54 Another application can be found in Schinasi (1981).55 Cf. Lorenz (1987e) for the following model.

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56 Chapter 2

The solution of such a second-order differential equation with constant coefficientsis discussed in Appendix A.1.1; the eigenvalues of (2.3.11) are

λ1,2 =−A1 ±

√(A1)2 − 4A2

2(2.3.12)

with A1 =(α(1 − c) + β − αβv

)and A2 = αβ(1 − c). The eigenvalues are com-

plex conjugate when the discriminant is negative. Equation (2.3.10) then exhibitspersistent oscillations whenA1 = 0, i.e., when the eigenvalues are purely imaginary.

In order to transform (2.3.11) into a Lienard equation the assumption of con-stant coefficients has to be abandoned. Formally, a Lienard equation can easily beobtained. For example, let β = h(y) be a smooth function depending on incomein the way illustrated in Figure 2.19, i.e., investment responds nonlinearily to gapsbetween the desired and the actual capital stock. While a strong reaction to thesegaps is assumed for income levels near the fixed point y = 0, investment respondssluggishly if the deviation of income from its fixed-point level is large. The invest-ment function (2.3.7) therefore turns into a kind of Kaldorian investment functionwith the typical sigmoid shape.

The Investment Coefficient β = h(y)Figure 2.19

With β = h(y), equation (2.3.11) becomes

y +(α(1 − c) + h(y) − αh(y)v

)y + αh(y)(1 − c)y = 0. (2.3.13)

Set f(y) =(α(1 − c) + h(y) − αh(y)v

)and g(y) = αh(y)(1 − c)y. Under the

assumptions

• f and g are C1,

• β = h(y), h(y) > 0 ∀ y, h′(0) = 0, h′′(0) < 0,

• h(y) = h(−y),

• αv > 1,

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2.3.3. Unique Cycles in a Kaldor Model 57

• h(0) > −α(1 − c)1 − αv ,

equation (2.3.13) is indeed a Lienard equation which fulfills the assumptions of theLevinson/Smith theorem:

a) Assumed

b) There exist y1 < 0 and y2 > 0 such that f =(α(1 − c) + h(y) −αh(y)v

)< 0 for

y1 < y < y2 and f > 0 otherwise.

c) As g(y) = αh(y)(1 − c)y > (<) 0 for y > (<) 0, it follows that g(y)y > 0 ∀ y.

d) limy→∞F (y) = ∞ because f(y) increases for y > y2, and lim

y→∞G(y) =∫ y

0 g(ξ)dξ =

∞ because h(y) > 0 ∀ y.

e) f(y) = f(−y) by assumption, and g(y) = −g(−y) becauseαh(y)(1−c)y > (<)0for y > (<) 0.

Though equation (2.3.13) is therefore formally identical with a Lienard equationand fulfills the requirements of the Levinson/Smith theorem, it must be stressedthat the postulated function β = h(y) is purely ad hoc. While the general assump-tion that h(y) is bell-shaped can already be criticized, it is further necessary toassume the above relations between h(y) and the remaining coefficients in orderto obtain the desired result. The usual advantage of nonlinear cycle models overlinear models like the original Phillips model, namely that these models do not relyon precise parameter constellations in order to generate persistent fluctuations,therefore vanishes when the Levinson/Smith theorem is applied to this particularexample.

The following application of Theorem 2.6 demonstrates that the uniqueness oflimit cycles can be established in some cases without the introduction of additionaland restrictive assumptions.

2.3.3. The Asymmetric Case: Unique Cycles in a Kaldor Model

Consider once again the familiar Kaldor model that serves as a prototype modelin the course of this book. The general formulation of the model in its net valueversion is:

Y = α(I(Y,K) − S(Y,K)

),

K = I(Y,K).(2.3.14)

It is not possible to write this standard model immediately as a Lienard equationbecause the two-dimensional system obtained by differentiating the first or secondequation of (2.3.14) with respect to time and substituting for Y or K, respectively,will not be independent of the second variable: in either case partial derivatives willremain that depend on the second variable.

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58 Chapter 2

However, there are several ways to transform (2.3.14) into a Lienard equationby suitable assumptions on the functions I(·, ·) and S(·).56 The following modifica-tion is studied in Galeotti/Gori (1990).57 Assume that savings does not dependanymore on the capital stock, i.e., S = S(Y ), and that IK < 0 is constant. TheKaldor model (2.3.14), centered at the fixed-point values Y ∗ and K∗, then turnsinto

y = α(i(y, k) − s(y)

),

k = i(y, k),(2.3.15)

with y = Y −Y ∗ and k = K−K∗. The reader may verify that (2.3.15) still cannot bedirectly transformed into a Lienard equation with the help of the method describedabove. However, the variable transformation

u = y,

v = k − y

α,

(2.3.16)

transforms (2.3.15) into

u = α(i(u, v +

u

α) − s(u)

),

v = i(u, v +u

α) − u

α= s(u).

(2.3.17)

Combining the two equations yields

u = α[iuu+

izαu+ iz v − suu

],

= α(iu +izα

− su)u+ αizs(u),(2.3.18)

with iz as the partial derivative of i(u, v + (u/α)

)with respect to the second argu-

ment.When iz is constant, (2.3.18) is obviously a Lienard equation. In order to apply

Theorem 2.6 to this system, it is desirable to write (2.3.18) in a two-dimensionalform comparable to the form (2.3.4) in Theorem 2.6. Write iz/α = a. The so-called Lienard transformation introduces a new variable w = u−α

∫ u

0 (iξ + a− sξ) dξ

56 A couple of modifications of (2.3.14) are described in Lorenz (1987e). Galeotti/Gori(1990) demonstrate that many of these modifications can be reduced to a common formby appropriate transformations.

57 The authors use a slightly different version of Zhifen’s theorem which can be shown tobe identical with Theorem 2.6 presented above. Galeotti/Gori prove the existenceof a limit cycle with the Filippov theorem.

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2.3.3. Unique Cycles in a Kaldor Model 59

such that (2.3.18) is transformed into the system of first-order equations

u = w + α

∫ u

0(iξ + a− sξ) dξ,

w = u− α(iu + a− su)u = αizs(u).(2.3.19)

In (2.3.19), the expressions corresponding to F (x), f(x), φ(y) and g(x) of Theo-rem 2.6 are 58

F (u) = −α∫ u

0(iξ + a− sξ) dξ,

f(u) = F ′ = −α(iu + a− su),φ(w) = w,

g(u) = −αizs(u).

(2.3.20)

In order to apply Theorem 2.6 to the system (2.3.19), the functions i(u, z) and s(u)have not been specified precisely enough yet. Assume that s(u) and i(u, ·) displaythe sigmoid shapes known from the original Kaldor model with limu→±∞ su = ∞and limu→±∞ iu = 0.59

In addition to the sigmoid shapes of i(u, z) and s(u) it will be assumed that thelower partial equilibrium point u1 is closer to zero than u2. Figure 2.20 depictsthis scenario on the goods market. Furthermore, the functions f(u) and g(u) areassumed to intersect twice, such that for the assumed limits of su and iu, the ratiof(u)/g(u) is increasing for all values of u �= 0 (cf. Figure 2.21 for the shapes off(u), g(u), and the ratio f(u)/g(u)).

With this geometric specification of the function f(u) and g(u) and the assump-tion iz < 0 and constant, Theorem 2.6 can be applied to (2.3.19):

1. a) Assumed; g(u) is C∞;b) −uαizs(u) > 0 ∀ u �= 0;

c) G(∞) =

∫ ∞

0

(−αizs(ξ)) dξ = ∞ =

∫ −∞

0

(−αizs(ξ)) dξ = G(−∞),

2. a) Assumed;b) F (0) = −α(i(0, 0) − s(0)

)= 0;

58 Of course, these expressions can also be determined by direct inspection of (2.3.18)and comparing it with x = −φyg(x) − f(x)x derived form (2.3.4).

59 Cf. Kaldor (1940) for a justification of the shape of s(u). While the nonlinearity inKaldor’s investment function does not represent a really controversial assumption, anever-increasing savings rate off the equilibrium point does not seem to be very con-vincing. Nonetheless, Kaldor’s original shape of s(u) will be assumed in the followingbecause a linear function s(u) does not fulfill the requirements of Theorem 2.6. Thereader may verify that a linear s(u) or a savings rate converging to a finite value implyan eventually declining ratio f(u)/g(u).

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60 Chapter 2

An Asymmetric Kaldor ScenarioFigure 2.20

The Functions f(u), g(u), and the Ratio f(u)/g(u)Figure 2.21

c) Cf. Figure 2.21;

d) ” ” ,

3. a) Obvious, since φ(w) = w ;

b) φ(w)w = w2 > 0 ∀ w �= 0;

c) dφ/dw = 1;

d) Obvious, since φ(w) = w and φ′(w) = 1;

e) φ′φ′ = 1 �= 0.

All requirements of Theorem 2.6 are fulfilled and possible limit cycles of (2.3.19)are unique and stable. The existence of this limit cycle can easily be demonstratedwith the help of Theorem 2.7: as the theorem concentrates on the function F (u),the existence of a limit cycle follows immediately from the assumed form of f(u) andhence F (u) in Figure 2.21. All other requirements are covered by the propertiesmentioned above.

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2.4.1. The Dynamics of Conservative Dynamical Systems 61

2.4. Predator-Prey Models

The dynamic models presented thus far are able to exhibit limit cycles. If a systemhas a single limit cycle, then the trajectories starting at initial points in the basin ofattraction are attracted by this cycle. In addition to these limit cycle systems thereexists another type of a dynamical system which is able to generate oscillations butwhich is characterized by a different dynamic behavior.

2.4.1. The Dynamics of Conservative Dynamical Systems

Consider the two-dimensional dynamical system

x = f1(x, y),

y = f2(x, y),(2.4.1)

with the Jacobian matrix

J =

∂f1

∂x

∂f1

∂y

∂f2

∂x

∂f2

∂y

. (2.4.2)

Assume that the determinant of the matrix J is positive for all (x, y). It is shown inthe Appendix A.1.1 and in the models presented thus far that the sign of the traceof the Jacobian then plays a dominant role in determining the kind of oscillatingbehavior of a two-dimensional dynamical system. The question therefore ariseswhether a qualitative description of the meaning of the trace of J can be provided.In fact, in some physical applications of systems like (2.4.1) it is possible to assignthe existence of dampening or friction to the negative value of the trace.60 Thefollowing heuristic reflection may be helpful in understanding dynamical systemswhich exhibit closed orbits.

Consider a dynamic model like the Kaldor model to which the Poincare-Bendix-son theorem can be applied. The fixed point has to be unstable, i.e., the trace ofthe Jacobian has to be positive. In other words, there exists a tendency away fromthe fixed point in all directions, which may be interpreted as a negative friction. Ifthis were the case for every point in the phase space, the flow of the system wouldspiral toward the outer bounds of the phase space and no closed orbit could exist.However, it is demonstrated in the Bendixson criterion, i.e., Theorem 2.4, that thetrace of the Jacobian must change its sign if limit cycles are to be generated. Anegative trace corresponds to a positive friction such that the formerly explodingbehavior will be dampened for points sufficiently far away from the equilibrium.

60 Cf., e.g., the original Lienard equation in Section 2.3., where f ′(x)x represents a damp-ening term.

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62 Chapter 2

A closed orbit therefore emerges where the exploding and imploding forces bothtend toward zero, i.e., where the trace vanishes.

Dynamical systems with this kind of behavior are called dissipative systems.61 Mosteconomic models discussed in this book belong to this class of dynamical systems.However, there exists another class of systems which has received interest especiallyin classical mechanics, namely the so-called conservative dynamical systems. In aconservative system there is neither an additional input nor a loss of energy, imply-ing that no friction exists. According to the preceding heuristic reflection on thequalitative meaning of the trace of a Jacobian, this absence of friction is equivalentto a zero trace for all points in the phase space.62 The zero trace implies that the(possibly multiple) fixed points can be only saddles or centers.

One such conservative dynamical system which can be of economic interest isthe predator-prey system investigated by Lotka (1925) and Volterra (1931) inan early attempt to understand biological and ecological phenomena by means ofmathematical analysis. The model is concerned with the dynamic relations betweentwo interdependent species acting as predator and prey, respectively, within anecosystem.63 The dynamical system consists of the two-dimensional differentialequation system

x = ax− bxy,

y = −cy + dxy,a, b, c, d > 0, (2.4.3)

with x as the total prey population and y as the predator population. The preyare the only food source available to the predator. Thus, if x = 0, the predatorpopulation decreases exponentially at the rate c. If y = 0, the prey populationgrows exponentially to infinity at the rate a.

System (2.4.3) has two fixed points with x = y = 0, namely (x∗, y∗) = (c/d, a/b)and the trivial fixed point (0, 0) (a saddle point). The Jacobian matrix of (2.4.3),evaluated at the non-trivial fixed point, is

J =

(a− by −bxdy −c+ dx

)=

( 0 −bc/dda/b 0

)(2.4.4)

and has det J = ac > 0 and tr J = 0, i.e., the eigenvalues are purely imaginary. Thefixed point is therefore neutrally stable, implying that no conclusion on the dynamicbehavior of (2.4.3) can be drawn from the inspection of the Jacobian (2.4.4).

61 The term stems from considerations of physical systems with a permanent input of en-ergy which dissipates through the system. If the energy input is interrupted, the systemcollapses to its equilibrium state.

62 The obvious physical example of a conservative dynamical system is the perfect pendu-lum where no friction is involved. Note that the harmonic oscillator, shortly mentionedin Appendix A.1.1, is an example of a conservative system.

63 Cf. Clark (1976) for a survey of economic approaches to biological phenomena.

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2.4.1. The Dynamics of Conservative Dynamical Systems 63

In order to study the global dynamic behavior of a system like (2.4.3), it is usefulto introduce the concept of the first integral :64

Definition 2.6: A continuously differentiable function F : R2 → R issaid to be a first integral of a system x = f(x), x ∈ R2, if F is constant forany solution x(t) of the system.

When such a first integral exists it is not unique, i.e., when F (x) is a first integralthen F (x) + C is a first integral as well.

Level Curves in a System with a First Integral (The Dashed Line is Impossible)Figure 2.22

The constancy of F (x) for any solution can be expressed as dF (x)/dt = 0. Theconstant expressions F (x)+C define level curves for different values of the constantC. When a saddle is the only fixed point, the level curves are given by the unstableand stable manifolds and the associated hyperbolic trajectories. When a uniquefixed point is a center the level curves are closed orbits. Any initial point (exceptthe fixed points) is then located in a closed orbit. This can be visualized by aninspection of Figure 2.22.65 The closed curves L1, L2, and L3 represent examplesof level curves for different values of C. Each level curve is characterized by theproperty that dF (x)/dt = 0. Consider a point x(0) located in the level curve L2.The trajectory passing through this point is Φt(x(0)) (for t > 0 and t < 0). As x(0)is a point in a level curve, it can be described by the constant F

(x(0)

). The point

Φt(x(0)), t > 0, must also be located in this level curve because otherwise the termF (x) would not be constant for any solution. It follows that F (x(0)) = F

(Φt(x(0))

)∀ t > 0 and t < 0. The trajectory indicated by the dashed line in Figure 2.22 thuscannot exist when the system has a first integral. All initial points are located in oneof the infinitely many level curves characterized by different values of C.

64 Cf. Andronov/Chaikin (1949), pp. 99ff., and Arrowsmith/Place (1982), pp. 101ff.and 144ff., for the following ideas. The term is a relic of early inquiries into the behaviorof differential equations, cf. Arnold (1973, p. 75.

65 Cf. Arrowsmith/Place (1982), p. 101, for this argument.

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64 Chapter 2

In order to examine whether (2.4.3) possesses a first integral66, eliminate timefrom the system by dividing both equations, i.e.,

dy

dx= −(c− dx)y

(a− by)x. (2.4.5)

Rearranging, dividing by xy, and integrating yields

−a ln y + by − c lnx+ dx = A, (2.4.6)

where A is a constant. Equation (2.4.6) can be written as

y−aebyx−cedx = B. (2.4.7)

Set y−aebyx−cedx = F (x, y). The function F (x, y) is a first integral of (2.4.3), whichcan be seen from differentiating it with respect to time:

d

dtF (x, y) =

∂F (x, y)∂x

x+∂F (x, y)

∂yy. (2.4.8)

The partial derivatives of F are

∂F (x, y)∂x

= F (x, y)(− c

x+ d

), (2.4.9)

and

∂F (x, y)∂y

= F (x, y)(−a

y+ b

), (2.4.10)

respectively, such that

d

dtF (x, y) = F (x, y)

(− c

x+ d

)(a− by

)x+ F (x, y)

(−ay

+ b)(−c+ dx

)y

= 0. (2.4.11)

The function F (x, y) is therefore a first integral.The following theorem summarizes the discussion of the dynamic behavior of

the system (2.4.3):

Theorem 2.8 (Hirsch/Smale):67 Every trajectory of the Lotka/Vol-terra equations (2.4.3) is a closed orbit (except the fixed point (y∗, x∗)and the coordinate axes).

66 Cf. Gandolfo (1983), pp. 450ff.67 Cf. Hirsch/Smale (1974), p. 262.

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2.4.1. The Dynamics of Conservative Dynamical Systems 65

It follows that the closed orbits cannot be limit cycles. Otherwise, the trajectorieswhich approach a limit cycle are not closed orbits. As each point in the phase spaceis located in a closed orbit, the initial values of x and y at t = 0 therefore determinewhich of the infinitely many closed orbits describes the actual dynamic behavior ofthe system (cf. Figure 2.23).

Stylized Closed Orbits in a Predator-Prey SystemFigure 2.23

The predator-prey system (2.4.3) was classified as a conservative system with thehelp of the first integral. An alternative definition of conservative dynamical systemsconcentrates on the evolution of initial points contained in a subset of the phasespace. Assume that a dynamical system has infinitely many closed orbits and thatevery initial point is located in such a closed orbit. Consider the area A in Figure2.24. Initial points contained in this subset of the plane move to the area B underthe action of the flow. If the area A is identical with the area B, the dynamical systemis called area preserving (or volume preserving when the system is higher-dimensional(n ≥ 3)). A dynamical system can be called conservative if it is area preserving.

In contrast, dissipative systems contract areas (or volumes) when trajectories ap-proach an attractor. Figure 2.25 shows two trajectories starting at different initialpoints and approaching a fixed point attractor. The area between the two trajecto-ries is continuously getting smaller and approaches zero when the trajectories areclose to the fixed point.

Formally, the property of area preservation of a system x = f(x), x ∈ Rn, can beexamined with the help of the Lie derivative or the divergence of the vector field definedas

V

V= div f =

∑i

∂fi∂xi

, i = 1, . . . , n,

with V as the “volume” (i.e., the n-dimensional analog of an area) and div f as thedivergence of f. The Lie derivative is negative when the system is dissipative, i.e., ifit contracts area, and it vanishes when the system is conservative.

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66 Chapter 2

Area Preservation in a Conservative SystemFigure 2.24

Area Contraction in a Dissipative SystemFigure 2.25

In most examples from classical mechanics the two alternative definitions ofconservative and dissipative systems, i.e., via the existence of a first integral or thearea preservation property, lead to identical classifications.68 The predator-preysystem (2.4.3) is a peculiar system because it has a first integral but the Lie derivativediffers from zero:

V

V= a− by − c+ dx �= 0

68 Cf. Arnold (1973), pp. 198f., and Arrowsmith/Place (1982), pp. 103f., for identicalresults in Hamiltonian systems.

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2.4.2. The Goodwin Model 67

for all (x, y) except the fixed point. When x and y change during the motion ina closed orbit, the sign of the Lie derivative changes. An answer to this puzzlecan heuristically be delivered by inspecting the area covered by the closed orbits inFigure 2.23: in the region where the orbits come close to each other the derivativeis positive. Areas between the orbits are extended and imply the larger distancebetween the orbits in other regions. A negative Lie derivative in those regionsimplies the contraction toward the initial region with a short distance between theorbits.

2.4.2. Goodwin’s Predator-Prey Model of the Class Struggle

Conservative dynamical systems are really rare in economics. A remarkable excep-tion is Goodwin’s 1967-model of the class struggle which leads to the same formalframework as the predator-prey model of Lotka and Volterra and which will bepresented in the following.Consider an economy consisting of workers and capitalists. Workers spend all theirincome on consumption, while capitalists save all their income. The following list ofabbreviations, definitions, and relations describes the framework of the economy.For convenience, the goods price is normalized to unity.

Output: YLabor: LCapital: KWage rate: wGoods price: p = 1Labor productivity: Y/L = a = a0e

φt, φ = constantLabor income: wLLabor income share: u = wL/Y = w/aCapital income: Y − wLProfit share: 1 − w/aSavings: (1 − w/a)YCapital output ratio: K/Y = σ, σ = constantLabor supply: N = N0e

nt, n = constantEmployment rate: v = L/N

If investment, I = K, equals savings, the growth rate of the capital stock, K/K, isgiven as K/K = (1 − w/a)Y/K = (1 − w/a)/σ. The growth rate, K/K, equalsthe growth rate of income, Y /Y , when the capital-output ratio is constant. With anexogenously determined labor productivity, a, employment, L, is given by L = Y/a.Logarithmic differentiation of this equation yields

L/L = Y /Y − φ,

= (1 − w/a)/σ − φ.(2.4.12)

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68 Chapter 2

The above presentation can be summarized in the following set of growth ratesinvolved in the model:

N

N= n,

a

a= φ,

Y

Y=

K

K,

K

K= (1 − w

a)Y

K= (1 − w

a)/σ,

L

L= (1 − w

a)/σ − φ =

Y

Y− φ.

The central variables in the Goodwin model are the employment rate, v, and the la-bor bill share, u. Consider first the evolution of the employment rate v: logarithmicdifferentiation and substitution yields

v/v = L/L− N/N,

= Y /Y − φ− n,

= (1 − w/a)/σ − (φ+ n),

=1 − u

σ− (φ+ n),

(2.4.13)

or

v =(1 − u

σ− (φ+ n)

)v, (2.4.14)

which is a differential equation in the two variables v and u. The labor bill share, u,develops according to

u/u = w/w − a/a = w/w − φ. (2.4.15)

Goodwin assumed that the wage rate changes according to a standard Phillips curve,i.e.,

w/w = f(v), limv→1

f(v) = ∞, limv→0

f(v) = ω < 0,∂f

∂v> 0. (2.4.16)

For simplicity, (2.4.16) is linearily approximated by w/w = −γ + ρv, yielding

u/u = −γ + ρv − φ, (2.4.17)

or

u =(−γ + ρv − φ

)u. (2.4.18)

Equations (2.4.14) and (2.4.18) have the same formal structure as the Lotka/Volterra equations (2.4.3):

v =(1/σ − (φ+ n) − u/σ

)v,

u =(−(φ+ γ) + ρv

)u.

(2.4.19)

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2.4.3. Predator-Prey Structures in Dissipative Systems 69

The employment rate v serves as the prey while the wage bill share acts as thepredator. When there is no employment, the wage bill tends to zero. When thewage bill tends to zero, the employment rate increases since no relevant labor costsoccur.

System (2.4.19) has two fixed points, namely the trivial fixed point at the originand

v∗ =φ+ γ

ρ,

u∗ = 1 − σ(φ+ n).(2.4.20)

The Jacobian, evaluated at the non-trivial fixed point, is

J =

(0 −(φ+ γ)

σρ

ρ(1 − σ(φ+ n)

)0

). (2.4.21)

As the equations (2.4.19) and the Jacobian (2.4.21) are formally identical with theLotka-Volterra equations (2.4.3) and the associated Jacobian (2.4.4), every initialpoint in the Goodwin model is located in a closed orbit.69

This result supports the idea that a capitalist economy is permanently oscillat-ing. While the dynamic behavior of the Kaldor model outlined in Section 2.2.2.depends on the sign of the trace of the associated Jacobian, the trajectories of theGoodwin model describe closed orbits independent of any special magnitude ofthe derivatives. It may be that this oscillation property, together with the suggestedanalogy between predator-prey interdependence and the class struggle, constitutesthe main reason why the Goodwin model found attention especially among politi-cal economists.70 However, the analogy is superficial and does not refer directly tothe functional income shares of capitalists and workers or even to their populationsize. Further, the Goodwin model can be criticized along the same lines as was thecase with the original Lotka/Volterra system in biology, namely that the model isput together as an isolated set of assumptions which might not necessarily reflectrelevant influences. It may therefore be useful to investigate whether the Goodwinmodel is robust when facing modifications.

2.4.3. Other Examples and Predator-Prey Structures in Dissipative Systems

The Goodwin model constitutes the most prominent economic example of a pred-ator-prey structure. Other examples do exist but usually very specific functional

69 Goodwin (1967) investigated the solution to (2.4.10) by means of graphical integra-tion. Cf. Gabisch/Lorenz (1989), pp. 153ff., and Gandolfo (1983), pp. 448ff., for apresentation of Goodwin’s method.

70 Further developments of Goodwin’s model can be found in a variety of papers, includingDesai (1973), Flaschel (1984), Glombowski/Kruger (1987), Ploeg (1983, 1985),Pohjola (1981), Velupillai (1979), and Wolfstetter (1982).

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70 Chapter 2

forms are assumed. An example of a predator-prey model with a fairly generalset of assumptions is contained in v. Tunzelmann (1986) who re-considered theMalthusian population dynamics. The population growth rate increases with anincreasing real wage rate, w, and decreases when the real wage rate is zero, i.e.,

P /P = −c+ γw, c, γ > 0. (2.4.22)

The growth rate of the real wage rate decreases when the population increases, i.e.,

w/w = a− αP, a, α > 0. (2.4.23)

The equation reflects the basic Malthusian assumption that food supplies cannotgrow as fast as the population. An increasing population thus decreases per-capitafood supply which can be considered the real wage rate (in this simple scenariowith workers paid in the form of food). Multiplying the equations by P and w,respectively, yields the standard predator-prey form (2.4.3.).

The Lotka/Volterra system (2.4.3) and its economic equivalents (2.4.9) and(2.4.22) – (2.4.23) are dynamical systems whose behavior is very sensitive to vari-ations in their functional structure. Dynamical systems which change the characterof their dynamic behavior under small perturbations are called structurally unstablesystems.71 In order to demonstrate the effect of small perturbations, a basicallyarbitrary modification of the original Goodwin model will be performed in thefollowing.72 Instead of assuming that the rate of change of the wage rate, w, de-pends only on the employment rate, v, according to the usual Phillips relation, letthis rate additionally be influenced by the labor bill share, u:

w/w = f(v) + g(u), (2.4.24)

and assume that g(u) > 0∀u and g′(u) < 0, i.e., wage claims increase if workers areat a disadvantage in the functional income distribution. The derivative g′(u) canbe taken as being arbitrarily small.

The consideration of this modified Phillips curve in the Goodwin model leadsto

v =(1/σ − (φ+ n) − u/σ

)v,

u =(−(φ+ γ) + ρv + g(u)

)u.

(2.4.25)

71 Compare the discussion in Section 3.1 for precise definitions of various notions of struc-tural stability.

72 Economically more reasonable modifications can be found, e.g., in Wolfstetter (1982)in an investigation of the influence of stabilization policies in the Goodwin model andin an elaborate discussion of Wolfstetter’s results in Flaschel (1987). However, theeffects of these modifications are not as easily to trace as the simple perturbation givenhere.

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2.4.3. Predator-Prey Structures in Dissipative Systems 71

The new non-trivial fixed point is

v∗ =φ+ γ − g(u)

ρ=φ+ γ − g

(1 − σ(φ+ n)

,

u∗ = 1 − σ(φ+ n).(2.4.26)

The Jacobian, evaluated at this fixed point, is

J =

0 g(u∗) − (φ+ γ)

σρ

ρ(1 − σ(φ+ n)

)g′(u∗)

(1 − σ(φ+ n)

) . (2.4.27)

The determinant of J is not unambiguously positive anymore. Suppose that g(u∗)> 0 is sufficiently small such that det J is indeed positive. The trace of J will bedifferent from zero even for a seemingly negligible magnitude of g′(u∗)u∗ �= 0.As the derivative is assumed to be negative, the trace is negative. The real partsof the complex conjugate eigenvalues are therefore negative and the fixed point islocally asymptotically stable. System (2.4.25) therefore possesses an attractor andhas turned into a dissipative system.

Cugno/Montrucchio (1982b) investigated a similar modification of the Good-win model with an extended Phillips curve f(u, v) and were able to provide globalstability results. Other modifications of the original Goodwin model can easilybe constructed. Samuelson (1971, 1972) demonstrated that the consideration ofdiminishing returns in a general Lotka-Volterra framework can destroy the conser-vative character of the system.73

Actually, any additional term that influences the growth rate of a variable andwhich depends on the value of this variable is equivalent to the introduction ofa dampening effect. The conservative dynamics of the original Lotka/Volterraequations will then be destroyed and the emerging system turns into a dissipativedynamical system.

The Goodwin model suffers from its inherent structural instability (as is the casewith any conservative dynamical system) and is therefore sensitive to (even numeri-cally small) modifications in its structure. As soon as a dissipative structure prevails,a modified Goodwin model can exhibit converging or diverging oscillations as wellas limit cycles depending on the assumed dampening or forcing terms. While theoriginal model is structurally unstable, modifications thereof nevertheless still allowfor an oscillating behavior of the economically most relevant magnitudes like theunemployment rate and the labor income share.

It is possible to increase the dimension of the original model by consideringadditional state variables (like variable capital-output ratios or variable growth rates

73 Samuelson did not refer specifically to the Goodwin model but to the biologically ori-ented Lotka-Volterra framework. In the Goodwin model, diminishing or increasing re-turns to scale can be taken into account by assuming that the capital-output ratio σchanges with Y .

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72 Chapter 2

of the labor supply and labor productivity). It is, however, also possible to increasethe dimension by introducing particular lag structures. As an example, considerthe Phillips curve

w

w= −γ + ρv, (2.4.28)

assumed in the original Goodwin model. In (2.4.28), the growth rate dependsonly on the present value of the employment rate. Alternatively, it can be suspectedthat past values of the employment rate have an influence on this growth rate aswell. The most recent values will probably have the strongest influence on thegrowth rate, and the influence of values realized in the past should cease when therealization dates are far away from the present date. An example of a lag structurethat represents such a vanishing influence of past realizations of a variable is givenby a continuously distributed lag (cf. Appendix A.3 for details). In modifications ofGoodwin’s original model, Brody/Farkas (1987) and Chiarella (1990) assumedthat the growth rate of the real wage rate is determined by

w

w= −γ + ρx, (2.4.29)

where x is defined as

x =

∫ t

−∞

e−(t−ξ)/T

Tv(ξ)dξ = e−t/T

∫ t

−∞

eξ/T

Tv(ξ)dξ. (2.4.30)

It is shown in the Appendix A.3 that differentiating an equation like (2.4.30) withrespect to t yields the ordinary differential equation

x =v − x

T. (2.4.31)

Together with the previously derived equations (2.4.19), the Goodwin model thusturns into the three-dimensional system

v =(1/σ − (φ+ n) − u/σ

)v,

u =(−(φ+ γ) + ρx

)u,

x =v − x

T.

(2.4.32)

It has been demonstrated by Chiarella (1990a), pp. 73ff., that the system (2.4.32)generates limit cycles in the variables v and u. The result could be obtained byinvestigating the dynamic behavior of the system on its center manifold (cf. Ap-pendix A.2. for details). The conservative character of the original two-dimensional

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2.5. Relaxation Oscillations 73

Goodwin model has disappeared through the introduction of an exponential lagstructure and a dissipative system has emerged.74

2.5. Relaxation Oscillations

In the previous sections diverse two-dimensional models have been consideredwhich allow for oscillations in the two state variables. Like the Kaldor model, manyother dynamical systems include equations of the form

xi = αifi(x1, x2, . . . , xn), i = 1, . . . , n, (2.5.1)

with αi as adjustment coefficients. In many investigations of n-dimensional, economicdynamic models involving equations of the form (2.5.1) it has been assumed thatone or several of the adjustment coefficients are very large. The assumption isusually justified with the observation that well-organized markets (like financialmarkets with extensive information flows) react much faster to disequilibria thanothers. The interesting consequence of this assumption consists in the fact that theeffective dimension of the dynamical system can be reduced with this procedure.For example, if a single αi in an n – dimensional systems tends toward infinity thedimension of the remaining system can be reduced to n−1 when the motion of thefast variable is bounded. However, when the motion is bounded it is not definedon Rn−1 but on an n−1 – dimensional manifold in Rn.75

As an example, consider once again the Kaldor model (2.2.2)76 but (for the sakeof simplicity) assume that there is no capital depreciation and that savings dependsonly on income:

Y = α(I(Y,K) − S(Y )

),

K = I(Y,K).α > 0 (2.5.2)

74 Brody/Farkas (1987) estimated the parameter values in the original Goodwin modeland the modified model (2.4.32) for the Hungarian economy. A comparison betweenthe resulting equilibrium values of employment and wages in both models uncoveredthat the modified version is better suited for generating actual, empirical values.

75 Cf. Andronov/Chaikin (1949), Chapter 12, for a detailed discussion of the theory ofrelaxation oscillations. Compare also Hairer/Nørsett/Wanner (1982), pp. 107ff., fora description of relaxation oscillations in the van-der-Pol oscillator.

76 Other dynamical systems than this particular Kaldor model are actually better suited forillustrating the phenomenon of relaxation oscillation, cf. Chiarella (1990a), pp. 25ff.,or Guckenheimer/Holmes (1983), pp. 68ff. The Kaldor model is nevertheless beingused because a few standard problems in transforming given systems to a form requiredfor an application of an established mathematical result will become obvious. Othereconomic examples of relaxation oscillations can be found in Chiarella (1990b) andFranke/Lux (1992).

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74 Chapter 2

2.26.a 2.26.bThe Graphs of K = g(Y ) and Y = g−1(K)

Figure 2.26

With γ = 1/α, the first equation can be re-written as

Y =1γ

(I(Y,K) − S(Y )

), (2.5.3)

Assume that γ → 0.77 It follows that, in the limit, Y → ∞ except in a neighborhoodof those points which imply F (Y,K) ≡ I(Y,K) − S(Y ) = 0.

The consequence of this infinitely fast adjustment of Y on the dynamic behaviorof a system depends on the properties ofF (Y,K). WhenFY > 0∀(Y,K), the systemexplodes. However, in the considered Kaldor model, the derivative is negative forlow and high values of Y and positive for Y – values in the neighborhood of thefixed point. It follows that the motion is bounded and that Y rapidly converges toone of the possibly multiple goods-market equilibrium values of Y depending onthe initial conditions. It follows that F (Y,K) = 0 for almost all t on the time scalerelevant for the evolution of the slow variable K.

With γ → 0, the Kaldorian model (2.5.2) can thus be written as

F (Y,K) = I(Y,K) − S(Y ) = 0,

K = I(Y,K).(2.5.4)

Alternatively, (2.5.4) can also be written as

K = S(Y ), (2.5.5)

where the goods-market equilibrium condition has already been considered in thecapital-adjustment equation. However, (2.5.5) still mentions the two variables Kand Y , and cannot be called a 1D system. The state variable Y has to be replacedby an expression depending on K.

77 The parameter γ is also known as the time constant.

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2.5. Relaxation Oscillations 75

With the knowledge derived in Section 2.2.2 it is easy to provide a functionK = g(Y ). The equality I(Y,K) − S(Y ) = 0 has been derived in the presentcontext under the assumption γ → 0, but the equation has already been studied inSection 2.2.2. in the context of a description of the curve with the property Y = 0.Implicitly differentiating F (Y,K) = 0 yields

dK

dY=SY − IYIK

� 0. (2.5.6)

The same arguments as those provided in Section 2.2.2 apply for this case. Thus, afunctionK = g(Y ) exists for this model with a shape outlined in Figure 2.26.a. Theinverse of K = g(Y ) is shown in Figure 2.26.b. Inserting the relation Y = g−1(K)in (2.5.5) yields

K = S(g−1(K)

), (2.5.7)

i.e., a single equation of motion for the state variable K.

The Graph of (2.5.7)Figure 2.27

It might be suspected that the possibletypes of motion of this system are restricted tothose types known from linear 1D systems be-cause oscillatory behavior is known to emergeonly for 2D continuous-time systems. How-ever, due to the motion on a manifold in-stead of the entire phase space, a different dy-namic phenomenon can be observed. In or-der to get an idea of the possible motion con-sider the phase space of the remaining equa-tion (2.5.7) in Figure 2.27. The reduced sys-tem (2.5.7) has been derived from the orig-inal model (2.5.2); thus, it should imply thesame fixed point (stationary equilibrium) asthe original Kaldor model. According to the

construction of the relation Y = g−1(K), the stationary value of K is located in themiddle, upward sloping part of the graph in Figure 2.26.b. At the fixed point, onehas K = 0. Thus, the savings function should have such a form that the graph of(2.5.7) intersects the K – axis with its upward-sloping, middle part.

The system (2.5.7) with the associated graph in Figure 2.27 appears as a regularone-dimensional system. K increases (decreases) when the initial point on thegraph is located in the positive (negative) orthant. The fixed point C is obviouslyunstable; this should have been expected in this model because it is just a variant ofthe standard Kaldor model. If one chooses an initial point on the graph in a smallneighborhood of the fixed point, the motion is repelled from C and convergestoward B or D, respectively. The motion converges toward the points B and D alsowhen the initial point is located on the upper, downward-sloping part of the graphor the lower, downward-sloping part of the graph, respectively. However, neither B

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76 Chapter 2

The Kaldor Model with a Large Adjustment CoefficientFigure 2.28

nor D are fixed-points of the system (2.5.7). Thus, the motion of the system cannotcome to a halt at B or D.

In order to get an idea of what happens in this model when α = 1/γ → ∞consider Figure 2.28 which is a replicate of Figure 2.15 under the assumption thatα is large but finite. The long arrows which indicate the high adjustment speed inincome imply a nearly horizontal vector field. If an initial point is chosen whichis not located on the curve Y = 0 the system approaches this curve very rapidly.The pointsB′, C ′, andD′ in this Figure represent those points which are analogousto the points B, C, and D in Figure 2.27.78 At B′ and D′ the vector field pointsupwards or downwards, respectively. However, as soon as the curve Y = 0 is left,the strong influence of the large α dominates the motion of the system. It followsthat the system does not stop at B′ or D′ but that it is leaving the points verticallyand a sharp orientation change takes place shortly afterwards. Thus, a limit cyclebehavior can be established in this model. Instead of the smooth, elliptical cyclesknown from the discussion in the first sections of this chapter, limit cycles in thismodel are characterized by motions along the curve Y = 0 and very rapid motionsbetween points B′, D′ and the appropriate pieces of the Y = 0 – curve.

Keeping in mind that a direct analogy is not possible for the Kaldor model, theconsideration of the case of a large, but finite value of α in Figure 2.28 is helpful

78 A direct identification of the points is impossible in this Kaldor model. The reasonconsists in the fact that in this model the time derivative K is not identical with thesecond state variable. For example, Chiarella (1990a), p. 25, considered a system ofthe general form

x = f(x, y),y = x.

Guckenheimer/Holmes (1983), pp. 68f., investigate the van-der-Pol equation (2.3.3)which can be transformed to a form similar to Chiarella’s example.

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2.6. Irreversibility and Determinism 77

in explaining the motion at the points B and D in Figure 2.27. When the systemapproaches B or D, the trajectory jumps downwards or upwards to the points E orA, respectively. The overall motion indicated in Figure 2.27 can be called an oscil-latory motion though this motion is discontinuous at B and D.79 This oscillatorybehavior is indeed surprising because system (2.5.7) is effectively one-dimensional.However, the restriction of the system to a manifold and the involved nonlinearityare responsible for the emergence of cyclical behavior in this particular version ofthe Kaldor model.80

2.6. Irreversibility and Determinism in Dynamical Systems

This chapter concludes with a short discussion of time irreversibility inherent inmany dynamical systems. It was described in Section 2.4 that conservative dynam-ical systems are characterized by the presence of an infinity of closed orbits, i.e.,arbitrarily given initial points are located in one of these orbits.

Suppose that the motion of a conservative dynamical system starts at such anarbitrary initial point. As time is assumed to be continuous, the dynamical systemstarting at this point will continuously move in phase space and will eventually comeback to the initial point. Passing the initial point, the system will proceed in exactlythe same manner as during the first oscillation. If an observer of this motion knowsthe underlying differential equation system and the initial values of the state vari-ables at a given point in time, he/she will be (at least in principle) able to calculatethe location of the system in phase space by means of analytical or numerical meth-ods. Even if the initial point is not known precisely, the calculated trajectory startingat a slightly different point in phase space will stay close to the original trajectory.Equally important, if the dynamical system is conservative, it is possible to calculatethe history of a given point in phase space: as the system stays in a closed orbitforever, it also stayed in the orbit in the past. The past can be calculated by simplyreversing the direction of the time variable. Instead of counting time from t = 0to t = ∞ in predicting the future, time is assumed to run from t = 0 to t = −∞ indescribing the past. This property of conservative dynamical systems was responsi-ble for Laplace’s famous statement on the predictability question (cf. Chapter 1).In fact, many phenomena in celestial mechanics can be described by conservativedynamical systems with a high degree of accuracy, and it was the precision of sev-eral predictions in classical mechanics which encouraged the belief in the potentialpredictability of other dynamical systems in different fields.

Consider on the contrary a dissipative system which is characterized by the pres-ence of (negative or positive) friction. A dissipative system always possesses attrac-

79 The term relaxation oscillation can be explained by these discontinuous jumps. Imaginea rubber band spanned while the motion takes place on the manifold. At B and D therubber band is released implying a fast motion toward A and E.

80 For an indication that this type of dynamic behavior was not only known by mathemati-cians at the high tide of 2D systems at the end of the 1940s compare, e.g., Velupillai(1990), pp. 20ff.

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78 Chapter 2

tors or repellers, either in the form of fixed points, limit cycles, orbits in higher-dimensional phase space, or strange objects which are to be introduced later in thisbook. A completely unstable dynamical system can be viewed as being attracted byinfinity. In case of these dissipative systems it may still be possible to predict theevolution of the system in the future but it may be impossible to determine wherethe system started at a certain point of time in the past.

Irreversibility in Dissipative SystemsFigure 2.29

For example, let a dissipative system possess a unique fixed-point attractor ofthe focus type. If the observer exactly knows the underlying laws of motion, he orshe is able to predict the state of the system in the future for every arbitrary initialpoint. Consider the two different initial points A and B in Figure 2.29. In the limit,the trajectories belonging to the two initial points will approach each other and willspiral toward the fixed-point attractor. Assume that the two trajectories need thesame time until they enter a certain ball around the fixed point. If the system isclose to the attractor and the observer precisely knows the state of the system, thenit is possible to calculate the past of this point close to the fixed point. Moving back-wards on the trajectory belonging to point A for the same time span as the forwardmotion will carry the observer to point A again. However, a minor deviation of theestimated point from the actual point will imply a divergence of the calculated back-ward trajectory from the actual one, because an infinity of trajectories belonging todifferent initial points in phase space are located in the ball around the attractor.The observer may thus incorrectly calculate point B as the past of a point locatedin the ball around the fixed point.

If the initial points of a dissipative dynamical system are located on the attractor,the remarks on the past and future predictability of conservative systems apply aswell. For example, if an initial point is located in a limit cycle81 the trajectory startingat this point will eventually return to the initial point and complete prediction inboth time directions is possible. For all other initial points located on transients,

81 The case of an initial point identical with a fixed-point attractor is, of course, trivial.

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2.6. Irreversibility and Determinism 79

Time Reversible Time Irreversible

Dissipative Systems On an attractor On transients

Conservative Systems Everywhere Nowhere

Reversibility and Irreversibility in Dynamical SystemsTable 2.1

the determination of a point’s past is possible only if the coordinates of that pointare known with absolute precision.

These properties are summarized in Table 2.1. As dissipative systems are domi-nating economic dynamics, it can be concluded that backward prediction is practi-cally impossible in most economic models.

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Chapter 3

Bifurcation Theory and Economic Dynamics

This chapter deals with a subject that has become a major focus of research ineconomic dynamics during the last decade, namely bifurcation theory. Central

to this topic is the question whether the qualitative properties of a dynamical systemchange when one or more of the exogenous parameters are changing. In contrastto the physical sciences, it is usually impossible to assign a definitive, once-and-for-allvalid number to most parameters occurring in dynamical systems in economics. Pa-rameters are introduced into an economic model in order to reflect the influenceof exogenous forces which are either beyond the scope of pure economic expla-nation or which are intentionally considered as being exogenously given from thepoint of view of partial theorizing. It is desirable to determine whether the qual-itative behavior of a dynamical system persists under variations in the parameterspace. Thus, the results of bifurcation theory are especially important to dynamicmodelling in economics.

The bifurcation behavior of a dynamical system depends to some degree on theinvolved time concept, i.e., whether the system is designed in continuous or discretetime.1 As some kinds of bifurcation occur in only one of these two types of dynami-cal systems, this chapter is separated into two sections, one which presents the mostimportant bifurcations in continuous-time systems and one which surveys discrete-time systems. Though this may be viewed as being ponderous, the distinction formsa bridge between the presentation of the regular nonlinear continuous-time systems

1 Although discrete-time dynamical systems can also occur in the form of Poincare mapsin the study of continuous-time systems, it will be assumed in the course of this sectionthat a discrete-time system emerges generically from a discrete time concept.

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3.1. Preliminaries and Structural Stability Concepts 81

of Chapter 2 and the introduction of chaotic discrete-time systems in the next chap-ter.

Both subsections contain a short description of the fold bifurcation, the pitch-fork bifurcation and the transcritical bifurcation for the sake of relative complete-ness. Central to the presentation of the bifurcation behavior in both types of dynam-ical systems is the Hopf bifurcation which has recently gained the most attention ineconomic dynamics. The presentation of the flip bifurcation, which occurs only inone-dimensional discrete-time systems, will directly transfer to chaotic dynamics.

All types of bifurcations introduced in this chapter are local bifurcations in thesense that only the behavior of a dynamical system in the neighborhood of a singlefixed point is affected. The global bifurcation behavior of a dynamical system overthe whole range of admissible values for the state variables will be the subject ofparts of the following chapters.

3.1. Preliminaries and Different Concepts of Structural Stability

This section introduces some basic notations and discusses the concept of structuralstability at some length. The presentation of the standard definition of structuralstability relying on the notion of topological equivalence is followed by short de-scriptions of other possible definitions and a discussion of the usefulness of theconcept in economic theory.

Preliminaries

Consider the ordinary differential equation2

x = f(x, µ), x ∈ R, µ ∈ R, (3.1.1)

with µ as a parameter. Assume that (3.1.1), for µ = µ0, has a fixed point (x∗, µ0)such that 0 = f(x∗, µ0). The eigenvalue of the system (3.1.1) is given by λ =∂f(x, µ)/∂x, and it is well-known that the fixed point is locally asymptotically stableas long as λ < 0 at (x∗, µ0). Assume that, at (x∗, µ0), the eigenvalue is equal tozero. It follows from the implicit function theorem that the fixed points of (3.1.1)for values of µ different from µ0 can be expressed as a smooth function x∗ = x∗(µ)if λ �= 0 for µ �= µ0. The function x∗(µ) describes branches of fixed points. If,at (x∗, µ0), several branches of fixed points come together, the point (x∗, µ0) issaid to be a bifurcation point. The presentation of the branches of fixed points in(x∗ − µ) – space is called a bifurcation diagram. In Figure 3.1 the solid and dashedlines depict branches of fixed points. The solid lines represent stable fixed points,and the dashed line shows an unstable fixed point.

2 The generalization of the notation to the n-dimensional case is straightforward. Cf.Guckenheimer/Holmes (1983), pp. 118f., for the following definitions.

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82 Chapter 3

A Bifurcation DiagramFigure 3.1

As it can be seen from the bifurcation diagram, a former single fixed point splits(bifurcates) into several distinct fixed points at the bifurcation point. The value ofµ at which the bifurcation occurs is called the bifurcation value of µ. If no bifurcationoccurs at a fixed point (x∗, µ), the fixed point is said to be hyperbolic.

The bifurcation phenomenon can be related to the notion of structural stability.Roughly speaking, a dynamical system is called structurally stable if the qualitativedynamic properties of the system persist with small variations in its structure, i.e.,when varying the parameters or considering small changes in the functional forms.For example, if a dynamical system possesses a unique and asymptotically stablefixed point, structural stability implies that the fixed point is unique and asymp-totically stable for different parameter values as well. In other words, a dynamicalsystem is structurally stable if the two trajectories stay close together. A bifurcationvalue µ0 is therefore a value of µ for which the dynamical system is structurallyunstable.

Topological Equivalence and Structural Stability

The foregoing description of structural stability is superficial because nothing hasbeen said on the meaning of terms like “qualitative properties” or “close together”.Depending on the definition of these terms, different notions of structural stabilitycan be distinguished in a more careful description.3

The most widespread definition of structural stability is usually attributed to theRussian Gorki School (cf. Andronov/Chaikin (1949)). The similarity between twodynamical systems is expressed in terms of the so-called topological equivalence:

3 Extensive discussions of the following concepts can be found in Abraham/Marsden(1980), Arnold (1988), and Vercelli (1984).

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3.1. Preliminaries and Structural Stability Concepts 83

Definition 3.1: Two dynamical systems are topologically equivalent ifthere exists a homeomorphism4 from the phase space of the first sys-tem to the phase space of the second system that transforms the phaseflow of the first system to the phase flow of the second system.

Figures 3.2 and 3.3 illustrate the meaning of topological equivalence. Figure 3.2.adepicts an attracting circle, i.e., a limit cycle. The elliptic attracting orbit in Figure3.2.b differs from the circle in 3.2.a in a geometric sense, but the property of alimit cycle persisted under the transformation. Imagine that Figure 3.2.b has beengenerated by an appropriate stretching of Figure 3.2.a. The homeomorphism thattransforms 3.2.a to 3.2.b can be understood as a coordinate transformation.

3.2.a 3.2.bLimit Cycles in Topologically Equivalent Dynamical Systems

Figure 3.2

Figures 3.3.a and 3.3.b show examples of two systems that are not topologicallyequivalent. There does not exist a homeomorphism that can transform the limitcycle in 3.3.a to the fixed-point attractor in 3.3.b by stretching or squeezing the limitcycle.

The notion of topological equivalence suggests the following definition of struc-tural stability:5

Definition 3.2: A dynamical system is structurally stable if for every suf-ficiently small perturbation of the vector field the perturbed system istopologically equivalent to the original system.

The term “small perturbation” is usually interpreted in terms of the C1 norm: twodynamical systems are close at a point x if the associated images, e.g., f(x) and g(x),and the first derivatives, f ′(x) and g′(x), are close together.

4 A homeomorphism is a continuous map f : X → Y ; X,Y ⊂ Rn, with a continuous

inverse.5 Cf. Arnold (1988), p. 90.

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84 Chapter 3

3.3.a 3.3.bTopologically Non-Equivalent Dynamical Systems

Figure 3.3

In the two-dimensional, continuous-time case structural stability in the sense ofDefinition 3.2 can be established relatively easily. A theorem by Peixoto (1962)says (among other things) that a dynamical system

x = f(x), x ∈ R2 (3.1.2)

is structurally stable if 6

i) the fixed points, i.e., {x | x = 0}, are hyperbolic, orii) every closed orbit is either a periodic attractor or repeller, or

iii) no trajectory connects two saddle-points, oriv) the number of limit cycles is finite.

Definition 3.2 can be generalized when other descriptions of equivalence are used.For example, Arnold (1988) introduced the concept of orbital equivalence in orderto include closed orbits in the class of structurally stable systems which differ bytheir periods.

The Spectrum of Definitions and Generic Systems

Actually, an entire spectrum of varying definitions of structural stability can be im-agined.7 The end points of this spectrum are characterized by two contradictorypositions: on the one hand, the attempt is made to specify the considered systems

6 Cf. Guckenheimer/Holmes (1983, p. 60, and Hirsch/Smale (1974), p. 314.7 Cf. Vercelli (1984, 1989) for an intensive discussion.

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3.1. Preliminaries and Structural Stability Concepts 85

as precisely as possible; on the other hand, one tries to include as many systems aspossible in the class of structurally stable systems. The aforementioned definitionthat incorporates the notion of topological equivalence is unable to fulfill bothrequirements: too many dynamical systems turned out to be structurally unstablein the sense of Definition 3.2.

Two-dimensional dynamical systems are well-understood and the concept ofstructural stability in the sense of Definition 3.2 can be applied to these systemswithout any difficulties. After the extensive work of members of the Gorki Schoolon two-dimensional systems, the presumption emerged that Definition 3.2 couldserve as an instrument in describing so-called generic systems, i.e., systems with typi-cal and generally valid dynamic properties. The culminating point of work on thissubject can be seen in Peixoto’s theorem which (in addition to the aforementionedlist of properties of structurally stable systems) says that structurally stable systemsare generic in the two-dimensional phase space.

Many different definitions of structural stability were constructed in order to findsomething similar to Peixoto’s theorem in higher dimensions. However, the workof Smale (1963, 1967) has uncovered that structurally stable dynamical systems aregeneric only in the two-dimensional phase space. There exist higher-dimensionalsystems in whose neighborhood there is not a single structurally stable system. Infact, this work laid the foundations for the investigation of chaotic dynamical systemsto be introduced in the next chapters.

Possible Problems with Structural Stability Concepts in Economics

The concept of structural stability introduced above is a mathematical conceptthough the motivation for dealing with it originated in the natural sciences: for along time only those laboratory experiments were considered relevant which couldbe repeated at any time under (necessarily) slightly different environmental con-ditions. A mathematical model that describes this experiment should thereforepossess the same property of qualitatively similar results under small perturbations.

Mathematics does dot necessarily have problems with the notions of “slight” or“small” perturbations. It was mentioned above that the term is usually interpreted inthe sense of the C1 norm with infinitesimally small parameter variations. However,research in applied sciences like economics often does not deal with infinitesimalchanges but with finite variations in the parameters of a model. When such finitevariations are permitted in various definitions of structural stability, it turns out thatthe concept becomes vague.

Imagine that a dynamical system changes its dynamic properties at a particularbifurcation value µ0 of a parameter. When a given system is perturbed by varyingµ, the system should be called structurally stable in the sense of Definition 3.2 aslong as µ < µ0. However, when a larger variation is considered such that µ > µ0the system should be called structurally unstable. Equivalently, when the system isat µ0, an infinitely small variation in µ changes its qualitative properties and it isstructurally unstable. The present value of the parameter and the magnitude ofits variation determine whether the system is structurally stable or unstable. The

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86 Chapter 3

different variations in the parameter constitute a change in the norm underlyingthe idea of a perturbation, but applied science has to accept the magnitude of a(usually exogenous) parameter. In order to be formally correct, finite variationsin the parameters require a separate definition of the underlying norm in everyapplication of the concept of structural stability. From a practical point of view thisseems to imply a failure of the concept.

Economic theory encounters another problem which is seldom mentioned inthe mathematical and natural-science literature on structural stability. The conceptsmentioned above assume that the original and the perturbed systems possess thesame dimension. However, economic models are abstract pictures of real-life phe-nomena and only a few economic variables are taken into account in each model. Itis therefore important to know what happens when in a necessarily low-dimensionalmodel an additional variable is included in the list of interesting variables.

For example, consider the simple, linear, three-dimensional system

x1 = x2,

x2 = ax1 + bx2 + cx3,

x3 = dx1.

(3.1.3)

For c = 0, the first two equations of the system (3.1.3) can be combined in thelinear, second-order differential equation

x1 − bx1 − ax1 = 0. (3.1.4)

This second-order equation and the third equation of (3.1.3) still constitute a three-dimensional system, but the relevant system in the form of (3.1.4) is two-dimensional:the evolution of x3 follows that of x1, and x3 has no influence on the evolution ofthe first two variables at all. For c �= 0, the system (3.1.3) can be written as thethird-order differential equation

˙x− bx1 − ax− cdx1 = 0. (3.1.5)

The dimension of the relevant system is now identical with the dimension of (3.1.3).In general, it cannot be assumed per se that the dynamic behavior of (3.1.5) is similarto that of the de-coupled system (3.1.4). This is particularly true when nonlinearterms are involved on the r.h.s. of (3.1.3).8

Summarizing, the change in the dimension of the relevant system can imply achange in its qualitative dynamic behavior. Alternative definitions of structural sta-bility can take the change in the dimension into account: a system may be calledstructurally stable if a change in the dimension of the system does not change itsdynamic properties. In fact, parts of this idea are realized in the elementary catas-trophe theory which will be described in greater detail in Chapter 7.

8 If x3 = f(x1) with f(x1) as a logistic curve, the so-called Shil’nikov scenario emergeswhich can imply chaotic motion. Cf. Section 5.4.1 for details.

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3.2. Bifurcations in Continuous-Time Dynamical Systems 87

3.2. Local Bifurcations in Continuous-Time Dynamical Systems

This section deals with the change in the qualitative dynamic behavior of continu-ous-time dynamical systems when a parameter is changed at a fixed point. In theselocal bifurcations only the change in the stability properties of a fixed point or theemergence of closed orbits or additional fixed points in a small neighborhood of afixed point are considered.

The first section deals with the simplest local bifurcations in which a fixed pointchanges its stability properties and/or additional fixed points emerge. The emer-gence of closed orbits via a Hopf bifurcation is discussed in the second sub-section.

3.2.1. Fold, Transcritical, and Pitchfork Bifurcations

The following types of bifurcation will be presented only for the one-dimensionalcase, i.e., for dynamical systems of the type

x = f(x, µ), x ∈ R, µ ∈ R, (3.2.1)

though these bifurcations can occur in higher-dimensional systems as well.

Fold Bifurcation

Consider the differential equation (3.2.1) and let (x∗, µ0) = (0, 0) for simplicity.

Theorem 3.1 (Fold Bifurcation):9 Let f in (3.2.1) be C2 and assumethat there is a fixed point (x∗, µ0) = (0, 0). If

(1)∂f(0, 0)∂x

= λ = 0,

(2)∂2f(0, 0)∂x2 �= 0,

(3)∂f(0, 0)∂µ

�= 0,

then, depending on the sign of the expressions in (2) and (3), there are

i) no fixed points near (0, 0) if µ < 0 (µ > 0), and

ii) two fixed points near (0, 0) if µ > 0 (µ < 0).

9 Cf. Guckenheimer/Holmes (1983), pp. 146 ff., for a generalized version of the theo-rem for the case x ∈ R

n. The following version is related to Whitley’s (1983) formu-lation for discrete maps.

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88 Chapter 3

The fold bifurcation is sometimes also called a saddle-node bifurcation. Conditions(2) and (3) are called transversality conditions.10

Figure 3.4 illustrates the fold bifurcation for the prototype equation x = µ− x2.The signs of the transversality conditions (2) and (3) are negative and positive,respectively. If the parameter µ is lower than the bifurcation value µ0 = 0, no fixedpoint exists. For µ > µ0, two branches of fixed points emerge, one being stable andthe other being unstable. In other dynamical systems the bifurcation diagrams canlook differently. For example, when (3) has a negative sign, the bifurcation diagramappears mirror-imaged with respect to the x-axis. If (2) is positive, the stability ofthe two fixed-point branches is reversed.

3.4.a: The Phase Portrait 3.4.b: The Bifurcation DiagramThe Fold Bifurcation

Figure 3.4

As an economic example of the fold bifurcation consider a simple partial-analyt-ical model of the labor market. Let s(w) and d(w) be the supply of and demandfor labor, respectively, which both depend on the real wage w. The change in thereal wage rate is assumed to depend on the excess demand for labor in this market,i.e.,

w = β(d(w) − s(w)

), β > 0. (3.2.2)

Assume that the demand function is parameterized by µ and let d(w) = µ − bwin the following. Assume that the labor supply function reflects an inferiority suchthat it is bending backwards for high values of w (cf. Figure 3.5).

In detail, let d2s(w)/dw2 < 0 ∀ w and ds(w)/dw < 0 for w greater than avalue w0. Denote the right-hand side of (3.2.2) as f(w, µ) = d(w, µ) − s(w) andlet µ0 be the value of µ such that f(w, µ0) = 0 and ∂f(w, µ0)/∂w = 0, i.e., thereis a fixed point where the demand and supply functions are tangent. Obviously,

10 In the present context “transversality” should be read as “the most general descriptionof a family of functions at a bifurcation point”.

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A Labor Market with an Inferiority in the Labor SupplyFigure 3.5

∂f2(w, µ0)/∂w2 > 0 and ∂f(w, µ0)/∂µ > 0, and the conditions (2) and (3) ofTheorem 3.1 are fulfilled. Thus, a fold bifurcation occurs at the bifurcation valueµ0. For µ > µ0 no fixed point exists. If µ < µ0, two branches of fixed points emerge,

3.6.a: The Phase Portrait 3.6.b: The Bifurcation DiagramA Fold Bifurcation in the Labor Market

Figure 3.6

one being stable and the other being unstable. Figure 3.6 shows the phase portraitand the bifurcation diagram for this simple labor market model.

Transcritical Bifurcations

The fold bifurcation implies that no fixed point exists for parameter values smalleror larger (depending on the signs of (2) and (3)) than the bifurcation value. How-ever, it often occurs in practical applications that dynamical systems have at least a

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90 Chapter 3

so-called trivial fixed point at the origin. The transcritical bifurcation deals with theexchange of stability of a persisting fixed point. If the fixed point persists undervariations in µ, then f(0, µ) = 0 ∀µ. As this contradicts the transversality condition(3) in Theorem 3.1, that condition will be replaced by condition (3’) in the nexttheorem.

Theorem 3.2 (Transcritical Bifurcation): Let f in (3.2.1) be C2 andassume that there is a fixed point (x∗, µ0) = (0, 0). If

(1)∂f(0, 0)∂x

= λ = 0,

(2)∂2f(0, 0)∂x2 �= 0,

(3’)∂f2(0, 0)(∂µ∂x)

�= 0,

then, depending on the sign of the expressions in (2) and (3’),

i) the fixed point x∗ is stable (unstable) for µ < 0 (µ > 0), and

ii) the fixed point x∗ becomes unstable (stable) for µ > 0 (µ < 0) anda branch of additional stable (unstable) fixed points x(µ) emerges.

The transcritical bifurcation is thus characterized by an exchange of stability of theorigin. Figure 3.7 shows the phase portrait and the bifurcation diagram of thetranscritical bifurcation for the prototype equation x = µx − x2. The sign of thetransversality conditions (2) and (3’) are negative and positive, respectively. Forµ < µ0 = 0 the origin x = 0 is stable and a branch x∗(µ) of unstable, negative fixedpoints exists. If µ > µ0, the fixed point x = 0 becomes unstable and a branch ofstable, positive fixed-point emerges.

3.7.a: The Phase Portrait 3.7.b: The Bifurcation DiagramThe Transcritical Bifurcation

Figure 3.7

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3.2. Bifurcations in Continuous-Time Dynamical Systems 91

A “Neoclassical” Production Function with Incomplete Inada ConditionsFigure 3.8

If the sign of (3’) were negative (instead of the positive sign in the prototypeequation), the bifurcation diagram in Figure 3.7.b would appear mirror-imaged. If(2) had a positive sign instead, the stability of the fixed points for different µ wouldbe reversed.

A simple economic example of a transcritical bifurcation in a one-dimensionalsystem can be constructed from standard neoclassical growth theory. Consider thefamiliar adjustment equation in a one-sectoral growing economy

k = sy(k) − nk, (3.2.3)

with k as capital intensity, y as per-capita-output, n as the labor growth rate, and s asthe savings rate. The production function y(k) is usually assumed to fulfill the Inadaconditions, i.e., i) y(0) = 0, ii) y′(k) > 0, iii) y′′(k) < 0, and iv) y′(0) = ∞. Assumethat conditions i) - iii) hold but that the slope of y depends upon a parameter µwith y′µ(0)|µ=0 = 0 and ∂y′µ(k)/∂µ > 0 ∀ k > 0 (cf. Figure 3.8).

Define µ0 as the value of µ such that, for given n and s, sy′µ0(0) = n, i.e., the

eigenvalue is λ = 0. If µ < µ0, λ is negative and the origin is a stable fixed point. Letµ = µ0. The transversality conditions are fulfilled by assumption, i.e., y′′µ(k) < 0 ands∂y′µ(k)/∂µ > 0. Thus, a transcritical bifurcation occurs at µ = µ0 such that theorigin becomes unstable and new fixed points k∗ > 0 emerge in a neighborhoodof k = 0 for increasing µ. Figures 3.9.a and 3.9.b show the phase portraits for thetwo cases µ < µ0 and µ > µ0.

Pitchfork Bifurcation

A final example of a bifurcation in a one-dimensional continuous-time system is theso-called pitchfork bifurcation. This bifurcation can occur in dynamical systems ofthe form (3.2.1) with shapes of f similar to an odd function with respect to x, i.e.,

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92 Chapter 3

3.9.a: µ < µ0 3.9.b: µ > µ0Phase Portraits in a Neoclassical Growth Model for Different µ

Figure 3.9

f(x, ·) = −f(−x, ·).11 When f is an odd function, then the sufficient conditionsfor a transcritical bifurcation are not fulfilled since condition (2) in Theorem 3.1will be violated for at least one x. Condition (2) will be replaced by the requirementthat the third partial derivative with respect to x is different from zero.

Theorem 3.3 (Pitchfork Bifurcation): Let f in (3.2.1) be C3 andassume that there is a fixed point (x∗, µ0) = (0, 0). If

(1)∂f(0, 0)∂x

= λ = 0,

(2’)∂3f(0, 0)∂x3 �= 0,

(3’)∂2f(0, 0)(∂µ∂x)

�= 0,

then, depending on the sign of the expressions in (2’) and (3’),

i) the fixed point x∗ is stable (unstable) for µ < 0 (µ > 0), and

ii) the fixed point x∗ becomes unstable (stable) for µ > 0 (µ < 0)and two branches of additional stable (unstable) fixed points x(µ)emerge.

Figure 3.10 shows the phase portrait and the bifurcation diagram for the prototypeequation x = µx − x3. The signs of the transversality conditions (2’) and (3’) inTheorem 3.3 are negative and positive, respectively, such that a so-called supercritical

11 Cf. Section 2.3. for the relevance of odd functions in the Lienard equation. The trivialexample of a linear odd function is a straight line with nonzero slope passing throughthe origin.

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3.2. Bifurcations in Continuous-Time Dynamical Systems 93

3.10.a: The Phase Portrait 3.10.b: The Bifurcation DiagramThe Pitchfork Bifurcation

Figure 3.10

pitchfork bifurcation occurs with the bifurcating branches representing stable fixedpoints.

As in the case of the transcritical bifurcation, the bifurcation diagram in Figure3.10 would appear mirror-imaged if the sign of (3’) were reversed. If (2’) werepositive, then the two emerging additional fixed points would be unstable. In thatcase, a subcritical pitchfork bifurcation would occur.

As an economic example, consider an abridged version of the Kaldor modelpresented in Section 2.2.2. Assume that the investment function has the sameshape as in Figure 2.13, but let investment be independent of the capital stock.12

The model then reduces to the single goods market adjustment equation

Y = α(I(Y ) − S(Y )

)(3.2.4)

with the usual meaning of the symbols. Let Y ∗ denote the inner goods marketequilibrium in Figure 2.14, and formulate (3.2.4) in terms of the deviations fromthe appropriate I∗ and S∗ levels:

y = α(i(y) − s(y)

), (3.2.5)

with y = Y − Y ∗, i = I − I∗, and s = S − S∗. Assume further that the investmentfunction can be parameterized such that the slope of i(y) decreases for all y when aparameter µ is increased, i.e., the investment response to deviations from the equi-librium level Y ∗ is getting smaller. In formal terms, let ds(y)/dy be a constant andassume ∂i2(y, µ)/(∂y∂µ) > 0. The Kaldor assumption on the investment functionimplies ∂i3(y, µ)/∂y3 < 0 at y = 0.

Figure 3.11 shows the fixed-point constellations for values of the parameter suchthat the origin is unstable (solid line) and stable (dashed line).

12 This is, of course, the standard short-run macroeconomic approach, assuming that inthe short-run the influence of investment on the capital stock can be neglected.

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94 Chapter 3

A Parameterized Kaldorian Investment FunctionFigure 3.11

3.12.a: µ < µ0 3.12.b: µ > µ0

Phase Portraits of an Abridged Kaldor Model for Different µFigure 3.12

Define µ0 as that parameter value for which the eigenvalue of (3.2.5) is zero, i.e.,α(∂i(y, µ0)/∂y− ∂s(y)/∂y

)= 0. Then the conditions of Theorem 3.3 are fulfilled

and a pitchfork bifurcation occurs at µ0. Figure 3.12 shows the phase portraits fordifferent values of µ in this abridged Kaldor model.

Summary

The different bifurcation types with the associated transversality conditions and theprototype equations are summarized in Table 3.1 for the case n = 1. The table alsoincludes the Hopf bifurcation to be presented in the next section.

All three of these bifurcation types can occur in higher-dimensional continuous-time dynamical systems as well. The requirement λ = 0 in Table 3.1 then has to bereplaced by the condition that out of the n eigenvalues a single eigenvalue is zero

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3.2. Bifurcations in Continuous-Time Dynamical Systems 95

Eigenvalue Transversality PrototypeCondition Equation

Fold λ = 0 ∂f/∂µ �= 0 x = µ− x2

Bifurcation ∂2f/∂x2 �= 0

Transcritical λ = 0 ∂2f/(∂µ∂x) �= 0 x = µx− x2

Bifurcation ∂2f/∂x2 �= 0

Pitchfork λ = 0 ∂2f/(∂µ∂x) �= 0 x = µx− x3

Bifurcation ∂3f/∂x3 �= 0

Hopf λi, λi ∈ C n.a. x = −y+Bifurcation Reλi = 0 x

(µ− (x2 + y2)

)∂ Reλi∂µ

> 0 y = x+

y(µ− (x2 + y2)

)Bifurcation Types in Continuous-Time Dynamical Systems

Table 3.1

while k eigenvalues are positive and n−k−1 eigenvalues are negative. Furthermore,the conditions on the single partial derivatives must be replaced by the appropriatematrix expressions.13

3.2.2. The Hopf Bifurcation in Continuous-Time Dynamical Systems

The types of bifurcation presented in the foregoing section deal with the emergenceof additional branches of fixed points or with the exchange of stability betweentwo branches of fixed points. While these bifurcations are important dynamicalphenomena, another kind of bifurcation deserves attention in dynamical systemstheory, namely the bifurcation of a fixed point into a closed orbit in a neighborhoodof the fixed point. In contrast to the aforementioned bifurcations which can alreadyoccur in one-dimensional dynamical systems, the Hopf bifurcation in continuoustime, named after E. Hopf (1942), requires an at least two-dimensional system.14

Consider the continuous-time system

x = f(x, µ), x ∈ Rn, µ ∈ R. (3.2.6)

13 Cf. Sotomayor (1973) for further details.14 It should be remembered that linear one-dimensional systems can generate only mono-

tonic motion (cf. Appendix A.1.1). The same is true for the nonlinear analogs.

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96 Chapter 3

Assume that (3.2.6) possesses a unique fixed point x∗0 at the value µ0 of the param-eter, i.e.,

x = 0 = f(x∗0 , µ0). (3.2.7)

Furthermore, assume that the determinant of the Jacobian matrix J of (3.2.6), i.e.,

det J =

∣∣∣∣∣∣∣∣∣∣∣

∂f1

∂x1

∂f1

∂x2. . .

∂f1

∂xn...

.... . .

...∂fn∂x1

∂fn∂x2

. . .∂fn∂xn

∣∣∣∣∣∣∣∣∣∣∣, (3.2.8)

differs from zero for all possible fixed points (x, µ). Consider a neighborhoodBr(µ0) ∈ R of the parameter value µ0. Then the implicit function theorem ensuresthe existence of a smooth function x∗ = x∗(µ) for µ ∈ Br(µ0); i.e., for every µ inthe neighborhood there exists a unique fixed point x∗.

Assume that this fixed point is stable for small values of the parameter µ. (Itis also possible to consider a scenario with an unstable fixed point for µ < µ0; inthat case all of the following statements on µ ≷ µ0 must be reversed). The Hopfbifurcation theorem establishes the existence of closed orbits in a neighborhood ofa fixed point for appropriate values of the parameter µ.15

Theorem 3.4 (Hopf bifurcation – Existence Part): Suppose that thesystem (3.2.6) has a fixed point (x∗0 , µ0) at which the following propertiesare satisfied:

i) The Jacobian of (3.2.6), evaluated at (x∗0 , µ0), has a pair of pureimaginary eigenvalues and no other eigenvalues with zero real parts.

This implies that there is a smooth curve of fixed points (x∗(µ), µ) withx∗(µ0) = x∗0 . The complex conjugate eigenvalues λ(µ), λ(µ) of theJacobian which are purely imaginary at µ = µ0 vary smoothly with µ. Ifmoreover

ii)d(Reλ(µ)

)dµ |µ=µ0

> 0,

then there exist some periodic solutions bifurcating from x∗(µ0) at µ =µ0 and the period of the solutions is close to 2π/β0 (β0 = λ(µ0)/i).

When µ is increased from µ < µ0 to µ > µ0, the single fixed point changes itsstability because the real parts Re λ become positive. Figure 3.13 shows the Gaussian

15 There exist several versions of the Hopf bifurcation theorem. The following is a trun-cated version of Guckenheimer/Holmes (1983), pp. 151ff. For other versions see,e.g., Alexander/Yorke (1978) and Marsden/McCracken (1976). Compare also Has-sard/Kazarinoff/Wan (1981).

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3.2. Bifurcations in Continuous-Time Dynamical Systems 97

The Eigenvalues in the Hopf BifurcationFigure 3.13

3.14.a: µ < µ0 3.14.b: µ = µ0 3.14.c: µ > µ0

The Emergence of a Closed Orbit in the Hopf BifurcationFigure 3.14

plane with complex conjugate eigenvalues before and after a Hopf bifurcation. Thephase portraits for different parameter values are shown in Figure 3.14.

Theorem 3.4 establishes only the existence of closed orbits in a neighborhood ofx∗ at µ = µ0, and it does not say anything about the stability of the orbits. Indeed,the closed orbits may arise on either side of µ0. Consider first the so-called subcriticalcase in which closed orbits arise at µ < µ0. Closed orbits encircle stable fixed pointsx∗(µ). For µ > µ0, the fixed points are unstable and no orbits exist. Figure 3.15illustrates this subcritical Hopf bifurcation in the two-dimensional case. All pointson the µ axis represent fixed points (x∗1 , x∗2 ) = (0, 0) of the system. For each µ < µ0in a neighborhood of µ0, a closed orbit exists. The union of these orbits forms theparaboloid which is tangential to the planar cross section at µ0.

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98 Chapter 3

Trajectories starting at initial values in a neighborhood of the orbits are repelledfrom these orbits: initial points outside the orbits spiral away from the orbits, initialpoints inside the orbits are attracted by the appropriate fixed points.16

The Subcritical CaseFigure 3.15

In the second case of the so-called supercritical Hopf bifurcation the orbits arisefor µ > µ0. The fixed points x∗(µ) are unstable, and the orbits are attracting. Forµ ≤ µ0 the fixed points are stable and no orbits exist (cf. Figure 3.14).

As a formal example of the occurrence of the Hopf bifurcation, consider theprototype differential equation system

x = −y + x(µ− (x2 + y2)

),

y = x+ y(µ− (x2 + y2)

).

(3.2.9)

System (3.2.9) possesses a fixed point at x∗ = y∗ = 0. The Jacobian matrix of(3.2.9) is

J =

(µ− 3x2 − y2 −1 − 2xy

1 − 2xy µ− 3y2 − x2

), (3.2.10)

16 Cf. Benhabib/Miyao (1981) for economic interpretations of subcritical bifurcations.The fact that the closed orbits define basins of attraction can be used to relate thesub-critical Hopf bifurcation to the notion of corridor stability: as long as an initial pointx(0) is located inside a region bounded by the closed orbit, x(t) will stay in the corridordefined by the orbit and will eventually converge toward x∗.

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3.2. Bifurcations in Continuous-Time Dynamical Systems 99

The Supercritical CaseFigure 3.16

which, evaluated at the fixed point, is

J =

(µ −1

1 µ

). (3.2.11)

The determinant of (3.2.11) is det J = µ2 + 1, and the trace is tr J = 2µ. It followsthat the eigenvalues are λ1,2 = µ ±

√µ2 − µ2 − 1 = µ ± √−1. For µ = 0, the

eigenvalues are purely imaginary, and ∂ (Reλi)/∂µ = 1 > 0. The requirements i)and ii) of Theorem 3.4 are therefore fulfilled and system (3.2.9) undergoes a Hopfbifurcation at (0, 0) if µ = µ0 = 0.

While the existence of closed orbits via the Hopf bifurcation theorem can rel-atively easily be established in most cases, the distinction between the sub- andsupercritical Hopf bifurcation is much more difficult. The usual procedure in de-termining which case prevails will be demonstrated with the prototype equations(3.2.9).17

When the bifurcation value µ0 = 0 is taken into account, the dynamical system(3.2.9) can be written as

(xy

)=

(0 −11 0

)(xy

)+

(−x3 − xy2

−x2y − y3

)

= L ·(xy

)+ g(x, y).

(3.2.12)

17 Cf. Guckenheimer/Holmes (1983), pp. 150-156, or Marsden/McCracken (1976), pp.63ff. and pp. 137ff. for discussions of this procedure.

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100 Chapter 3

If the matrix L possesses a form as in (3.2.12), the dynamical system is said tobe written in normal form. As most generic dynamical systems do not appear in thisform, a transformation into normal form is necessary for the stability analysis below.An example will be provided in the next subsection.

The stability properties of the closed cycles depend on the nonlinear termsg(x, y) because in the Hopf bifurcation the real parts of the eigenvalues of J, i.e., ofthe linear approximation, vanish. It can be shown that the stability of the emerg-ing cycle depends on up to third-order derivatives of the nonlinear functions g in(3.2.12). Consider the expression18

b =116

(g1xxx + g1

xyy + g2xxy + g2

yyy

)+

116 β

(g1xy(g1

xx + g1yy) − g2

xy(g2xx + g2

yy) − g1xxg

2xx + g1

yyg2yy

),

(3.2.13)

with g(x, y) =(g1(x, y), g2(x, y)

)Tand the subscripts denoting the partial deriva-

tives with respect to the arguments x and y, respectively. The emerging cycle isattracting if b < 0; it is repelling if b > 0.

With g1(x, y) = −x3 − xy2 and g2(x, y) = −x2y − y3, the partial derivatives are

g1xx = −6x, g1

yy = −2x, g1xy = −2y,

g2yy = −6y, g2

xx = −2y, g2xy = −2x,

g1xxx = −6, g1

xyy = −2, g2xxy = −2, g2

yyy = −6. (3.2.14)

It follows that b = −16/16 = −1 < 0. The emerging cycle of system (3.2.9) istherefore attracting, i.e., a supercritical Hopf bifurcation occurs in this example.

This procedure can imply technical difficulties during the necessary transfor-mation of the generic system to the normal form (3.2.12). Furthermore, in then-dimensional case (n ≥ 3) a reduction of the dynamical system to its center mani-fold19 must be performed, which in most cases is impossible in face of the typicallynumerically unspecified economic models.

Summary

In order to demonstrate the existence of a Hopf bifurcation in a concrete system itis thus sufficient to show that by increasing the parameter µ:

• complex eigenvalues exist or emerge,• the real parts of the pairs of complex conjugate eigenvalues are zero at the

bifurcation value µ = µ0,• all other real eigenvalues differ from zero at µ = µ0,

18 Cf. Guckenheimer/Holmes (1983), p. 152. The expression β is the square root inλ = α + βi. In the Jacobian matrix (3.2.11), β is equal to 1.

19 Cf. Guckenheimer/Holmes (1983), pp. 123 ff., and the Appendix A.2.

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3.2. Bifurcations in Continuous-Time Dynamical Systems 101

• the real parts of the complex conjugate eigenvalues differ from zero for µ > µ0.

In addition, the stability of the emerging cycles should be studied with the help ofthe method mentioned above.

Though applications of the Hopf bifurcation theorem (and especially its ex-istence part) are generally not restricted to low-dimensional dynamical systems,conditions i) and ii) in Theorem 3.4 can be shown to be fulfilled without diffi-culty only in two- and three-dimensional cases. In higher-dimensional systems withn ≥ 4 the bifurcation values µ0 can often be calculated only by means of numericalalgorithms.

The following two sub-sections contain two economic examples of the emer-gence of a Hopf bifurcation. Other applications can be found in, e.g., Benhabib/Miyao (1981) and Zhang (1990), Chapter 3, who re-considered a monetary growthmodel, Dockner (1985), Dockner/Feichtinger (1989, 1991), Feichtinger/No-vak/Wirl (1991), Feichtinger/Sorger (1986) containing optimal control prob-lems from various economic fields, Feichtinger (1988) who studied an advertise-ment model, Semmler (1986) who investigated a macroeconomic model with finan-cial crises in the Minsky tradition, Diamond/Fudenberg (1989) and Lux (1992)who established cycles in a search-and-barter model with rational expectations, andZhang (1988) who elaborated upon multisector optimal growth. The cyclical be-havior in a Keynes-Wicksell monetary growth model was studied by Franke (1992).

3.2.2.1. The Hopf Bifurcation in Business-Cycle Theory

This section describes the application of the Hopf bifurcation theorem to twoKaldorian-type, descriptive business cycle model. The two-dimensional model dis-cussed in the first part of this section represents the standard Kaldor model alreadyknown from Section 2.2.2. The three-dimensional model in the second part in-cludes an interest-rate dynamics and can be considered an IS – LM growth-cyclemodel.

The case n = 2

Recall the familiar Kaldor model outlined in Section 2.2.2, which serves as a proto-type model in nonlinear economic dynamics:

Y = α(I(Y,K) − S(Y )

),

K = I(Y,K) − δK.(3.2.15)

In order to avoid possible conflicts with the assumptions of some theorems, performa coordinate transformation such that the system is centered at the fixed point(Y ∗,K∗). Let y = Y ∗ − Y , k = K∗ −K, i = I∗ − I, and s = S∗ − S. The system(3.2.15) then turns into

y = α(i(y, k) − s(y)

),

k = i(y, k) − δk.(3.2.16)

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102 Chapter 3

Assume that s(y) is linear, and that i(y, k) can be separated such that i(y, k) =i1(y) + i2(k). The part i2(k) is assumed to be linear. For the derivatives of i1(y)assume that i1y(0) > 0, i1yy(0) = 0, and i1yyy(0) < 0.

The Jacobian of (3.2.16) is

J =

(α(iy − sy) αik

iy ik − δ

), (3.2.17)

with the determinant

det J = α(iy − sy)(ik − δ) − αiyik, (3.2.18)

and the trace

tr J = α(iy − sy) + (ik − δ). (3.2.19)

The characteristic equation is

λ2 + aλ+ b = 0, (3.2.20)

with a = − tr J and b = det J. The eigenvalues are

λ1,2 = −a/2 ±√a2/4 − b, (3.2.21)

and it follows that the fixed point is locally stable if and only if the real parts arenegative. In order to exclude a saddle point, the determinant (3.2.18) is assumedto be positive (i.e., b > 0 ). The fixed point is then asymptotically stable if a =− tr J > 0 =⇒ tr J < 0:

α(iy − sy) + (ik − δ) < 0. (3.2.22)

According to Theorem 3.4, a Hopf bifurcation occurs if the complex conjugateroots cross the imaginary axis. Apparently, the roots are complex conjugate withzero real part if a = 0. As there are no other real roots in this two-dimensional exam-ple, the consideration of the existence of closed orbits is complete if the eigenvaluescross the imaginary axis with nonzero speed at the bifurcation point.

Though there may exist several possibilities to parameterize the Kaldor model,the choice of the adjustment coefficient α on the goods market as the bifurcationparameter seems to be obvious.20 With (iy − sy) > 0 at the fixed point y = 0 andik = constant, it can directly be seen that there exists a value α = α0 for which

α0(iy − sy) + (ik − δ) = 0, (3.2.23)

20 Cf. Dana/Malgrange (1984) for an investigation of the effects of different values of αin a discrete-time version of the Kaldor model. Compare also Section 4.2.2.

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3.2. Bifurcations in Continuous-Time Dynamical Systems 103

implying that the complex conjugate roots cross the imaginary axis. As, for α > α0,the real parts are becoming positive, α0 is indeed a bifurcation value of the Kaldormodel.

Inspection of (3.2.17) shows that the model is not expressed in its normal form.Evaluated at the bifurcation point, i.e., taking (3.2.23) into account, the centeredKaldor model can be written as

(y

k

)=

(−(ik − δ) −ik(ik − δ)/(iy − sy)iy ik − δ

)(y

k

)+ g(y, k), (3.2.24)

with g(y, k) as nonlinear terms which can be derived from a Taylor expansion of(3.2.16). As the expression (3.2.13) contains up to third-order derivatives, thefunction g(y, k) must be at leastC3. As it was assumed that i(y, ·) is the only involvednonlinearity, the nonlinear part g(y, k) reduces to

g1(y) = α0(i(y) − s(y)

)− L1(i(y)

)= α0

(i(y) − s(y)

)+ (ik − δ)y,

g2(y) = i(y) − L2(i(y)

)= i(y) − iyy,

(3.2.25)

with Li

(i(y)

)as the linear parts expressed by the matrix terms in (3.2.16).

In order to transform (3.2.24) into the desired normal form, consider the coor-dinate transformation21

(yk

)= D

(uv

)with D =

(d11 d12

d21 d22

). (3.2.26)

The entries of D are

d11 = 0,d12 = 1,

d21 =

√− 1

4 (f11 − f22)2 − f12f21

f12,

d22 = −f11 − f22

2f12,

with fij as the entries in the Jacobian (3.2.17), evaluated at the bifurcation point.The inverse of D is

D−1 = − 1d21

(d22 −1

−d21 0

). (3.2.27)

21 The following transformation is adopted from Herrmann (1986), pp. 89ff.

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104 Chapter 3

The matrix D transforms the coordinate system (y, k) into a new coordinate system(u, v). The linear part of (3.2.24) is transformed into(

uv

)= D−1J D

(u

v

),

=

(0 −f12d21

f12d21 0

)(u

v

),

(3.2.28)

i.e., into the normal form with f12d21 =√

(ik − δ)2 + ikiy(ik − δ)/(iy − sy). Thenonlinear terms gi(y) are transformed in the following way. The variables (y, k) areexpressed in the (u, v) system according to (3.2.26). Multiplication yields y = v;the expression for k is not needed in this example because the nonlinear functionsgi do not involve terms in k. Finally, the vector D−1g has to be calculated:22

(g1(v)g2(v)

)= D−1

(α0(i(v) − s(v)

)− L1(i(v)

)i(v) − L2

(i(v)

) ), (3.2.29)

= − 1d21

d22

(α0(i(v) − s(v)

)− L1(i(v)

))− i(v) + L2(i(v)

)−d21

(α0(i(v) − s(v)

)− L1(i(v)

)) .

The functions

g1(v) = −d22

d21

(α0(i(v) − s(v)

)+ (ik − δ)v

)+

1d21

(i(v) − iyv

),

=1 − α0d22

d21i(v) +

1d21

(−iy − d22(ik − δ))v +

d22

d21α0s(v),

g2(v) = α0(i(v) − s(v)

)+ (ik − δ)v,

(3.2.30)

depend only on the new variable v. Expression (3.2.13) therefore reduces to

b =116g1vvv +

116ω

g1vvg

2vv. (3.2.31)

The partial derivatives of g1 in (3.2.31) are

g1vv =

1 − d22α0

d21ivv = 0,

g2vv = α0ivv = 0,

g1vvv =

1 − d22α0

d21ivvv .

22 The inverse matrix has to be multiplied with the vector g because D originally appearson the left-hand side of (3.2.24) when the original transformation is applied.

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3.2. Bifurcations in Continuous-Time Dynamical Systems 105

As ivvv is negative by assumption, the expression b is negative if (1 − d22α0)/d21 ispositive. The reader may verify that the assumptions made thus far are not sufficientto establish a positive sign of the coefficient. Whether or not the cycle is indeedattracting depends on the particular numerical specification of the model.

In the two-dimensional case the use of bifurcation theory actually provides nonew insights into known models. The existence of closed orbits in the Kaldor modelcan also be established via the Poincare-Bendixson theorem. In many applications,however, it may be easier to use bifurcation theory rather than, e.g., the Poincare-Bendixson theorem, because it may be more difficult to find the necessary invariantset on whose boundary the vector field points toward the interior of the set than tocalculate the bifurcation values.

The case n ≥ 3

In the three- and higher-dimensional case the Poincare-Bendixson theorem cannotbe applied anymore. The Hopf bifurcation theorem may constitute the only toolto establish the existence of closed orbit.

As an example consider an augmented IS-LM business-cycle model:23

Y = α(I(Y,K, r) − S(Y, r)

),

r = β(L(r, Y ) −M

),

K = I(Y,K, r) − δK,

(3.2.32)

with r as the interest rate, L(r, Y ) as the money demand, and M as the constantmoney supply. The model can also be considered a Kaldor model augmented byan interest-rate dynamics. In particular, it will be assumed that investment dependson income in the typical Kaldorian, sigmoid form.

The Jacobian matrix of (3.2.32) is

J =

α(IY − SY ) α(Ir − Sr) αIK

βLY βLr 0

IY Ir IK − δ

, (3.2.33)

with the characteristic equation

λ3 + aλ2 + bλ+ c = 0, (3.2.34)

23 Compare Boldrin (1984, 1988) for a similar model. Compare also Section 5.2.2 con-taining a few remarks on the possibly inappropriate specification of the interest-rateadjustment equation.

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106 Chapter 3

and

a = − tr J = −(α(IY − SY ) + βLr + (IK − δ)

),

b = βLr(IK − δ) + α(IY − SY )(IK − δ) − αIY IK

+ αβ(IY − SY )Lr − αβLY (Ir − Sr),

c = −det J = −(αβLr(IY − SY )(IK − δ)

− αβLY (Ir − Sr)(IK − δ) + αβIK(LY Ir − IY Lr

).

(3.2.35)

The coefficient b represents the sum of the principal minors of the Jacobian J.In case of a third-order polynomial like (3.2.34) it is a little bit more difficult to

examine the qualitative properties of the eigenvalues. Equation (3.2.34) has onereal and two complex conjugate eigenvalues if the discriminant,

∆ = A2 +B3, (3.2.36)

is positive with

A =a3

27− ab

6+c

2and B =

b

3− a2

9. (3.2.37)

While in the two-dimensional case the stability of the fixed point is determinedby the sign of the trace of J, the three-dimensional case is slightly more difficult toanalyze. A very helpful criterion in proving the local stability of a dynamical systemis the Routh-Hurwitz criterion.24 In the three-dimensional case the real parts of theeigenvalues are negative if

a, b, c > 0 and ab− c > 0. (3.2.38)

Making use of the root theorem of Vieta, i.e.,

3∑i=1

λi = −a and3∏

i=1

λi = −c,

it can be shown that the real parts of the complex conjugate eigenvalues are zeroand that there is no other real eigenvalue which equals zero if

a, b, c �= 0 and ab− c = 0. (3.2.39)

Assume that the discriminant ∆ in (3.2.36) is always positive in order to assurethat the three eigenvalues consist of one real and two complex conjugate roots.

24 See, e.g., Dernburg/Dernburg (1969), pp. 214ff., Gandolfo (1983), p. 248ff., or theremarks in Appendix A.1.1.

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3.2. Bifurcations in Continuous-Time Dynamical Systems 107

Let α be the bifurcation parameter and assume an initial value of α such that theRouth-Hurwitz conditions are fulfilled. An increase in α implies ∂a/∂α < 0, and,eventually, a will be equal to zero. An increase in α implies an increase in c because∂c/∂α = c/α; when c is positive by assumption (in order to exclude a saddle point)for low values of α it will stay positive for high values of α. The sign of ∂b/∂α isambiguous but the existence of a value α0 with the consequence ab − c = 0 cannevertheless be demonstrated. With ∂a/∂α < 0, the product ab will eventually beequal to zero at a value α, implying that ab− c = −c < 0. It follows that there mustbe a value α0 < α at which ab− c = 0 and a > 0, i.e., α0 is a bifurcation value. Thesum ab − c is a quadratic expression in α implying that two bifurcation values α1

0and α2

0 exist. The sign of ∂b/∂α is responsible for the number of positive bifurcationvalues.

The fact that a pair of purely imaginary eigenvalues and a non-zero real eigen-value exists at the bifurcation value(s) α0 can be seen from Orlando’s formula:25 Theexpression ab−c (which is actually the determinant of one of the Hurwitz matrices)equals

ab− c = −(λ1 + λ2)(λ1 + λ3)(λ2 + λ3).

As the product of all three eigenvalues equals −c < 0 according to Vieta’s formula,it is impossible to encounter a real zero eigenvalue. When the case of a saddle pointis explicitly excluded,26 a pair of real eigenvalues cannot come with opposite signs.It follows that ab − c = 0 can only be fulfilled when a pair of eigenvalues is purelyimaginary. For values of α > α0, the expression ab− c becomes negative accordingto the above consideration. Thus, the conjugated pair of complex eigenvalues λi, λjwhich assures ab− c = 0 cannot still imply λi + λj = 0. It follows that the real partsof the complex conjugate eigenvalues differ from zero for α > α0. This completesthe demonstration of the emergence of a Hopf bifurcation in system (3.2.32). Thesystem possesses closed orbits in a neighborhood of the bifurcation point.

In order to perform a stability analysis of the emerging cycles by means ofthe same method as in the last subsection, it is necessary to reduce the three-dimensional system (3.2.35) to its center manifold (cf. Appendix A.2).27 The calcula-tions are tedious, and there is not much hope to derive simple stability conditions.

3.2.2.2. Closed Orbits in Optimal Economic Growth

The Hopf bifurcation theorem can be applied to economic models in other fieldsthan business cycle theory which is explicitly attempting to model oscillatory mo-

25 Cf. Gantmacher (1954), Chapter 16.7.26 Unfortunately, a positive c is necessary but not sufficient for excluding a saddle point.27 Cf. Guckenheimer/Holmes (1983), pp. 123ff. Economic examples of the use of center

manifolds in the investigation of higher-dimensional systems can be found in Chiarella(1990) and in Reichlin (1987).

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108 Chapter 3

tions. It can be shown that closed orbits exist in several models which have tradi-tionally been characterized by more or less monotonic time paths of its variables.

The literature on optimal control in the 1960s and 1970s was dominated by thesearch for the assumptions necessary and/or sufficient for the saddle-point stabilityof an optimal control trajectory: one and only one trajectory exists such that all ini-tial points located on this trajectory eventually converge to a stationary equilibriumpoint. If an initial point is not precisely located on this saddle, it will never reachthe equilibrium. Actually, the saddle-point characteristic of most optimal controlmodels constitutes a negation of the practical controllability of an economy becauseit shows that the probable inaccurateness of the involved information will preventa political institution from hitting the exact saddle-trajectory. Nevertheless, thesaddle-point property of optimal control trajectories has found attention especiallyamong Rational Expectations theorists because the existence of a single optimal tra-jectory which converges to an equilibrium is compatible with the concept of perfectforesight.

However, the saddle-point stability/instability property of a fixed point does notrepresent the only possible dynamic phenomenon in optimal control models. Ben-habib/Nishimura (1979) and Medio (1987) have demonstrated that it is possibleto establish (at least locally) the existence of closed orbits in models of optimaleconomic growth. Consider the general, multi-sector optimal growth problem for-mulated by Benhabib/Nishimura (1979):28

maxy

∫ ∞

0e−(δ−n)U

((T (y, k)

))dt

s.t. ki = yi − nki, i = 1, . . . , n,(3.2.40)

with y = (y1, . . . , yn) as the vector of per-capita outputs yi in sector i, k = (k1,. . . , kn) as the vector of per-capita stocks of capital, T (y, k) = c as the macroeco-nomic consumption frontier, U(·) as the utility derived from consumption, δ as thediscount rate, and n as the population growth rate.

The Hamiltonian function of problem (3.2.40) is

H(y, k,λ) = e−(δ−n){U(T (y, k))

+ λ(y − nk)}. (3.2.41)

By the maximum principle and the assumption of perfect competition, i.e.,

∂c

∂yj=∂T

∂yj= pj ,

∂c

∂kj=∂T

∂kj= wj ,

(3.2.42)

28 Compare also Zhang (1988).

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3.2. Bifurcations in Continuous-Time Dynamical Systems 109

with yj ∈ y, kj ∈ k and pj ∈ p and wj ∈ w as the product prices and rental prices ofgood j, respectively, it follows that

kj = yj − nkj ,

λj = −Uk′wj + δλj ,

λj = Uk′pj ,

(3.2.43)

or

kj = yj(k,p) − nkj ,

pj = −wj(k,p) + δpj ,(3.2.44)

if U ′ = 1.29 The Jacobian matrix of (3.2.44) is

J =

( (∂y/∂k) − nI (∂y/∂p)

−(∂w/∂k) −(∂w/∂p) + δI

), (3.2.45)

which, under some additional assumptions on the technology set and competition,can be written as

J =

(Bk − nI (∂y/∂p)

0 −Bk′ + δI

), (3.2.46)

with B and I as × – matrices. As the Jacobian (3.2.46) is quasi-triangular, the char-acteristic roots are given by the roots of the matrices B−nI and B′+δI, respectively.Assume that the determinants of both matrices are positive, and that the eigenval-ues are complex conjugate. If there is a value δ = δ0 such that the roots are purelyimaginary, and if the real parts of the eigenvalues are increasing for increasing δ, aHopf bifurcation occurs at δ0 implying that closed orbits arise in a neighborhoodof the fixed point with yj = pj = 0 ∀j.

Depending on the value of the bifurcation parameter δ, it is thus possible thatthe optimal control trajectory is oscillating. On a first glimpse, this appears to be atheoretical curiosity. For example, it may be argued that the discount rate is one ofthe parameters of the model which can be influenced relatively easily by politicalinstitutions. A political institution which is aware of the possible oscillating behav-ior of a control trajectory can circumvent this phenomenon by suitably choosingthe discount rate. However, in some practical cases it may not be possible to manip-ulate the discount rate. The rate can be predetermined by a social consensus andinstitutional arrangements.

The usual argument in justifying governmental interventions into the marketprocesses points out that in some cases

29 Cf. Benhabib/Nishimura (1979), p. 424, for details.

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110 Chapter 3

• the market is not able to realize the predetermined welfare criteria, and that• the economy, if left to itself, may be characterized by fluctuations which consti-

tute divergences from a monotonic time path.

If it is optimal for the instrumental tools of a political institution to behave in abasically oscillating manner according to the optimal program described above,this basic paradigm of economic policy interventions is challenged. Suppose thatan uncontrolled economy does not fluctuate. Then it may happen that an economystarts oscillating when the political institutions intervene in the economic process.In other words, the optimality criterion of the institution requires that an economycharacterized by monotone, but unoptimal time paths starts to oscillate after theinstallation of the policy. Fluctuations usually considered as non-optimal emerge asthe result of an optimization process.30

3.3. Local Bifurcations in Discrete-Time Dynamical Systems

This section deals with discrete-time dynamical system which either emerge gen-uinely in dynamic models with a discrete, finite time concept, or which can beinterpreted as Poincare maps31 of continuous-time dynamical systems. Consider aone-parameter, discrete-time, one-dimensional map f : R × R → R: 32

xt+1 = f(xt, µ), x ∈ R, µ ∈ R. (3.3.1)

Let x∗ be a fixed point of (3.3.1), i.e., x∗ = f(x∗, µ). The asymptotic stability ofthe fixed point x∗ depends on whether the slope of f , evaluated at the fixed point,lies within the unit circle, i.e., whether |df(x∗)/dx| = |λ| < 1. Bifurcations, i.e.,changes in the qualitative behavior of (3.3.1) can therefore occur only when theeigenvalue λ takes on the value +1 or −1.

3.3.1. Fold, Transcritical, Pitchfork, and Flip Bifurcations

The first three bifurcation types are essentially equivalent to their analogs in contin-uous-time dynamical systems. The appropriate theorems represent adaptions of thecontinuous-time versions to the discrete-time case and are therefore only brieflymentioned in the following.

30 Compare also Foley (1986).31 Cf. Section 5.1 for details.32 The following presentation is to a large degree stimulated by the survey in Whitley

(1983).

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3.3. Local Bifurcations in Discrete-Time Dynamical Systems 111

Fold, Transcritical, and Pitchfork Bifurcations

These bifurcations types can occur in dynamical systems having an eigenvalue λ =+1 at the bifurcation point. The possible bifurcation types are summarized in Table3.2. The transversality conditions for the different bifurcation types are the samefor continuous-time and discrete-time dynamical systems. In the graphical presen-tations of the different types of bifurcation, the phase portraits have to be replacedby the appropriate graphs of the mappings. The intersections of the graphs f(x, µ)with the 450 line represent the fixed points of the mappings. Figure 3.17 shows thegraph of the prototype mapping xt+1 = µ−x2

t for the fold bifurcation and differentvalues of µ. Figure 3.18 contains a description of the transcritical bifurcation in theprototype equation xt+1 = µxt − x2

t . The pitchfork bifurcation in the prototypeequation xt+1 = µxt − x3

t is shown in Figure 3.19.It is easily possible to modify the economic examples provided in Section 3.2.1

such that they fulfill the requirements of the appropriately modified theorems fordiscrete-time systems.

Flip Bifurcation

A bifurcation type which is unique to discrete-time dynamical systems is the flipbifurcation. Assume that a fixed point x∗ exists, i.e., f(x∗, µ0) = x∗, and that itseigenvalue is equal to −1.

Theorem 3.5 (Flip Bifurcation)33 Let fµ: R → R be a one-parameterfamily of mappings such that fµ0 has a fixed point x∗ with eigenvalue−1. If, at (x∗, µ0),

(1)(∂f

∂µ

∂2f

∂x2 + 2∂2f

∂x∂µ

)�= 0

(2) −2(∂3f

∂x3

)− 3

(∂2f

∂x2

)2

= a �= 0,

then, depending on the signs of the expressions in (1) and (2),

i) the fixed point x∗ is stable (unstable) for µ < µ0 (µ > µ0), andii) the fixed point x∗ becomes unstable (stable) for µ > µ0 (µ < µ0),

and, additionally, a branch of stable (unstable) fixed points of order2 emerges which enclose x∗.

A fixed point of order 2 is a fixed point of the second iterate of (3.3.1), i.e., of themapping

xt+2 = f(xt+1) = f(f(xt)

). (3.3.2)

33 Cf. Whitley (1983)

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112 Chapter 3

The Fold Bifurcation in a One-Dimensional Map, xt+1 = µ− x2t

Figure 3.17

The Transcritical Bifurcation in a One-Dimensional Map, xt+1 = µxt − x2t

Figure 3.18

The Pitchfork Bifurcation in a One-Dimensional Map, xt+1 = µxt − x3t

Figure 3.19

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3.3. Local Bifurcations in Discrete-Time Dynamical Systems 113

Denote the second iterate of the mapping as f ◦ f = f (2). A fixed point of order 2is therefore a fixed point of the mapping f (2), i.e., x∗ = f (2)(x∗).

For obvious reasons the flip bifurcation is often also called a period-doubling bi-furcation. If the sign of the expression in (2) is negative, the emerging fixed pointsof order 2 are stable, i.e., xt permanently switches between two values x1

t and x2t .

In that case, the bifurcation is called a supercritical flip bifurcation. The prototypeequation of the flip bifurcation is xt+1 = µxt − µx2

t (cf. Figure 3.20 with a < 0).34

Note that though the bifurcation diagram looks similar to that of the pitchfork bi-furcation, both are essentially different. In the pitchfork bifurcation two separateadditional fixed points (of order 1) emerge, while in the case of the flip bifurcationtwo components of a fixed point of order 2 emerge.

The Supercritical Flip BifurcationFigure 3.20

If a > 0 in Theorem 3.5, the fixed point x∗ is stable and the emerging fixedpoint of order 2 is unstable. In that case, the bifurcation is said to be a subcriticalflip bifurcation.

Consider the following very simple economic example from population eco-nomics.35 In nearly all economic models in which the population size changes overtime it is assumed that the population growth rate is constant, i.e.,

Nt+1 −Nt

Nt= n, (3.3.3)

with Nt as the size of the population in period t. This assumption, which is usu-ally assigned to Malthus (1798), implies that a positive growth rate n leads to apermanent and unbounded increase in the population.

34 The sign of (2) in Theorem 3.5 can be related to the Schwarzian derivative which willbe introduced in Chapter 4: if a < 0, then the Schwarzian derivative is also negative.

35 Compare West (1985), pp. 150ff., for the following model.

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114 Chapter 3

The assumption of an unrestricted population growth was criticized relativelyearly.36 Empirical reasoning suggests that the population growth rate may insteaddepend on the population level such that the rate decreases when the popula-tion level increases. For simplicity, assume that a linear relation exists between thegrowth factor 1 + n and the population level, i.e.,

1 + n = µ(1 −Nt/M), µ > 0, M > 0, Nt ≤M ∀ t. (3.3.4)

The constant M serves as a saturation level of the population: if the populationincreases, the growth factor decreases and eventually approaches 1, i.e., the growthrate n is zero. If the population is equal to the level M , the growth factor n + 1 isequal to zero, i.e., the growth rate n reaches its lower bound of −100%.

3.21.a: The Growth Factor 3.21.b: The Mapping (3.3.6)Population Growth Depending on the Population Level

Figure 3.21

Substitution for n in (3.3.3) yields

Nt+1 −Nt

Nt= µ(1 −Nt/M) − 1, (3.3.5)

or

Nt+1 = µNt(1 −Nt/M). (3.3.6)

The growth factor (3.3.4) and the mapping (3.3.6) are illustrated in Figure 3.21.Obviously, the coefficient µ in (3.3.6) stretches the graph vertically. Denote theright-hand side of (3.3.6) as f(Nt, µ).37 Let µ0 be the value of µ such that there

36 Cf. Verhulst (1845, 1847) for an early critique of Malthus’ assumptions. Cf. also West(1985), p. 101, for a discussion.

37 The variable M is assumed to be constant.

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3.3. Local Bifurcations in Discrete-Time Dynamical Systems 115

is a fixed point of the mapping, i.e., N∗ = f(N∗, µ0), with an eigenvalue λ =∂f(N∗, µ0)/∂N = µ−2µN/M = −1. Simple calculation shows that the conditions(1) and (2) of Theorem 3.5 are fulfilled, namely

(1)

(∂f

∂µ

∂2f

∂x2 + 2∂2f

∂x∂µ

)=

(N − N2

M

)(−2µM

)+ 2

(1 − 2N

M

)

=−6 − 2µ2

4µ< 0 ∀µ > 0

(2) −2(∂3f

∂x3

)− 3

(∂2f

∂x2

)2

= −2(0) − 3(−2µM

)2

< 0

Thus, a flip bifurcation occurs at µ = µ0, . Instead of the monotone populationgrowth under the Malthusian assumption, the time path of xt is now characterizedby a permanent period-2 cycle in the population level. From an empirical pointof view, this may be considered as artificial as the former hypothesis. However, itwill be demonstrated in Chapter 4 that there may be a sequence of flip bifurca-tions such that the time path of xt can eventually be described as irregular.38 Theflip bifurcation can therefore be viewed as a transition to more complex dynamicphenomena.

Summary

The different bifurcation types with the appropriate transversality condition andthe prototype equations are summarized in Table 3.2 for the case n = 1.

3.3.2. The Hopf Bifurcation in Discrete-Time Dynamical Systems

Most mathematical statements on closed orbits in dynamical systems refer to con-tinuous-time systems. An exception to this rule is the Hopf bifurcation theorem formappings in R2. Unfortunately, a generalization of the theorem to n-dimensionalsystems does not exist. The following result is essentially due to Ruelle/Takens(1971):39

Theorem 3.6 (Hopf bifurcation – Existence Part): Let the mappingxt+1 = F(xt, µ), xt ∈ R2, µ ∈ R, have a smooth family of fixed points

38 Note that in the above model there may occur different types of bifurcation as well. Forexample, a transcritical bifurcation occurs at the origin for low values of µ.

39 The following theorem is a truncated version of Iooss (1979) and Guckenheimer/Holmes (1983).

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116 Chapter 3

λ Transversality PrototypeCondition Equation

Fold λ = 1 ∂f/∂µ �= 0; xt+1 = µ− x2t

Bifurcation ∂2f/∂x2 �= 0

Transcritical λ = 1 ∂2f/(∂µ∂x) �= 0; xt+1 = µxt − x2t

Bifurcation ∂2f/∂x2 �= 0

Pitchfork λ = 1 ∂2f/(∂µ∂x) �= 0 xt+1 = µxt − x3t

Bifurcation ∂3f/∂x3 �= 0

Flip λ = −1 (∂f/∂µ)(∂2f/∂x2)+ xt+1 = µxt − µx2t

Bifurcation +2(∂2f/(∂x∂µ)

) �= 0−2

(∂3f/∂x3

)−−3

(∂2/∂x2

)2 �= 0

Bifurcation Types in Discrete-Time Dynamical SystemsTable 3.2

x∗(µ) at which the eigenvalues are complex conjugate. If there is a µ0such that

mod λ(µ0) = 1 but λn(µ0) �= ±1, n = 1, 2, 3, 4

and

d(mod λ(µ0)

)d µ

> 0,

then there is an invariant closed curve bifurcating from µ = µ0.

A comparison of Theorem 3.6 with Theorem 3.4 uncovers the analogy of this theo-rem with the Hopf bifurcation theorem for the continuous-time case. The require-ment that the eigenvalues cross the imaginary axis is replaced by the condition thatthe complex conjugate eigenvalues cross the unit cycle, i.e., that mod λ = 1 at thebifurcation point µ = µ0. Furthermore, it is required that the roots do not becomereal when they are iterated on the unit circle: the first four iterations λn must also becomplex conjugate. Finally, the eigenvalues must cross the unit cycle with nonzerospeed for varying µ at µ0.

Theorem 3.6 establishes only the existence of closed orbits in systems that un-dergo a Hopf bifurcation. The stability of the orbits can be demonstrated in a waysimilar to the procedure described for continuous-time systems.40

40 See Guckenheimer/Holmes (1983), pp. 162-165 for details on stability proofs.

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3.3. Local Bifurcations in Discrete-Time Dynamical Systems 117

The value of the modulus can be determined by the following simple consider-ation. The characteristic equation is:

λ2 + aλ+ b = 0 (3.3.7)

with the solution

λ1,2 = −a/2 ±√a2/4 − b. (3.3.8)

In the case of complex eigenvalues, (3.3.8) can be written as λ1,2 = β1 ± β2i withβ1 = −a/2 and β2 =

√b− a2/4. The modulus is defined as

mod(λ) =√β2

1 + β22 .

It follows that the modulus equals the square root of the determinant b:

mod(λ) =√a2/4 + b− a2/4 =

√b. (3.3.9)

As a pedagogical example, consider once again the Kaldor model. Replacing thedifferential operator d/dt in (2.2.2) by finite differences yields

∆Yt+1 = Yt+1 − Yt = α(I(Yt,Kt) − S(Yt,Kt)

),

∆Kt+1 = Kt+1 −Kt = I(Yt,Kt) − δKt,(3.3.10)

or

Yt+1 = α(I(Yt,Kt) − S(Yt,Kt)

)+ Yt,

Kt+1 = I(Yt,Kt) + (1 − δ)Kt.(3.3.11)

The Jacobian matrix of (3.3.11) is

J =

(α(IY − SY ) + 1 α(IK − SK)

IY IK + (1 − δ)

), (3.3.12)

with

det J =(α(IY − SY ) + 1

)(IK + 1 − δ

)− αIY (IK − SK). (3.3.13)

The eigenvalues are complex conjugate if

det J >(tr J)2

4. (3.3.14)

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118 Chapter 3

Assume that the inequality holds. A Hopf bifurcation occurs at a value α = α0 ifdet J|α=α0 = 1:

(α(IY − SY ) + 1

)(IK + 1 − δ

)− αIY (IK − SK) = 1

=⇒ α0 =δ − IK

(IY − SY )(IK + 1 − δ) − IY (IK − SK). (3.3.15)

Note that it is not assured that the bifurcation value α0 is economically reasonablebecause the denominator can be positive. In that case the calculated bifurcationvalue would be negative.

The modulus crosses the unit circle with nonzero speed when the parameter αis changed:

d |λ(α)|dα |α=α0

=d(√

det J)

= 1/2((α(IY − SY ) + 1

)(IK + 1 − δ) − αIY (IK − SK)

)−1/2

((IY − SY )(IK + 1 − δ) − IY (IK − SK)

)=δ − IK

2α0> 0. (3.3.16)

Provided that the iterates λn, n = 1, . . . , 4, on the unit circle remain complex con-jugate roots, the requirements of Theorem 3.6 are fulfilled, and a Hopf bifurcationoccurs when α = α0. Without inspecting the sign of a specific expression contain-ing third-order derivatives of the nonlinear parts in (3.3.11), nothing can be saidabout the stability of the closed orbit.

Recently, the Hopf bifurcation theorem for discrete-time systems has been ap-plied to several economic models. For example, Cugno/Montrucchio (1984)studied a discrete version of Goodwin’s predator-prey model, augmented by a mark-up pricing relation. An overlapping-generations model with production can befound in Reichlin (1986), who also provides stability conditions. Governmentalpolicy in an overlapping-generations model is studied by Farmer (1986).

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Chapter 4

Chaotic Dynamics in Discrete-Time

Economic Models

The presentation of nonlinear dynamical systems in the preceding two chaptersuncovered a variety of mathematical concepts which allow one to establish en-

dogenous oscillations in economic applications. In these models, cyclical behaviorcan prevail for large ranges of the parameters while persistent oscillations in lineardynamical systems usually occur only for a particular parameter constellation. Itseems natural, therefore, to refer to nonlinear approaches when cyclical motion isto be modeled in economics. In other words, cyclical behavior is synonymous withthe presence of nonlinearities in most cases.

Even so, the recent interest in nonlinear dynamical systems cannot be attributedsimply to the possibility for easily generating cyclical patterns like limit cycles. Non-linear dynamical systems can exhibit a behavior of the variables that strongly resem-bles a random process. This means that the generated time series look erratic andthat it is not possible to predict the future development of the variables with preci-sion. Even if a model is completely deterministic with respect to the specificationof the structure and initial values, a pair of initial values located arbitrarily closetogether may lead to completely different time series though they are generated bythe same dynamical system. Figure 4.1 illustrates this kind of dynamic behavior fora one-dimensional difference equation. This unexpected property of some non-linear deterministic dynamical systems is responsible for the label chaotic behavior ,deterministic chaos, deterministic noise, or just chaos. However, it should be noted fromthe beginning that several definitions of “chaos” exist which emphasize differentaspects of the dynamic behavior in a given system.

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120 Chapter 4

Stylized Chaotic Time SeriesFigure 4.1

Chaotic motion can exist in very simple nonlinear dynamical systems. Thereforethe question arises of why this behavior has found attention in nearly all formallyoriented scientific disciplines only during the last two decades. In fact, many el-ements of the modern theory of chaotic dynamical systems were known to suchdistinguished mathematicians as H. Poincare at the turn of this century and P. Fa-tou and G. Julia in the 1920s. It was the increasing usage of modern computingdevices which enabled a fast numerical generation of time series and their graph-ical presentations in systems already known to possess unconventional behavioralpatterns. On the other hand, the numerical investigation of dynamical systems,which became popular in the late 1950s, uncovered unexpected behavior in sys-tems which had been studied for quite different reasons. When E.N. Lorenz, whonowadays is usually cited as the initiator of the current research in the field,1 be-gan to perform numerical experiments with a fluid convection model in the early1960s, the discovered presence of a so-called strange attractor in a three-dimensionalcontinuous-time system could not have been foreseen (and in fact was not honoreduntil the mid-1970s). In any case, the fascination that can arise in investigations ofchaotic dynamical systems can only be understood once the actual emergence of astrange attractor has been followed on a graphics terminal.2

1 Y. Ueda actually presented an earlier example of a strange attractor, cf. Ueda (1992).2 Any reader without programming experience but who has access to a microcomputer is

strongly advised to examine the Phaser program by Kocak (1986). The program allowsthe inspection of the behavior of all standard examples in dynamical systems theory ina simple and relatively fast way. A faster and more sophisticated program is Dynam-ics, written and circulated by J.A. Yorke. The program also includes algorithms for thecalculation of (still) more esoteric things like basin boundaries, saddle-straddle trajec-tories, etc. Readers with some experience in Fortran programming should inspect theDynamical Systems Software package which represents the state-of-the-art in non-linear systems software. The most important numerical calculations can be performedwith the Insite program (cf. Parker/Chua (1989)). A speedy, highly integrated, and

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4.1. Chaos in One-Dimensional, Discrete-Time Models 121

It is obvious that random-like behavior in deterministic nonlinear systems is atleast theoretically relevant to economics. Actual economic time series do not showthe regular and harmonic motion known in linear dynamical systems. Exogenouslygiven random influences are therefore assumed to being superimposed on regularmotion in linear systems for reasons of a realistic description of actual series and/orthe persistence of cycles in otherwise dampened oscillations. The chaos propertyof some nonlinear dynamical systems can provide an alternative to this resort tonon-economic forces in descriptive explanations of actual time series. In addition,the presence of chaotic motion can contribute to an explanation why economicprognoses have been notoriously bad.

The aim of this chapter is to provide an overview of the emergence of chaoticmotion in dynamic economic models. The presentation of the mathematical con-cepts necessary for understanding these economic applications is thus limited, yethopefully sufficient. The relevant literature will be given as often as possible forreaders interested in more in-depth mathematics.3 This chapter deals with chaoticdynamics in discrete-time dynamical systems. Chaotic motion in one-dimensional,discrete-time models in dynamic economic models is introduced in the first section.A short overview of chaotic properties of higher-dimensional dynamical systems anda discussion of economic applications are contained in Section 4.2. Discussionsof the properties of continuous-time dynamical systems and numerical techniquesused to describe chaotic motion in a more quantitative manner are postponed tothe following Chapters 5 and 6.

4.1. Chaos in One-Dimensional, Discrete-Time Dynamical Systems

Recent mathematical studies of one-dimensional, discrete-time, nonlinear systemsshow that even very simple systems can behave in a very complicated dynamicalmanner. Though this complicated behavior can also occur in higher-dimensional,discrete-time and continuous-time systems, there are three motivations for elabo-rating on one-dimensional systems at greater length:

easy-to-use program with high-quality graphics is the DMC program described in Medio(1993). A nice collection of graphical illustrations of the behavior of nonlinear dynam-ical systems can be found in the multivolume book by Abraham/Shaw (1983). Usersof the Mathematica program should consult Anderson (1993) and Eckalbar (1993).

3 In recent years a large number of introductory texts on chaotic dynamics has beenpublished. Introductions to chaotic nonlinear models can be found in Berge et al.(1986), where the mathematical concepts are illustrated with many applications fromthe natural sciences, in Devaney (1992) with a lot of geometrical illustrations, in Ru-elle (1991) who provides a non-technical survey and discusses essential implicationsof nonlinear dynamical systems, and in Schuster (1984). More advanced expositionscan be found in Collet/Eckmann (1981), Devaney (1986), Guckenheimer/Holmes(1983), Ruelle (1989, 1990), Wiggins (1988), and, with an emphasis on economics,Medio (1993). Good survey articles are, e.g., Eckmann (1981) or Ott (1981). Econom-ically motivated introductions to the definitional framework can be found in Baumol/Benhabib (1989), Brock (1986), Brock/Dechert (1991), Brock/Hsieh/LeBaron(1991), Chen (1988a), Kelsey (1988), and Samuelson (1990).

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122 Chapter 4

• The mathematical properties of one-dimensional dynamical systems are muchbetter understood than the properties of higher-dimensional systems.

• Many phenomena typical for higher-dimensional, discrete-time and continuous-time dynamical systems can be illustrated with one-dimensional maps.

• Most economic examples of complicated dynamical behavior are framed in one-dimensional difference equations.

This section therefore outlines the mathematical properties of one-dimensional,discrete-time maps and provides several economic examples from descriptive andoptimal economic growth theory. A short survey of other examples can be foundin Section 4.1.4.

4.1.1. Basic Concepts

This section is divided into two parts. The first part describes some simple phenom-ena observable in a family of one-dimensional maps and attempts to familiarize thereader with basic ideas of chaotic motion. This part concentrates on the geometri-cal aspects of successive bifurcations. In a second part, a more exact definition ofchaos and several theoretical results are presented.4

A Heuristic Introduction to One-Dimensional Chaos

Consider the one-dimensional, discrete-time system

xt+1 = f(xt, µ), xt ∈ R, µ ∈ R, (4.1.1)

with xt as the state variable and µ as a parameter. Assume that there are valuesa and b such that f(a, ·) = f(b, ·) = 0, i.e., the graph of f crosses the xt-axistwice. Furthermore, assume that there is a critical value xc for which f ′(xc) = 0 andf ′(xt) > (<) 0∀xt < (>)xc. A map with these properties is called a unimodal map.For example, let (4.1.1) be the concave quadratic function

xt+1 = f(xt, µ) ≡ µxt(1 − xt), xt ∈ [0, 1], µ ∈ [0, 4], (4.1.2)

which is the so-called logistic equation or Verhulst dynamics, already introduced inSection 3.3.1.5 This one-dimensional map is non-invertible, i.e., while xt+1 is un-ambiguously given for a certain xt, the inverse xt = f−1(xt+1) yields two values of

4 For detailed treatments of one-dimensional, discrete-time systems compare Collet/Eckmann (1980), Devaney (1986, 1992), Grandmont (1988), Lauwerier (1986), Pre-ston (1983), Singer (1978), and Whitley (1983).

5 Irregular and seemingly stochastic motion has been known to exist in this equation fora long time. As early as in 1947, Ulam/Neumann (1947) mention the possibility ofusing the logistic equation (4.1.2) with µ = 4 as a quasi-random-number generator oncomputers. The recent interest in the equation was stimulated by May’s (1976) famousNature article.

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4.1.1. Basic Concepts 123

xt for a single xt+1. The map is said to be an endomorphism. For µ ∈ [0, 4], theinterval [0, 1] of the state variable is mapped onto itself. The graph of the functionf(xt, µ) = µxt(1 − xt) is stretched upwards when µ is increased, while the pointsof intersection with the xt axis do not change (cf. Figure 4.2).

The Logistic Equation for Different Values of µFigure 4.2

The map (4.1.2) possesses two fixed points, namely the origin, x∗ = 0, and

x∗ = 1 − 1µ, µ > 0. (4.1.3)

When 0 < µ < 1, the second fixed point in addition to the origin is negative (andis thus located outside of the interval [0, 1]. For µ = 1, a transcritical bifurcationoccurs at the origin (cf. Section 3.2.1.): while the origin is stable for µ < 1, itbecomes unstable for µ > 1. The second fixed point turns from an unstable fixedpoint (x∗ < 0) into a stable fixed point (x∗ > 0). This stable fixed point increaseswith an increasing µ, i.e., x∗ = x∗(µ).

As was demonstrated in Section 3.3.1., a non-invertible map like (4.1.2) under-goes a flip bifurcation when µ is sufficiently large: the fixed point x∗(µ) > 0 isstable as long as the slope of f(xt, µ) at x∗(µ) is absolutely smaller than 1. As theabsolute value of the slope increases everywhere (except at the critical point) whenµ is increased, there will be a value of µ (possibly outside of the interval [0, 4])such that the fixed point x∗(µ) becomes unstable (cf. Figure 4.3). The slope of thegraph of equation (4.1.2) is

df(xt)dxt |x=x∗

= µ(1 − 2x∗) = 2 − µ, (4.1.4)

implying that a flip bifurcation occurs for µ = 3 ∈ [0, 4]. The formerly stable fixedpoint becomes unstable and a new stable fixed point of period 2 (also called period-2

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124 Chapter 4

cycle or fixed point of order 2) emerges: the state variable xt switches permanentlybetween the two components of the fixed point, i.e., x1 = x3, x2 = x4, x3 = x5, etc.,but xt �= xt+1.

4.3.a. stable 4.3.b. unstableLoss of Stability in the Logistic Equation

Figure 4.3

The emergence of a fixed point of period 2 in a map like (4.1.2) can also bedemonstrated in an alternative way.6 Define the second iterate as

f (2): R × R → R : xt+2 = f(xt+1, µ) = f(f(xt, µ), µ

)≡ f (2)(xt, µ).

(4.1.5)

For the logistic equation, the second iterate f (2)(xt, µ) is

f (2)(xt, µ) = xt+2 = µ(xt+1 − x2t+1) and xt+1 = µ(xt − x2

t)

=⇒ xt+2 = µ(µ(xt − x2

t) − µ2(xt − x2t)

2). (4.1.6)

Two graphs of the map f (2) for different values of µ are depicted in Figure 4.4. Thegraphs possess the same points of intersection with the xt axis, they are symmetricwith respect to a vertical line at the critical point xc = 0.5, and display two peaks anda valley. When µ is small, the peaks and the valley are not very pronounced. Thetwo peaks are stretched upwards and the valley is deepened when µ is increased.

If µ is small and a single non-trivial fixed point (x∗, µ) is stable, the graph of f (2)

can intersect the 450 line only once at the fixed point x∗ > 0 (cf. Figure 4.4.a). Thisis necessarily the case because a fixed point of the map f(xt, µ) with the propertyx1 = x2 = x∗ is also a fixed point of the map f (2)(xt, µ) with the property x1 = x3,

6 The following period-doubling scenario is very clearly described in Baumol/Benhabib(1989) and Devaney (1992). Baumol/Benhabib (1989) also present numerically exactplots of the graphs of f (2) and higher iterates.

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4.1.1. Basic Concepts 125

4.4.a. 4.4.b.The Second Iterate of the Logistic Equation for Different Values of µ

Figure 4.4

x2 = x4, etc. When µ is increased, the peaks are stretched upwards and the valleyis deepened, implying that the graph of f (2)(xt, µ) will eventually be tangent to the450 line at the fixed point x∗ with a slope of f (2)(xt, µ) = +1.

The relation between this positive slope of f (2)(xt, µ) and the slope of f(xt, µ)follows from

df (2)(xt, µ)dxt

=df(f(xt, µ), µ

)dxt

=df(xt+1, µ)dxt+1

df(xt, µ)dxt

, (4.1.7)

where use has been made of the chain rule of differentiation. As xt = xt+1 = x∗ atthe fixed point, the slope of f (2)(xt, µ) therefore equals f (2)′ = f ′f ′ with a primedenoting the partial derivative with respect to the first argument. It has alreadybeen shown that the flip bifurcation occurs at µ = 3 with a slope of f ′ = −1 at thefixed point (cf. (4.1.4)). Consequently, the slope of f (2) at this bifurcation point is|f ′f ′| = 1.

When µ is larger than its bifurcation value for the flip bifurcation, the graphof f (2)(xt, µ) intersects the 450 line three times for positive xt, namely at the nowunstable fixed point of f(xt, µ) and at the two components xi∗, i = 1, 2, of theperiod-2 fixed point with the property that xi∗ = f (2)(xi∗, µ), i = 1, 2.

The period-2 fixed point of the map f (2)(xt, µ) is stable as long as the slope ofthe graph, evaluated at the components of the period-2 fixed point, is absolutelysmaller than one, i.e., if∣∣∣∣df (2)(xt, µ)

dxt

∣∣∣∣ < 1. (4.1.8)

It follows from (4.1.7) that the slope of f (2)(xt, µ) at the components xi∗, i =1, 2, equals the product of the derivatives f ′ evaluated at the two components, i.e.,

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126 Chapter 4

f (2)′(xj∗, µ) = f ′(xi∗, µ) · f ′(xj∗, µ). The slopes of f (2) at the two components aretherefore identical. In Figure 4.4.b the graph has been drawn in such a way thatthe period-2 fixed point is stable.

4.5.a. 4.5.b.The Fourth Iterate of the Logistic Equation for Different Values of µ

Figure 4.5

When µ is increased even further, the slope of the graph of f (2) at the twocomponents xi∗, i = 1, 2, will eventually be (absolutely) larger then one, and thestability condition (4.1.8) will be violated. In order to demonstrate what kind ofbifurcation behavior occurs at the value of µ for which the period-2 fixed pointbecomes unstable, it is useful to consider the fourth iterate, i.e., the system definedby

f (4): R × R → R : f (4)(xt, µ) = xt+4 = f

(f(f(f(xt, µ)

))). (4.1.9)

A graph of f (4)(xt.µ) is shown in Figure 4.5.a for a value of µ such that the period-2 cycle is stable. The graph intersects with the 450 line at the two componentsof the period-3 cycle and the previous unstable fixed point. The same argumentsas above on the slope of the graph and the stability of the fixed points apply inthis case. When µ is increased, several new peaks and valleys appear in the graphof f (4)(xt, µ) which are stretched upwards or downwards, respectively. Eventually,seven points of intersection exist (in addition to the origin) (cf. Figure 4.5.b). Thetwo components of the previous period-2 cycle and the very first unstable fixed pointof f are described by the intersections with a positive slope of f (4). The remaining

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4.1.1. Basic Concepts 127

four points of intersection represent a stable period-4 cycle, i.e., a cycle describedby x1 = x5, x2 = x6, x3 = x7, x4 = x8, etc., xi �= xi+m, m < 4.

This procedure can be continued in order to describe the emergence of a period-8 cycle, period-16 cycle, and so on. This scenario of the emergence of a stable cyclewith order 2n, n = 0, 1, . . ., the splitting of each branch into two new stable branchesand an unstable fixed point, etc., persists for increasing µ in an interval. Figure 4.6shows a stylized bifurcation diagram of this period doubling scenario. The interestingobservation in this bifurcation diagram consists in the fact that the sequence of

Stylized Period-Doubling BifurcationsFigure 4.6

bifurcation values µ for which a period-doubling bifurcation occurs converges toa cumulation point µc. Feigenbaum (1978) made the important observation thatin the logistic equation this sequence of period-doubling bifurcation values followsthe rule7

limn→∞

(µn − µn−1

µn+1 − µn

)= δ ≈ 4.6692 . . . . (4.1.10)

If two successive bifurcation values are known, the next bifurcation value can becalculated from (4.1.10).8 In this way, all bifurcation values of the logistic map canbe determined. It turns out that the limit point of the period-doubling sequenceis µc ≈ 3.5699 . . . . More important, it has been shown that δ is a universal constantbecause it characterizes the period-doubling behavior in many one-dimensionalnon-invertible maps. In addition, other universal constants can be derived from

7 Cf. Collet/Eckmann (1980), p. 37.8 For example, if the first two bifurcation values µ1 = 3. and µ2 = 3.449 are known,

applying (4.1.10) yields((1 + δ)µ2 − µ1

)/δ = µ3 ≈ 3.54.

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128 Chapter 4

the logistic equation and can also be found in similar equations.9 It is this proto-type character of the logistic equation which justifies its usually long expositions intextbooks. In addition, the basic qualitative properties of the logistic equation canbe found in dynamical systems generated by a coordinate transformation of the orig-inal system. Consider the two one-dimensional maps f : X → Y and f∗: X∗ → Y ∗,X,Y,X∗, Y ∗ ∈ R, and let h: R → R be a diffeomorphism.10 When the relationbetween the two systems can be expressed in the form of the diagram

Xf

−−−−−−−−−−−−−−−−→ Y

h−1

��h h−1

��h

X∗ −−−−−−−−−−−−−−−−→f∗

Y ∗

(i.e., when the diagram commutates) then f∗ = h ◦ f ◦ h−1 is a dynamical systemconjugated with f by the diffeomorphism h.11

For values of µ above the critical value µc, phenomena other than period dou-bling can be observed. Figure 4.7 contains a numerical plot of the bifurcationdiagram of the logistic equation.12 The majority of µ values has a large number ofassociated xt values. In addition to 2n cycles, fixed points with all even periods k canemerge for appropriate µ values. Furthermore, when µ is sufficiently large, fixedpoints with odd periods occur. Figure 4.8 illustrates the emergence of a period-3fixed point which will be of interest in some theoretical results presented below.Most astonishing, there may be sequences of xt which do not possess any period atall, i.e., for which xt+n �= xt ∀ n > 0.

9 For example, let dn denote the distance between that element of a period-2n cycle whichis closest to the critical value xc and the element of a period-2n−1 cycle which is closestto the critical value. Then the ratio dn/dn+1 = −α ≈ −2.50 . . . is another universalconstant.

10 Cf. Section 4.2.1 for a precise definition. Roughly speaking, a diffeomorphism is a con-tinuous map with a continuous inverse.

11 For example, the diffeomorphism h(x) =(2 arcsin

√x)/π transforms the logistic equa-

tion f(x) = 4x(1 − x), x ∈ [0, 1], into the tent map (cf. Figure 4.12)

x∗t+1 =

{2x∗ if 0 ≤ x∗ < 0.5,2(1 − x∗) if 0.5 ≤ x∗ ≤ 1.

Important properties like the value of the Lyapunov exponent (cf. Chapter 6) are iden-tical for both systems.

12 Such a diagram is generated in the following way: fix a certain µ and an initial valuex0 and calculate the sequence {xt}T0 , T large, and drop the first elements such thattransients do not appear in the diagram. Then repeat the procedure for other equallyspaced values of µ.

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4.1.1. Basic Concepts 129

A Numerical Plot of the Bifurcation in the Logistic Equationxt+1 = µxt(1 − xt), µ ∈ [2.8, 4]

Figure 4.7

While for many µ values in Figure 4.7 it is impossible to determine by visualinspection whether the vertical xt-values represent a stable cycle of order k, k large,or aperiodic behavior, the diagram uncovers structure. The cloud of xt valuesdisappears for several intervals of µ values, and low-order periodic cycles prevail.These regions of µ values are called windows.

4.8.a: xt+1 = f(xt, µ) 4.8.b: xt+3 = f (3)(xt, µ)Period-3 Fixed Points in the Logistic Map

Figure 4.8

Without providing a precise definition at this place, the simultaneous presenceof periodic cycles of order k and of aperiodic cycles will synonymously be called deter-ministic chaos, deterministic noise, or complex behavior in the following. The parameterregime µc < µ ≤ 4 in the logistic equation is called the chaotic regime.

In the chaotic regime in Figure 4.7 the x values belonging to a given µ seemto be equally distributed over an interval. This impression can be verified by the

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130 Chapter 4

A Histogram of a Fictitious Time SeriesFigure 4.9

following experiment: calculate the time series from a given map, e.g., the logisticequation, with a sufficient number of elements, say 10,000 data points. Dividethe admissible x interval [0, 1] into m subintervals of equal length. For example,consider 20 subintervals, each of which is 1/20 of the total length of the admissiblex interval. Then, count the number of points in the calculated time series fallinginto subinterval h, h = 1, . . . ,m. In the histogram in Figure 4.9, the number of datapoints of a fictitious time series falling into each of the 20 subintervals is plotted onthe ordinate against the x values of the subintervals.

If a dynamical system possesses a stable fixed point, the data points will accu-mulate in an interval containing the fixed point. If transients are excluded fromthe time series (e.g., the first 200-500 data points of the time series), the histogramwill display only one point in a single interval. If the system possesses a stable or-bit of a low order (say, e.g., of order 4), the histogram will exhibit a finite numberof nonzero ordinate values in the different subintervals. Finally, if a time series ischaotic and no stable orbit exists, each interval will be visited by the time series witha more or less equal probability. Figure 4.10 is a histogram of the logistic equationxt+1 = 4xt(1− xt) for 50,000 data points. The interval [0, 1] has been divided into200 sub-intervals. It can be seen that the connection of the ordinate values in eachinterval forms a nearly continuous curve. As no distinguished peaks are observedin this curve, it can be concluded that the time series visits nearly every subinter-val with the same probability, i.e., that the deterministic time series behaves like apurely stochastic time series.13

13 Occasionally, the binned series can be approximated by continuous density functions.For example, the ‘curve’ in the histogram in Figure 4.10 can be approximated by thedensity function

g =a

π(x(1 − x)

)1/2.

Cf. Collet/Eckmann (1980), p. 16, and Day/Pianigiani (1991) for details.

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4.1.1. Basic Concepts 131

N

x

A Histogram for the Logistic Equation xt+1 = 4xt(1 − xt); 200 IntervalsFigure 4.10

xt+1

xt

Ergodic Behavior in the Logistic Equation; µ = 3.99Figure 4.11

The dynamic behavior depicted in the histogram above is called ergodic behavior.Roughly speaking, a system is said to exhibit ergodic behavior if the majority of

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132 Chapter 4

initial points visit every region in phase space with about equal probability. Figure4.11 demonstrates this behavior in the familiar (xt, xt+1) diagram: for sufficientlylarge µ, the entire diagram will be filled by the trajectory when the number ofiterations tends toward infinity.

One-dimensional maps have become

The Tent MapFigure 4.12

popular not only because of the ergodiccharacter of some cycles, but because of aphenomenon which has already been in-dicated in Figure 4.1: two initial pointswhich are close together develop in acompletely different way as time passes.The difference between the initial statesmay be arbitrarily small, but neverthelessthe trajectories belonging to the two ini-tial points may converge to cycles of dif-ferent period k or may behave aperiodi-cally. When the difference between theinitial states is smaller than the precisionof a calculator, it is impossible to preciselycalculate the sequence {xt} belonging toan initial value x0. This phenomenon iscalled sensitive dependence on initial conditions. When a dynamical system possessesthis property, its behavior is called mixing. Theoretically it may be difficult to es-tablish whether or not a map like (4.1.2) displays a mixing behavior. A map that isparticularly suited for analytical investigations is the tent map

xt+1 =

{axt if 0 ≤ xt ≤ 0.5b(1 − xt) if 0.5 < xt ≤ 1. (4.1.11)

Figure 4.12 shows the graph of this map for the parameter values a = 2 and b = 2.For these parameters the interval [0, 1] is mapped to itself. The next section pro-vides an introduction to some analytical methods that allow for establishing ergodicand mixing behavior. It will be shown that measures like Lyapunov exponents caneasily be calculated for this map and that the map indeed displays mixing behavior.

Deterministic Chaos

Ergodicity Mixing�� ��Stochasticity Sensitive Dependence

Table 4.1

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4.1.1. Basic Concepts 133

This brief survey ends with a description of a phenomenon which can be ob-served in several time series generated by deterministic dynamical systems.14 Whilethe time series in Figure 4.1 are characterized by a sawtooth pattern, i.e., a per-manent increase and decrease in successive data points, some series occasionallyseem to settle down to a stationary value, but eventually show a sawtooth behavioronce again. Figure 4.13 illustrates this intermittent behavior with a map, the graphof which comes close to the 450 line in a tangential way. When an initial point ismapped into that region, the sequence {xt} will stay in the region for a while andwill exhibit only minor changes from one iteration to the other. When the trajec-tory has left the intermittency region the typical large variations in xt can again beobserved.15

Intermittency in a One-Dimensional MapFigure 4.13

Some Results for One-Dimensional Maps

The foregoing presentation has uncovered that the dynamic behavior of one-dimen-sional maps can be rather complicated. This section provides a short overview ofsome analytical results. The question of whether the dynamic behavior is sensitiveto initial conditions will be elaborated upon at some length.

It was mentioned above that the presence of a period-3 cycle is of particularimportance to complex behavior in one-dimensional maps. The following theoremof Sarkovskii (1964) provides an answer to why period-3 cycles play a dominantrole for chaotic dynamics:16

14 Cf. Berge et al. (1984), pp. 226ff., for details.15 Occasionally, the term intermittency is also used to describe the windows in a bifurcation

diagram: intervals of µ values with associated clouds of xt values are superseded byintervals of µ with low-order cycles in xt.

16 Cf. Guckenheimer/Holmes (1983), p. 311.

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134 Chapter 4

Theorem 4.1 (Sarkovskii (1964)): Consider the following orderingof all positive integers:

1 ≺ 2 ≺ 4 ≺ 8 ≺ 16 . . . ≺ 2k ≺ 2k+1 ≺ . . .

. . . . . .

. . . ≺ 2k+1(2n+ 1) ≺ 2k+1(2n− 1) ≺ . . . ≺ 2k+15 ≺ 2k+13 ≺ . . .

. . . ≺ 2k(2n+ 1) ≺ 2k(2n− 1) ≺ . . . ≺ 2k5 ≺ 2k3 ≺ . . .

. . . . . .

. . . ≺ 2(2n+ 1) ≺ 2(2n− 1) ≺ . . . ≺ 2 · 5 ≺ 2 · 3 ≺ . . .

. . . ≺ (2n+ 1) ≺ (2n− 1) ≺ . . . ≺ 9 ≺ 7 ≺ 5 ≺ 3.

If f is a continuous map of an interval into itself with a period p andq ≺ p in this ordering, then f has a periodic point of period q.

The odd integers starting with the number 3 have received the highest ranks inthis ordering, followed by the odd integer times 2, 22, 23, etc. This ranking coversall integer numbers except the powers of 2. These last integers have received thelowest ranks. Consider an arbitrary integer in this ordering. For example, if thisnumber is 4, then the theorem implies that a mapping with a periodic point ofperiod 4 also has a periodic point of period 2 and a periodic point of period 1 (i.e.,a single stable equilibrium point).17 If this number is 12 (= 223), then all cycles oforder 2k, (k = 0, . . . ,∞), and cycles of order 20, 28, 36, 44, . . ., of order 24, 40, 56,72, . . ., etc. exist. As soon as a period-three cycle has been detected, it follows thatthere are periodic points with every possible period.

The Sarkovskii theorem can also be interpreted the other way round: for exam-ple, if it can be shown that no period-2 cycles exist, then no higher-order periodsexist as well because the latter implied the existence of the former according to theSarkovskii theorem.

A related theorem is the renowned Li/Yorke theorem:

Theorem 4.2 (Li/Yorke):18 Let J be an interval and let f : J → J becontinuous. Assume there is a point a ∈ J for which the points b = f(a),c = f (2)(a), and d = f (3)(a) satisfy

d ≤ a < b < c (or d ≥ a > b > c).

Then

i) for every k = 1, 2, . . . there is a periodic point in J having period k.

Furthermore,

17 Alternatively, a period-4 point does not necessarily imply the existence of a period-8point, which is ranked higher in this ordering.

18 Cf. Li/Yorke (1975), p. 987.

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4.1.1. Basic Concepts 135

ii) there is an uncountable set S ⊂ J (containing no periodic points),which satisfies the following conditions:

A. For every p, q ∈ S with p �= q

lim supn→∞

| f (n)(p) − f (n)(q) |> 0,

and

lim infn→∞ | f (n)(p) − f (n)(q) |= 0.

B. For every p ∈ S and periodic point q ∈ J,

lim supn→∞

| f (n)(p) − f (n)(q) |> 0.

The properties described in (ii A.) and (ii B.) of the theorem express the aforemen-tioned sensitive dependence on initial conditions (SDIC):

• No matter how close two distinct aperiodic trajectories come to each other, theymust eventually move away from each other.

• Every possible aperiodic trajectory moves arbitrarily close to every other one.

• If an aperiodic cycle approximates a cycle of order k for a while, it must moveaway from that cycle.

A one-dimensional map displaying the properties of i) and ii) of Theorem 4.2 willbe called a chaotic map in the Li/Yorke sense. If a map possesses a period-3 cycle,Theorem 4.2 implies the existence of Li/Yorke chaos.19

Chaos in the Li/Yorke sense is also called topological chaos. This type of “chaoticbehavior” does not exclude that the observable motion is indeed regular, i.e., “non-chaotic” without any sensitive dependence on initial condition when (for a givenvalue of µ) most initial points converge toward a period-k cycle. The Li/Yorketheorem implies the existence of a scrambled set S of initial points with aperiodicorbits and SDIC but it does not say anything on the size of this set. In fact, theset of initial points with “true” chaotic properties (i.e., with the properties of thescrambled set) can have Lebesgue measure zero, meaning that the initial points inthe interval [0, 1] which imply aperiodic orbits and SDIC are isolated and do notform sub-intervals on the line. When the map possesses a stable period-k cycle,almost every initial point (in the sense of Lebesgue measure) converges toward theperiod-k cycle.20

19 Subsequent results by Li/Misiurewicz/Pianigiani/Yorke (1982) have extended theperiod-3 requirement to the case of odd periods ≥ 3.

20 Cf. Guckenheimer (1979) and Nusse (1987).

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136 Chapter 4

Aperiodic points do exist, but they do not necessarily attract initial points. Itfollows that “true” chaos might be unobservable even though the set S is uncount-able.21 From a practical point of view this fact may not be extremely important.The presence of the scrambled set influences the evolution of initial points in thetransient phase (cf. Section 4.3 for details) which eventually converge to a periodicorbit. For example, the periodic orbits displayed in the windows in the bifurca-tion diagram in Figure 4.7 are typically approached in the form of a complicatedtransient which often cannot be distinguished from true chaotic motion (cf. Figure4.14). Furthermore, as the bifurcation diagram has been obtained from a numeri-cal experiment it can also not be excluded that the apparent chaotic regions in thediagram represent nothing else than extremely long transients.

Y

Time

Transient Chaotic Motion in a Period-3 Window of the Logistic Map; µ = 3.83.Figure 4.14

Theoretical attempts to provide conditions under which true chaotic motion isindeed observable make use of measure theory. It can be shown that true chaos isobservable when, for a given value of the parameter µ, a chaotic set S has a positiveabsolutely continuous invariant measure (with respect to the Lebesgue measure). Whenthis measure is positive, the set of initial points converging toward the chaotic setS will have positive Lebesgue measure as well.22 It has been demonstrated thatseveral one-dimensional maps indeed possess such positive measures.23 These mapsinclude the tent map displayed in Figure 4.12 and the logistic map with µ = 4. Infact, the density g = a/

(π√x(1 − x)

)mentioned above in the context of Figure

4.9 and 4.10 is an absolutely continuous, invariant, positive measure for the logistic

21 Occasionally, chaos in the Li/Yorke sense with most initial points converging towardstable orbits is called “thin” chaos; observable chaos with SDIC is also called “thick”chaos, cf. the discussion between Day (1986) and Melese/Transue (1986).

22 For detailed discussions of this measure-theoretic approach to chaotic dynamics com-pare Collet/Eckmann (1980), pp. 149ff., Day/Pianigiani (1991), Eckmann/Ruelle(1985), Medio (1993), Chapter 2, and Ruelle (1990), Part II.

23 Cf. Jacobson (1981), Lasota/Yorke (1973), and Pianigiani (1981).

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4.1.1. Basic Concepts 137

map (with µ = 4).24 In the rest of this book, the measure-theoretic approach willnot be deepened. Instead, a method will shortly be described that allows to excludethe existence of stable periodic cycles in some cases. When no stable periodic cycleexists for a particular parameter value, the aforementioned remarks on the possibleunobservability of true chaotic motion with SDIC are not valid anymore.

In order to determine whether a discrete-time, one-dimensional map has oneor several stable orbits introducing the following notion is useful.25

Definition 4.1 (Schwarzian derivative): Consider a C3-continuousone-dimensional map

xt+1 = f(xt), x ∈ R.

The derivative fS(xt) at a point x with f ′ �= 0, defined as

fS(xt) =f ′′′(xt)f ′(xt)

− 32

(f ′′(xt)f ′(xt)

)2

,

is called the Schwarzian derivative of f .

The Schwarzian derivative preserves its sign under composition, i.e., if, for example,fS(xt) < 0, then the derivative f (n)S(xt) of the nth iteration is negative as well. Therelevance of this Schwarzian derivative becomes obvious in the following theorem:

Theorem 4.3 (Singer (1978)):26 Consider the map xt+1 = f(xt)which maps a closed interval I = [0, b], b > 0, onto itself. If

i) f is C3,

ii) f has one critical point c with f ′(x) > 0 ∀x < c, f ′(c) = 0, andf ′(x) < 0 ∀ x > c,

iii) f(0) = 0 and f ′(0) > 1, i.e., the origin is a repelling fixed point.

iv) fS(xt) ≤ 0 ∀ x ∈ I \ {c}.

then f has at most one stable periodic orbit in the interval I.

If conditions i)-iv) hold true the map f is sometimes also called S-unimodal.27 As anexample, consider the logistic equation (4.1.2). Obviously, f is C∞ and the originis a repelling fixed point. Furthermore, as f ′ = µ − 2µxt, f ′′ = −2µ, and f ′′′ = 0,

24 Cf. Day/Pianigiani (1991).25 Cf. Singer (1978) and Collet/Eckmann (1980), Chapter II.4, and Preston (1983),

pp. 60ff., for details on the following ideas.26 See also Nusse (1986).27 Cf. Collet/Eckmann (1981), pp. 94f.

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138 Chapter 4

4.15.a 4.15.bOrbits Starting at the Critical Point of a Map

Figure 4.15

the Schwarzian derivative is negative for all x ∈ [0, 1] except at the critical pointxc = 0.5. Therefore, f has at most one stable orbit according to Theorem 4.3.28

This does not mean that the map in question does indeed have a stable orbit.The following theorem suggests a simple method for establishing the existence ofa stable periodic orbit:29

Theorem 4.4: If a map f has a stable periodic orbit, then the criticalpoint xc will be attracted to it.

Figures 4.15.a and 4.15.b show two examples of iterations of the critical point xcin the quadratic map. In Figure 4.15.b the critical point happens to be located ina stable period-4 cycle. It is obvious that the critical point can never be mappedto the origin because f(xc) is always smaller than xmax = 1 in this case. In Figure4.15.a, the critical point xc is mapped to the origin within two iterations. As theorigin is a repeller, the system therefore does not possess a stable periodic orbit. Inthe quadratic map (4.1.2), this situation can only occur if µ = 4.

4.1.2. Chaos in Descriptive Growth Theory

This section presents the two probably simplest ways to model economies with com-plex dynamic behavior. It will be shown that standard models in descriptive growththeory can be reformulated such that their dynamic equations are similar to theunimodal maps discussed in the preceding section.

28 Cf. also Guckenheimer et al. (1977), pp. 140-142.29 Cf. Collet/Eckmann (1981), p. 14.

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4.1.2. Chaos in Descriptive Growth Theory 139

As a pioneer in detecting chaotic dynamics in economic systems, Day (1982) re-considered the standard neoclassical growth model. In discrete time and assumingthat the capital stock exists for exactly one period30 the model is expressed as

Yt = Ct + It,

It = Kt+1,

St = Yt − Ct = sYt, s > 0,Yt = F (Kt, Lt),

Lt = (1 + n)tLo, n > 0,

(4.1.12)

with the usual meaning of the symbols, n as the constant growth rate of the pop-ulation, and s as the constant marginal savings rate. The production function islinear-homogeneous, implying that the model can be reduced to

Kt+1

Lt= sF (Kt, Lt)/Lt or kt+1(1 + n) = sf(kt), (4.1.13)

with kt = Kt/Lt as the capital-labor ratio.Under the usual convexity assumption, the map possesses two fixed points: a

repelling fixed point at the origin and an asymptotically stable fixed point k∗ whichsolves k∗ = sf(k∗)/(1 + n).

In contrast to the usual neoclassical assumption, let the production functionhave the following form:

YtLt

= f(kt) = Bkβt (m− kt)γ , kt ≤ m = constant. (4.1.14)

The term (m− kt)γ reflects the influence of pollution on per-capita output. Whenthe capital intensity increases, pollution increases as well. Suppose that resourceshave to be sacrificed in order to avoid this pollution. The maximum output whichcan be produced with a given capital stock is then smaller than the output in thestandard textbook case for each value of k. The constant term m acts like a satura-tion level, implying that per-capita production falls to zero when kt = m. Substitut-ing for the production function in (4.1.13) yields

kt+1 =sBkβt (m− kt)γ

(1 + n). (4.1.15)

Consider first a simplification and let β = γ = m = 1. Equation (4.1.15) reducesto

kt+1 =sBkt(1 − kt)

(1 + n). (4.1.16)

30 This assumption does not have essential consequences. The shape of the resulting mapis unaltered when a gradual depreciation is assumed.

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140 Chapter 4

Let µ = sB/(1 + n). Equation (4.1.16) is then formally identical with the logisticequation (4.1.2), and all properties of (4.1.2) apply to (4.1.16) as well.

Consider next the general five-parameter equation (4.1.15). The graph of(4.1.15) can be modified by variations in the parameters. For example, increas-ing the parameter B stretches the graph upwards. B therefore plays essentially thesame role as µ in (4.1.2). In order to apply the Li/Yorke theorem to (4.1.15), con-sider the following three distinguished values of k. Let kc be the critical point of themap (4.1.16), i.e., the value of kt that implies the highest possible capital intensityin the next period:

dkt+1

dkt=

sB

1 + n

(βkβ−1

t (m− kt)γ − kβt γ(m− kt)γ−1) = 0

=⇒ βkβt (m− kt)γ

kt= kβt γ

(m− kt)γ

m− kt

=⇒ kc =βm

γ + β.

(4.1.17)

When B is sufficiently large, kc is lower than the fixed point k∗. Next, let kb be theresult of the backward iteration kb = f−1(kc). When kc < k∗, kb will be smallerthan kc.

A Neoclassical Growth Model with PollutionFigure 4.16

Finally, let km denote the maximum attainable capital intensity, i.e., the intersec-tion of the graph of (4.1.15) with the abscissa. Variations in B eventually imply thatthe graph of (4.1.15) is stretched upwards such that km is the forward iteration ofkc : f(kc) = sB/(1 + n)(kc)β(m − kc)γ = km (cf. Figure 4.16). As km is mappedto the origin, the following relations between the k values result:

0 < kb < kc < km

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4.1.2. Chaos in Descriptive Growth Theory 141

⇒ f(km) < kb < f(kb) < f(kc)⇒ f3(kb) < kb < f(kb) < f2(kb).

Thus, the requirements of the Li/Yorke theorem are fulfilled. The map (4.1.15)is chaotic in the Li/Yorke sense for appropriate values of the parameters. Further-more, applying Theorem 4.4 shows that for specific values of B there is no stableperiodic orbit: in Figure 4.16 the graph of the map is drawn such that the criticalpoint is mapped into the origin k = 0. As the origin is a repelling fixed point ofthe map, the map cannot have a stable period orbit. In this case, there may existinitial points with a sensitive dependency. When B is lower than the value assumedin Figure 4.16, Theorems 4.3 and 4.4 cannot be applied because the Schwarzianderivative is not unambiguously negative31 and because the forward orbit of thecritical point is not as simple as in Figure 4.16.

This neoclassical growth model (which now can be called a prototype modelin chaotic, discrete-time, dynamical economics) is a modification of the standardtextbook approach to growth theory, and a generalization of the results found forthis modified version is, of course, impossible. However, noninvertible maps canbe shown to exist in basic traditional models without any modifications of the func-tional forms. An example was provided by Stutzer (1980).

Stutzer’s model was one of the very first economic investigations of chaoticdynamics. In particular, Stutzer stressed the possibly fundamental differences be-tween continuous-time and discrete-time dynamical systems. Consider the growthcycle model studied by Haavelmo (1956) with

Y = KNa, K > 0, 0 < a < 1 (4.1.18)

as the production function with a fixed capital stock.32 The growth rate of employ-ment is assumed to be

N

N= α− β

N

Y, α, β > 0, (4.1.19)

i.e., the growth rate increases when per-capita output (income) increases.Combining (4.1.18) and (4.1.19) yields the first-order nonlinear differential

equation

N = αN − βN2−a

K. (4.1.20)

31 For example, a somewhat tedious calculation shows that the Schwarzian derivative is

fS(kt) =34m2(mkt − k2

t

)−2(m− 2kt

)−2(m2

2− 4mkt + 4k2

t

)for the specific values γ = 0.5 and β = 0.5.

32 The assumption of a fixed capital stock is, of course, an oversimplification, and it issurely problematic to speak of a growth model in this context.

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142 Chapter 4

Equation (4.1.20) can be interpreted as a special form of the so-called Bernoulli dif-ferential equation33, which fortunately can be solved explicitly. The stationary equi-librium (the particular solution) is

N∗ =(αKβ

) 11−a , (4.1.21)

and the general solution is 34

N(t) =

((KN(0)a−1 − β

α)eα(a−1)t + β

αK

) 1a−1

. (4.1.22)

Equation (4.1.22) describes a monotonic convergence toward the stationary equi-librium level N∗ because the coefficient of the exponential term in the nominatorrepresents the initial deviations from equilibrium.

A common procedure in many numerical investigations of differential equa-tions consists in replacing the differential operator in (4.1.20) by finite differences.Substituting Nt+1 −Nt for N in (4.1.20) yields

Nt+1 = (1 + α)Nt − βN2−a

K, (4.1.23)

which, by means of the transformation

Nt =

(K(1 + α)

β

) 11−a

xt, (4.1.24)

can be written as

xt+1 = (1 + α)xt(1 − x1−a

t

). (4.1.25)

As 0 < a < 1, the transformed difference equation (4.1.25) has qualitatively thesame structure as the logistic equation: it is a one-humped, noninvertible mapxt+1 = f(xt) defined on the unit interval with f(0) = f(1) = 0 and a critical valuexc located to the left or to the right of xt = 0.5 depending on the parameter a. Byincreasing the parameter α, the graph of (4.1.25) is stretched upwards. Equation(4.1.25) therefore exhibits chaos in the Li/Yorke sense for appropriately chosen

33 The general form of a Bernoulli differential equation is

x + g(t)x + f(t)xa = 0.

In eq. (4.1.20), g(t) and f(t) are constants.34 The solution mentioned in Stutzer (1980) is slightly incorrect but does not alter the

qualitative results.

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4.1.3. Chaos in Discrete-Time Models of Optimal Economic Growth 143

parameter values.35 As equation (4.1.24) is only a positive monotonic transforma-tion, chaos can thus prevail in the discrete-time version of the Haavelmo growthcycle model as well.

The example shows that a standard economic model can exhibit chaotic dy-namics without further (and maybe ad hoc) economically motivated modificationsof the functional forms when time derivatives are replaced by finite differences.This expands a result already known from linear dynamical systems: for constantparameter values, the replacement procedure always implies a different dynamicalbehavior in the differential and difference equations. If the parameters are notproperly adjusted (by taking account of possibly involved stocks and flows), thereplacement procedure can lead to an incorrect result (as measured with respectto the original continuous-time model). If the equations are nonlinear, a smoothmonotonic behavior of the differential equation may be transformed into erraticoscillations.

4.1.3. Chaos in Discrete-Time Models of Optimal Economic Growth

Some of the most interesting results on detecting chaotic trajectories in economicdynamics have been achieved by investigating discrete versions of optimal controlmodels. It has already been demonstrated in Chapter 3 that it may be theoreti-cally optimal to design a control program with cyclical trajectories. In the contextof a discrete-time, competitive, two-sector model, Benhabib/Nishimura (1985)provided sufficient conditions for the existence of period-2 cycles. While these re-sults touch on the validity of the usual welfare-oriented paradigm of a desirabledampening of fluctuations by means of policy measures, it can be shown that anoptimal economic control model may even involve the emergence of chaotic behav-ior. Surveys of the relevant literature can be found in Boldrin (1991), Boldrin/Woodford (1990), and Guesnerie/Woodford (1991).

Boldrin/Montrucchio (1986) have demonstrated that arbitrary, discrete-timedynamical systems can be the outcome of an optimal control program.36 Given aspecific C2 – differentiable map, it is possible to re-construct an optimal controlproblem that implies the given map as an optimal policy function when the discountrates are small enough. As this map may be unimodal, optimal control can lead tochaotic behavior. In the following, a specific numerical example of the presence ofchaotic motion in optimal control will be outlined.

35 The Schwarzian derivative of (4.1.25), namely

fS(xt) =(1 − a)(2 − a)

2(1 − (2 − a)x1−a

t

)2

(2ax−a−1

t − a2 + 5a− 6)

is negative when the last term in parentheses is negative. This is always true when xt isnot too close to 0. Theorems 4.3 and 4.4 can therefore be applied to (4.1.25).

36 Cf. Montrucchio (1988) for an extension to the continuous-time case.

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144 Chapter 4

Consider a standard, two-sector, optimal-growth model with a single capital goodkt and homogeneous labor t available in period t. Capital and labor serve as inputsin the production of consumption goods ct in sector 1 and in the production ofcapital goods in sector 2. Let k(1)

t and k(2)t denote the amounts of capital in sectors

1 and 2, respectively, the sum of which must fulfill k(1)t +k(2)

t ≤ kt. Equivalently, (1)t

and (2)t , (1)

t + (2)t ≤ L, denote the amounts of labor in both sectors with L as the

total labor force.Let ct = f (1)(k(1)

t , (1)t ) be the production function in the consumption goods

industry. The maximal production in the investment goods industry is, for givenk(2)t and (2)

t described by kt+1 = f (2)(k(2)t , (2)

t ), i.e., the output is only availablein the next period. Both functions f (1) and f (2) are assumed to be continuous,monotonically increasing, and strictly concave.

The optimal control problem can be formulated as follows:

max�(1)t ,�(2)

t ,k(1)t ,k(2)

t

∞∑t=0

βtu(ct)

s.t. ct ≤ f (1)(k(1)t , (1)

t )

kt+1 ≤ f (2)(k(2)t , (2)

t )

(1)t + (2)

t ≤ L

k(1)t + k(2)

t ≤ kt

k0 = k

(4.1.26)

with β > 0 as the discount factor and u(ct) as a standard, strictly concave utilityfunction.

The optimization problem (4.1.26) is equivalent to the problem

maxkt

∞∑t=0

βtV (kt, kt+1) s.t. (kt, kt+1) ∈ D

k0 = k,

(4.1.27)

with D as the admissible set and V as the consumption frontier. The assumptionsconcerning f (1), f (2), and u imply that

V1 > 0; V2 < 0; V11 < 0; V22 < 0; V11V22 − V 212 > 0.

The maximum possible consumption in each period depends on the existing capitalstock inherited from the last period and on the decision to produce capital goodsin the current period. For a given capital stock, a high output in the investmentgoods industry implies a low output in the consumption goods industry, i.e., thereis a trade-off between the production of consumption goods and capital goods ineach period.

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4.1.3. Chaos in Discrete-Time Models of Optimal Economic Growth 145

An optimal control problem like (4.1.26) can be investigated by dynamic pro-gramming techniques (cf. Bellman (1957) or Blackwell (1962) for introduc-tions). The value function Wβ(kt) for a given β is defined as

Wβ(k0) = max∞∑t=0

βtV (kt, kt+1) s.t. (kt, kt+1) ∈ D. (4.1.28)

The value function W (·) satisfies Bellman’s equation, i.e.,

Wβ(kt) = maxkt+1

{V (kt, kt+1) + βWβ(kt+1)

}, (4.1.29)

which allows the construction of a so-called policy function kt+1 = hβ(kt), determin-ing kt+1 as a function of its predecessor kt.

It has been known for a long time37 that the policy function kt+1 = hβ(kt)possesses a stable fixed point for high values of β. Furthermore, for very smalldiscount rates, hβ(kt) = 0 for all admissible kt, i.e., the capital stock is entirelyengaged in the production of consumption goods. However, there exist β and β

such that for β ∈ (β, β), the map hβ may exhibit complex dynamics for some V .As a numerical example consider the following particular form of V (kt, kt+1)

which was studied by Deneckere/Pelikan (1986):

V (kt, kt+1) = ktkt+1 − k2tkt+1 − 1

3kt+1 − .075k2

t+1 +100

3kt

− 7k2t + 4k3

t − 2k4t .

(4.1.30)

Equation (4.1.30) fulfills the concavity requirements mentioned above for a givennumerical value of the endowment of labor, L, and the associated maximal produc-tion of capital goods.

For the particular choice of β = 0.01, the value function W (kt) of this exampleis W (kt) = 100/3 kt − 5k2

t such that Bellman’s equation reads

W (kt) = maxkt+1

{V (kt, kt+1) + 0.01

(1003kt+1 − 5k2

t+1

)}. (4.1.31)

Setting the partial derivative of the r.h.s. of (4.1.31) with respect to kt+1 equal tozero and solving for kt+1 yields the desired policy function38

kt+1 = h(kt) = 4kt(1 − kt). (4.1.32)

37 Cf. Brock/Scheinkman (1978) and McKenzie (1986).38 Substituting (4.1.32) for kt+1 in (4.1.31) verifies that W (kt) = 100/3kt− 5k2

t is indeedthe value function of the problem.

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146 Chapter 4

As (4.1.30) is identical with the logistic map (4.1.2) of Section 4.1.1 with µ = 4,the policy function of this optimal growth problem implies chaos in the sense ofLi/Yorke. Furthermore, the map is sensitive to initial conditions.39

As it was mentioned above, the presence of chaotic dynamics in optimal controlmodels implies that the associated policies cannot be observed in practice. Whenthe choice of the appropriate capital stock in the next period depends on the in-finitely precise numerical value of the current capital stock, it will not be possiblefor a central planner to determine the optimal policy. Even if the next period’scapital stock is calculated with a relatively high accuracy depending on the measur-ing devices, repeated application of imprecise measurements ultimately causes thecalculated time path to diverge drastically from the theoretically ideal and optimalgrowth path.

In addition, even if a central planner succeeds in keeping the actual growth pathclose to its theoretical and optimal ideal, the policy measures may be abandonedbecause the path looks too erratic. When a growth path without any control ismonotonic but unoptimal, an irregular but actually optimal growth path may inspirereflections on the correctness of the planner’s underlying model because complexmotion seems to be incommensurable with the usual ideas on policy design.

4.1.4. Other Economic Examples

The examples of chaotic dynamical models outlined above constitute only a smallsubset of the available literature, and reviewing all interesting approaches to irreg-ular dynamics is impossible. In the following, a short list of economic applicationsof the mathematical results on nonlinear, one-dimensional maps will be presented.Naturally, the list does not claim to be complete.

A paper by Benhabib/Day (1981) belongs to the first major investigation ofchaotic behavior in economic dynamics. The authors demonstrated that rationalchoice in a standard micro-framework can involve erratic behavior when prefer-ences depend on past experience (see also Benhabib/Day (1980), Day (1986)).Consider the Cobb-Douglas-type utility function

U(x, y) = xay1−a, a > 0. (4.1.33)

With the standard budget constraint of the form

pxx+ pyy = M, (4.1.34)

the demand for x and y is

x∗ =aM

pxand y∗ =

(1 − a)Mpy

, (4.1.35)

39 Replacing the term 100/3kt in (4.1.30) by 1/(3β)kt yields more general policy functionsrevealing the dependency on the discount rate.

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4.1.4. Other Economic Examples 147

respectively. Assume now that a is not a constant anymore but that its value in tdepends on the demands for x and y in the previous period in the following way:

at = βx∗t−1y∗t−1. (4.1.36)

As the utility function (4.1.33) is maximized over xt and yt, i.e., the values in thecurrent period, the xt−1 and yt−1 values can be treated as constants in the currentperiod t. Thus, the r.h.s. of (4.1.36) can be substituted for a in the demand functions(4.1.35). For example, the demand for x is calculated as:

x∗ =aM

px=βx∗t−1y

∗t−1M

px=βx∗t−1

M − pxxt−1py M

px

=βx∗t−1(M − pxxt−1)M

pxpy=

βM

pxpyx∗t−1(M − pxxt−1).

(4.1.37)

Equation (4.1.37) describes a one-humped curve like the logistic map (4.1.2). Forpx = py = M = 1, (4.1.37) it is identical with (4.1.2). It follows that the path-dependent preferences described by (4.1.33) and (4.1.36) imply the emergence ofchaotic motion for appropriate values of the parameter β. Of course, the equation(4.1.37) has been derived under the assumption of constant prices px and py. Whenprices are changing, completely different results can be expected to hold.40

Day/Shafer (1986) considered a standard IS-LM framework and illustratedthe emergence of chaotic dynamics when the investment function possesses a slight-ly unusual property. This result is particularly important because the IS-LM setup istraditionally used to motivate governmental activities to intervene in the economy.When chaos prevails a government may fail to determine the correct degree ofintervention and the precise timing of the activities. Cf. also Grandmont (1989b)for discussions of Keynesian aspects in nonlinear disequilibrium models.

It seems as if parts of the profession really became aware of the possible relevanceof chaotic dynamics in economics after the publication of Grandmont’s (1985,1986) work on cyclical behavior in a general equilibrium framework. While gen-eral equilibrium theory (or competitive equilibrium theory) concentrated for a longtime on the existence and (asymptotic) stability of a (hopefully) unique equilib-rium, Grandmont showed that it is possible to encounter complex dynamics in anintertemporal, overlapping-generations model (cf. also Benhabib/Day (1982) andGrandmont (1991). The key to the model is the presence of a backward-bendingoffer curve due to a high risk-aversion of the young generation. Even if individualshave perfect foresight, the economy may, in retrospect, be characterized by com-plex behavior.41 Woodford (1987, 1989) studied the effects of imperfect financial

40 An attempt to stabilize the chaotic dynamics generated by (4.1.37) can be found inHeiner (1989).

41 Concise introductions to complex behavior in overlapping-generations models can befound in Brock/Dechert (1991) and Kelsey (1988).

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148 Chapter 4

intermediation between workers and entrepreneurs in an intertemporal optimiza-tion problem. The emergence of cyclic behavior in competitive economies has beenthe subject of an increasing number of publications. Surveys of the relevant liter-ature can be found in Boldrin/Woodford (1990) and Guesnerie/Woodford(1991). This literature also includes the papers mentioned in Section 4.1.3 and, asfar as complex motion is concerned, parts of the literature on “sunspot” models, cf.,for example, Azariadis/Guesnerie (1986), Grandmont (1989a), and Woodford(1990).

Other contributions from various fields include Gaertner (1986, 1987) whostudied cyclical consumption patterns and Ploeg (1986) who investigated complexdynamics in a financial-markets model developed by Begg (1983). Chiarella(1986, 1990), Hommes (1990a), and Jensen/Urban (1984) demonstrated theemergence of irregular dynamics in the well-known microeconomic cobweb modelof sluggish supply adjustment. One of the very first examples of chaotic motion ineconomic systems was provided by Rand (1979) who considered a simple, game-theoretic Cournot-duopoly model (cf. also Puu (1992)). Chaos has also been shownto exist in Walrasian tatonnement processes, cf. Bala/Majumdar (1992) and Hahn(1992). Pohjola (1981) studied a discrete-time version of Goodwin’s growth cyclemodel (cf. Section 2.4.2). Gabisch (1984) elaborated upon a multiplier-acceleratormodel of the Samuelson-Hicks type, where only slight nonlinearities are involved,cf. also Nusse/Hommes (1990) for a discussion of this model. Candela/Gardini(1986) studied a nonlinear version of a Post-Keynesian growth model and con-trasted its analytical properties with empirical findings. The dynamic effects ofspeculative behavior in a market with ‘fundamentalists’ and ‘chartists’ were investi-gated in Chiarella (1992). Models of population dynamics with chaotic motioncan be found in Day/Kim (1987) and Prskawetz/Feichtinger (1992). Samuel-son (1990) dwelled on a problem in Bernoulli – von Neumann utility theory andestablished the existence of chaotic dynamics. Franke/Weghorst (1988) demon-strated the emergence of complex motion in a simple input-output model. Dopfer(1991) discussed the role of chaotic dynamics in evolutionary economics.

The paper by Day/Walter (1989) represents a unique approach to the possi-bly chaotic behavior of an economy. Instead of postulating a nonlinear dynamicalsystem which describes the motion of the economy in the relevant time interval, thedynamics of an economy over a very long time horizon is comprehended as a suc-cession of multiple dynamical systems, each of which is valid only for a limited timespan. Compare also Day/Pianigiani (1991) for a measure-theoretic discussion ofthe model.

Traditional textbooks on the economics of location present typical patterns ofthe spatial organization of villages, cities, or industrial and commercial locations. Atleast with respect to non-American urban areas, these regular patterns often consti-tute a sharp contrast to the observable and historically given spatial organizations.Aside from such major influences as political, social, and (of course, most impor-tantly) geographical ones, it is tempting to apply recent techniques in nonlineardynamics also to the economics of spatial organization. In fact, recent research onspatial chaos has demonstrated that it is possible to explain some irregularities inthe spatial organization of economic units with the help of nonlinear dynamics. For

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4.2. Chaos in Higher-Dimensional Discrete-Time Systems 149

example, Dendrinos (1985) delivered a framework for the modelling of irregularurban decline. Papers by White (1985) and Nijkamp (1987) and the material con-tained in Nijkamp/Reggiani (1992) uncover that chaotic dynamics may be a typicalphenomenon in models dealing with spatial evolution. Rosser (1991) provided asurvey of these approaches.

4.2. Chaos in Higher-Dimensional Discrete-Time Systems

One-dimensional, discrete-time dynamical systems in economics are surely suitedfor demonstrating the relative ease with which complex behavior can be modeled.However, there are several objections to the use of one-dimensional maps in eco-nomic dynamics.

• Most economic phenomena are originally framed in higher-dimensional dynam-ical systems. The methods presented above can only be applied when the gen-uine model is extremely simplified.

• In many cases, ad hoc assumptions are necessary to justify the presence of uni-modal maps because these maps do not seem to be generic in standard economictheory.

• One-dimensional maps typically display a sawtooth pattern in the generated timeseries. Other interesting phenomena like smooth but irregular time paths can-not be observed in these maps.

This section provides a short introduction to complex behavior in higher-dimen-sional maps. The emphasis will be on two-dimensional systems because results forn-dimensional systems seem to be vague or unpractical. A short overview of existingtools in the next subsection is followed by a demonstration of complex behavior inthe prototype Kaldor model.

4.2.1. Some Basic Ideas

In the higher-dimensional case it is difficult to find examples of dynamical sys-tems which can act as prototype systems, i.e., systems which display many importantproperties of systems belonging to particular families of dynamical systems. Thefollowing two systems in R2 cannot be called prototype systems but have been stud-ied in greater detail in the literature. Both systems represent interesting examplesbecause the one-dimensional logistic map can be derived as special cases.

Consider the system

xt+1 = f(xt) + yt

yt+1 = βxt,(4.2.1)

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with f(xt) as a noninvertible map.42 When f(xt) = 1−γx2t , the map is known as the

Henon map.43 When β = 0, equation (4.1.33) is identical with the one-dimensionalmap (4.1.1). When β �= 0, (4.1.33) is invertible and can explicitly be solved for xtand yt, namely xt = yt+1/β and yt = xt+1 − f(yt+1/β). As this map and the map(4.2.1) are continuous, the map (4.2.1) is a so-called diffeomorphism.44

Although invertible maps dominate the literature on the behavior of two-dimen-sional discrete-time systems, complex behavior can also be observed in non-inverti-ble maps. For example, Marotto (1978) studied the two-dimensional system

xt+1 = (1 − axt − byt)(axt + byt)yt+1 = xt.

(4.2.2)

When b = 0, system (4.2.2) is again qualitatively identical with the logistic equation.Figures 4.17 and 4.18 show the results of the numerical simulation of (4.2.2) forparticular parameter values. The variation of the parameters yields a large numberof similarly complex and astonishing geometric objects.

The single points (xt, yt) jump irregularly in these geometric objects, and thestructure becomes apparent only after a larger number of iterations. The objectsform attractors, i.e., initial points located far away from the objects approach themrapidly. The unusual forms of the attractors which are neither single points norclosed orbits is the reason why they are called strange attractors.45

An early theoretical result for the dynamics of n-dimensional maps comparableto the Li/Yorke theorem was provided by Phil Diamond in 1976. By replacingthe iterate of a one-to-one function by the iterate of a set, Diamond formulatedconditions for the existence of a scrambled set qualitatively equivalent to the Li/Yorke conditions. Early applications of Diamond’s theorem in economics haveturned out to be problematic and will not be discussed further. Instead, the restof this section concentrates on the notion of a snap-back repeller introduced by Ma-rotto (1978). Consider a discrete-time dynamical system

xt+1 = f(xt), x ∈ Rn (4.2.3)

with an unstable fixed point x∗ = f(x∗).

Definition 4.2 (Marotto (1978)): LetBr(x∗) denote the closed ballwith radius r in Rn centered at x∗. The point x∗ ∈ Rn is an expanding

42 Cf. Ott (1981), p. 659, for a discussion.43 The Henon map often serves as the standard example for chaotic dynamics in R

2.Though the map is almost always cited as an example of a two-dimensional, discrete-timechaotic system, the identification of its dynamical behavior as chaotic is not undisputed.Cf. Ott (1981), pp. 660ff., and Benedicks/Carleson (1991).

44 Cf. Devaney (1986), p. 9, for details. A Ck map f : X → Y , X,Y ∈ Rn, is a Ck–

diffeomorphism if it is invertible and if the inverse map f−1: Y → X is also Ck.45 Cf. Section 5.1 for details on the notion of a strange attractor.

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4.2. Chaos in Higher-Dimensional Discrete-Time Systems 151

y

x

A Simulation of System (4.2.2); a = 0.8, b = 3.0Figure 4.17

y

x

A Simulation of System (4.2.2); a = 2.5, b = 2.0Figure 4.18

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152 Chapter 4

fixed point of f(x) in Br(x∗) if f(x∗) = x∗ and all eigenvalues of theJacobian of f(x) exceed 1 in (Euclidian) norm for all x ∈ Br(x∗).

This definition does not imply that a time series moves away from x∗ everywhere. Ifx /∈ Br(x∗) for an arbitrary r, the eigenvalues may be less than or equal to 1. Oncea point outside Br(x∗) is reached, xt may jump back into Br(x∗) and even ontox∗. In that case the fixed point is called a snap-back repeller :

Definition 4.3 (Marotto (1978)): Assume that x∗ is an expandingfixed point of f in Br(x∗) for some r. Then x∗ is said to be a snap-backrepeller of f if there exists a point x0 ∈ Br(x∗) with x0 �= x∗, f(k)(x0) = x∗,and the determinant of the Jacobian of f(k)(x0) is different from zero forsome positive integer k.

Figure 4.19 illustrates this notion of a snap-back repeller: A trajectory which startsarbitrarily close to the fixed point x∗, i.e., at a point x0 in Figure 4.19, is repelledfrom this fixed point, but, after having left Br(x∗), suddenly jumps back to hit thefixed point exactly. A snap-back repeller is the discrete-time analog of a homoclinicorbit to be introduced in Section 5.4.

A Snap-Back RepellerFigure 4.19

Theorem 4.5 (Marotto (1978)): If f possesses a snap-back repeller,then (4.2.3) is chaotic.

Marotto’s definition of chaos is qualitatively identical with the chaos definition ofLi/Yorke for one-dimensional maps.

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4.2. Chaos in Higher-Dimensional Discrete-Time Systems 153

4.2.2. An Economic Example

The economic literature actually abounds in examples of the potential emergenceof complex motion in two-dimensional dynamical systems. As a rule of thumb,complex motion can almost always be observed (for appropriate parameter val-ues) in discrete-time, two-dimensional systems which are derived from originallycontinuous-time systems (permitting limit-cycle behavior) by substituting the dif-ferential operator by finite differences, i.e., by considering

∆xt+1 = xt+1 − xt = f(xt) instead of x(t) = f(x(t)

).

The following example is due to Herrmann (1985) who studied a two-dimensional,discrete-time business-cycle model with Kaldorian elements:46

∆Yt+1 = α(I(Yt,Kt) + C(Yt) − Yt

),

∆Kt+1 = I(Yt,Kt) − δKt,(4.2.4)

with I(Yt,Kt) = β(Kdt −Kt)+ δKt, β > 0, as gross investment, and δ > 0 as the de-

preciation rate. Net investment depends proportionally on possible discrepanciesbetween the desired and actual capital stock. If the desired capital stock dependslinearily on output, i.e., Kd

t = kYt, k > 0, and if the consumption function C(Yt)has a sigmoid shape similar to Kaldor’s investment function,47 equations (4.2.4)become

∆Yt+1 = α(β(kYt −Kt) + δKt + C(Yt) − Yt

),

∆Kt+1 = β(kYt −Kt),(4.2.5)

or, abbreviated,

Yt+1 = F1(Yt,Kt) + Yt =: G1(Yt,Kt),Kt+1 = F2(Yt,Kt) +Kt =: G2(Yt,Kt),

(4.2.6)

with

F1(Yt,Kt) = α(β(kYt −Kt) + δKt + C(Yt) − Yt

),

F2(Yt,Kt) = β(kYt −Kt).(4.2.7)

46 Another example of a slightly different version of the Kaldor model can be found inDana/Malgrange (1984).

47 Herrmann (1985) used the following numerical specification of the consumption func-tion:

C(Yt) = 20.0 +2π

10.0 arctan

(0.85π20.0

(Yt − Y ∗)

)

with Y ∗ = 22.22 as the equilibrium level of income.

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154 Chapter 4

K

Y

Chaos in a Discrete-Time Kaldor Model; α = 25.0, k = 2.0, β = 0.1, δ = 0.05Figure 4.20

The Jacobian of (4.2.6) is

J =

(F11 + 1 F12

F21 F22 + 1

)(4.2.8)

with

F11 = α(βk +

dC(Yt)dYt

− 1),

F12 = α(δ − β),F21 = βk,

F22 = −β.

(4.2.9)

The eigenvalues of J are

λ1,2 =F11 + 1 + F22 + 1

2±√

(F11 − F22)2 + F12F21

4. (4.2.10)

It is possible to find reasonable numerical specifications of the parameters such thatthe modulus of the eigenvalues is greater than one, implying that the eigenvalueslie outside the unit circle and that the fixed point is an expanding fixed point inMarotto’s term. As the model is nonlinear, the entries in the Jacobian change forvarying (Yt,Kt). Eventually, the eigenvalues change so that they lie within the unitcircle.

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4.2. Chaos in Higher-Dimensional Discrete-Time Systems 155

Y

Time

K

Time

Two Time Series With Slightly Different Initial ValuesFigure 4.21

For a certain parameter set Herrmann (1985) was able to detect an initial pointin a neighborhood Br of the fixed point that leaves the neighborhood during thefirst iterations and jumps to the fixed point in the 12th iteration. It follows thatthe discrete-time Kaldor model (4.2.6) has a snap-back repeller for the specific pa-rameter set. As Marotto (1978) pointed out, snap-back repellers persist undersmall perturbations of the model and variations in the parameter set. However, itshould be recalled that the result of any numerical example cannot be generalizedto hold true for the entire range of parameter values in a certain model. The con-cept of snap-back repellers requires a separate numerical study for each numericalspecification of a model.

The results of a numerical simulation of (4.2.5) can be found in Figures 4.20and 4.21. For low values of α, the time series converges to the unique fixed point.When α is increased, the sequence {(Yt,Kt)}Tt=1 is first located on an attractingclosed orbit. For values of α larger than a critical αc, a sequence of 50000 pointsin the (Yt −Kt) plane generates the cloud depicted in Figure 4.20. The object inFigure 4.20 is another example of a strange attractor.

The separate plotting of the time series Yt and Kt versus time shows the irreg-ularity as well as the sensitive dependence on initial conditions more clearly. InFigure 4.21, the time series of Yt and of Kt are shown for slightly different initial

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156 Chapter 4

values. After wandering together for a few periods, the two time series eventuallydiverge. This property of the system prevails when higher t-values are considered,i.e., the phenomenon is not restricted to the transient phase.

K

Y

The Basin Boundary of the Attractor of (4.2.5); α = 25.0Figure 4.22

The basin of attraction of the attractor is plotted in Figure 4.22. White areasrepresent initial points that converge toward the attractor. The grey-shaded areasrepresent initial points that converge toward infinity. In contrast to the case of regu-lar attractors where the basin of attraction is usually a connected and relatively wideset, the basin of attraction in Figure 4.22 is a complicated, disconnected set. Oc-casionally, the basin boundary comes very close to the attractor itself. However, noevidence of a fractal nature of the basin boundary could be found in this numericalexperiment.

————–

Other examples of chaotic motion in two-dimensional, economic systems can befound in Hommes (1992), Hommes/Nusse/Simonovits (1990), and Simonovits(1992). The authors studied the effects of Hicksian ceilings and floors (cf. Hicks(1950)) in the investment behavior in capitalist and socialist economies. Therole of corporate debt and investment confidence was studied in Delli Gatti/Gallegati/Gardini (1991). A discussion of a financial-crises model can be foundin Gardini (1991) together with a lot of information on the bifurcation behaviorin two-dimensional endomorphisms. Gaertner/Jungeilges (1988) investigated

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4.3. Complex Transients in Discrete-Time Dynamical Systems 157

the possibly complicated behavior in a model describing the consumption deci-sion of two households, both of which are influenced by the consumption decisionof the other household. Piecewise linear, non-Walrasian macroeconomic modelswere studied by Simonovits (1982) and Hommes/Nusse (1989). Bohm (1993)discussed the emergence of regular periodic motion and chaotic behavior in amicro-macro model with overlapping generations and rationing. Marotto’s pro-cedure in detecting a strange attractor has also been applied to a growth cyclemodel in the Keynes-Kalecki tradition by Jarsulic (1991). A high-dimensionaldiscrete-time system emerging in the study of an inventory management prob-lem (the “beer game,” cf. Mosekilde/Larsen (1989)) was studied in Thomsen/Mosekilde/Sterman (1992). Another high-dimensional system was investigatedby Silverberg/Lehnert (1992, 1993) who considered a Schumpeterian model ofembodied technical change.

4.3. Complex Transients in Discrete-Time Dynamical Systems

The examples of chaotic dynamics presented in the previous sections concentratedon the motion on chaotic attractors. However, complex behavior cannot only occuron such attractors but can also be a property of the motion in the transient phasebefore an attractor has been reached. A transient trajectory can display complexbehavior even if the attractor is non-chaotic.

Complex transient motion in nonlinear dynamical systems may be a particularlyinteresting property for economics. There seems to be no doubt that nothing likean eternal law of economic motion exists. Even if one believes that aspects of themotion of an actual economy can be approximated with the help of a deterministicsystem, one can hardly deny the eventual invalidity of a given system. The presenceof innovations and structural change can imply changes in the parameters of agiven system, functional forms may vary, or the dimension of a system may changein the course of the emergence of new products and sectors. When the behaviorof the actual economy is modeled with a deterministic system, it may thereforehappen that the actual economy has already changed (and the deterministic modelhas consequently become invalid) before a trajectory of the investigated model hasreached an attractor. Thus, it cannot be excluded that the motion on an attractornever depicts the motion of the actual economy although the dynamical systemrepresented a correct picture of reality in the initial phase. When a dynamicalsystems generates complex transient motion it is possible to encounter irregularbehavior during the time span the considered system is indeed valid. The fact thatcomplex transient motion can emerge in systems with a regular attractor like astable fixed point or an attracting closed orbit lets this type of motion appear evenmore interesting.

The following two sections attempt to illustrate the emergence of such complextransient trajectories in one- and two-dimensional maps. The basic phenomenonunderlying the complex transient behavior is the existence of Cantor sets. In thefollowing examples, these sets act as repellers. Trajectories starting at initial pointsclose to such a set can nevertheless stay in a neighborhood of the set for a rather

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158 Chapter 4

long time and can display chaotic behavior. The emergence of a Cantor set will bedemonstrated with the horseshoe map investigated by Smale (1963).48

4.3.1. Complex Transient Behavior in One-Dimensional Systems

The one-dimensional examples of chaotic motion in Section 4.1 all assumed thepresence of a unimodal map with a graph similar to that of the logistic equation(cf. Figure 4.2). It has always been assumed in these examples that the appropriatemaps have at most one nontrivial fixed point (in addition to the origin). In thefollowing this uniqueness property of the fixed point will be abandoned, and thecase of maps with multiple fixed points will be considered instead.

Figure 4.23

Assume that the graph of a one-dimen-sional map has a shape like the one inFigure 4.23.49 The map still has a singlecritical point xc like logistic maps andrepresents an endomorphism. How-ever, there exist two additional inflec-tion points which are responsible forthe existence of the two additional fixedpoints A and B. The map has beendrawn in such a way that the fixed pointC is unstable, i.e., the slope of the graphat C is absolutely larger than 1. Thefixed point A is obviously stable whilethe fixed point B is unstable. Trajecto-ries starting at initial points to the leftof A converge toward A in a monotonicfashion. The same is true for trajecto-

ries starting in the interval between the fixed points A and B and in the interval(x−1

B , xM). The intervals (0, xB) and (x−1B , xM) represent the immediate basin of

attraction for A. However, almost all remaining initial points in the interval (0, xM)are also attracted by A. The critical point xc is mapped to the point xc2 in the im-mediate basin of attraction within two iterations. Other initial points in the interval(xB, x−1

B ) may require a longer time before they are eventually mapped into theinterval (0, xB). The motion of these initial points can be extremely complicated.In fact, the interval (xB, x−1

B ) contains a strange repeller in the form of a Cantor setΛ.50 Initial points located in this set will stay there forever. Initial points located in

48 Details on the mathematical aspects of complicated transient motion can be found,e.g., in Grebogi/Ott/Yorke (1987a,b,c), Kantz/Grassberger (1985), McDonald/Grebogi/Ott/Yorke (1985a,b), or Nusse/Yorke (1989).

49 Compare Mira (1987) for an intensive discussion of this map.50 Cantor sets will be described in greater detail in Section 4.3.2 (in the form of the invari-

ant set in the horseshoe map) and in Section 6.2.2 (in the form of the Cantor middle-third set).

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4.3. Complex Transients in Discrete-Time Dynamical Systems 159

a neighborhood of the set can initiate the onset of a wild transient motion beforethe trajectory eventually leaves the neighborhood of the set and the initial point isfinally mapped to the interval (0, xB). Summarizing, the basin of attraction of thefixed point A consists of the set B(A) = (0, xM)\Λ.

As an economic example, consider a Walrasian tatonnement in a single market(cf. Walras (1954)). The standard textbook literature presents supply and de-mand functions as monotonically increasing and decreasing functions of the goodsprice, respectively. It has already been mentioned in Section 2.2.3 that a mono-tonicity in individual excess demand functions does not necessarily imply the samemonotonicity in the aggregate supply and demand functions. It is surprising thatnon-monotonic supply functions can not only be found in the literature emergingin the 1970s of this century but that Walras (1954) himself provided a hint forpossibly complicated motion.

Walras’ Tatonnement ExampleFigure 4.24

Walras’s original picture for the de-scription of the supply-demand scenarioin a single market is depicted in Fig-ure 4.24. The downward bended supplycurve (in a Marshallian coordinate sys-tem) reflects a possible inferiority but itcan also be considered the consequenceof an aggregation procedure. The sce-nario depicted in Figure 4.24 was inves-tigated by Day/Pianigiani (1991). Theauthors demonstrated the presence of achaotic attractor and the occurrence ofcomplex motion in the properly spec-ified price dynamics. However, with aslight modification, it can be shown thatthis example also implies the potential

emergence of complex transient motion even if the system possesses a regularfixed-point attractor.

Assume for that reason the supply and demand constellation depicted in Figure4.25. While the monotonicity properties of both functions are the same as thosein Figure 4.24, the bubbles in the functions for high values of p imply multiplefixed points. The excess demand z(p) for varying prices is depicted in the lowerdiagram of Figure 4.25 with the four zero roots A, B, C, and D. Assume a standardtatonnement process, i.e., prices are changed when a non-zero excess demand, z(p),is observed in the market:51

∆pt+1 = pt+1 − pt = ξ(z(pt)

). (4.3.1)

For simplicity, it will be assumed in the following that the function ξ(·) is linear, i.e.,

∆pt+1 = αz(pt), α > 0. (4.3.2)

51 Cf. Section 2.2.3 for another tatonnement process.

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160 Chapter 4

The Demand and Supply Function in Walras’ Example with Additional SlightNonlinearities (Upper Part) and the Excess Demand Function (Lower Part)

Figure 4.25

The map (4.3.2) is shown in Figure 4.26. It possesses the four fixed points A, B, C,and D. Obviously, C represents a stable fixed point while the remaining points A,B, and D are unstable.

Although the graph of (4.3.2) in Figure 4.26 has a mirror-imaged shape as com-pared with Figure 4.23, it generates the same dynamic behavior. The intervals(pB, pD) and (p−1

D , p−1B ) represent the immediate basin of attraction of the fixed

point C. A Cantor set exists in the interval (p−1B , pB), implying that the motion

starting at initial points in this set or in its neighborhood is complicated. As thecritical point pc is mapped out of the interval (p−1

B , pB), this complex motion isobservable only within a (possibly considerably long) limited time span.52

52 Other economic examples implying maps like the one in Figure 4.23 can be found withrelative ease. For example, a variation of Day’s (1982) growth-cycle model can be found

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The Graph of the Map (4.3.2)Figure 4.26

4.3.2. Horseshoes, Homoclinic Orbits, and Complicated Invariant Sets

This section deals with complex transient behavior in two-dimensional, discrete-time systems. The so-called horseshoe map introduced by Smale (1963, 1967) will bedescribed at some length because the construction of the invariant set in this mapis enlightening for the geometry of chaotic dynamical systems.53

A consideration of the horseshoe map is not only interesting for a description ofcomplex transient motion. Some statistical tools for a description of chaotic motion(like Lyapunov exponents, cf. Chapter 6) rely on the stretching and contracting ofsets of initial points which can most clearly be illustrated with the horseshoe map. Inaddition, Poincare maps of continuous-time dynamical systems occasionally possessinvariant sets similar to the invariant set in the horseshoe map (cf. the discussion ofthe Shil’nikov scenario in Chapter 5).

in Lorenz (1992b). However, it should be emphasized that even more so-called ad-hocassumptions may be necessary in order to generate the inflexion points.

53 For details on the following concepts compare Guckenheimer/Holmes (1983), pp.227-267, Lanford (1983), Mees (1981), pp. 51-60, Nitecki (1971), pp. 118-158, orThompson/Stewart (1986), pp. 245-253. A concise collection of the relevant conceptsis contained in Grandmont (1988), pp. 82ff.

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162 Chapter 4

The horseshoe map constitutes a two-dimensional, discrete-time dynamical sys-tem representing a Ck – diffeomorphism. Instead of providing a set of differenceequations, the map will be described verbally. Consider the set of initial pointslocated in the unit square S = [0, 1] × [0, 1] in Figure 4.27.a. Under the actionof a map G these initial points are transformed into a new geometric object repre-senting the location of the initial points after one iteration. The transformation isexecuted in a two-step procedure (cf. Figure 4.27.b):

• The square is horizontally contracted (compressed) by a factorα, and is verticallystretched by a factor β.

• The rectangle [0, α] × [0, β] generated by this contraction and stretching isfolded such that the form of a horseshoe emerges.

Depending on the factors α and β two cases can be distinguished: i) The horseshoeis entirely contained in the area covered by the original square S (in that case themap is either area-preserving or area-contracting), or ii) the intersection of thesquare and the horseshoe is only a subset of the area covered by the horseshoe.Smale (1963) assumed that the folded region of the horseshoe and parts of thehorseshoe’s legs are not mapped to the area covered by the square S. This meansthat a portion of the original square is not mapped to itself by G, i.e., some pointsleave the square under the action of G.

4.27.a 4.27.b 4.27.cThe Construction of a Horseshoe

Figure 4.27

In a second iteration the square in Figure 4.27.b with the two shaded verticalstrips is contracted and stretched to the rectangle in Figure 4.27.c. Folding the

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4.3. Complex Transients in Discrete-Time Dynamical Systems 163

rectangle yields the intersection with the square S shown in the right-most picturein 4.27.c. The two shaded vertical strips in Figure 4.27.b are transformed to 4 verticalstrips in Figure 4.27.c. Again, parts of the two strips in 4.27.b ultimately leave thesquare S. Successive iterations, Gn(S), generate infinitely many bended strips forn→ ∞. The set of points constituting the vertical strips that overlap with S is givenby Gn(S) ∩ S.

G−1(S ∩G(S))

G1(S) ∩ SFigure 4.28

The two vertical strips in 4.27.b are generated by mapping only a part of theoriginal square S to itself. Working backwards from 4.27.b to the original squareshows that the vertical strips correspond to two horizontal strips in S (cf. Figure4.28), i.e., the horizontal strips are given by G−1

(S ∩ G(S)

). The unshaded areas

in the original square are the parts of the emerging horseshoe that do not overlapwith the square. Equivalently, the four vertical strips in Figure 4.27.c are generatedby mapping points in 4 horizontal strips in the pre-image to S (cf. Figure 4.29).

G−2(S ∩G2(S))

G2(S) ∩ SFigure 4.29

After n iterations the set of points representing the horizontal strips is given byG−n

(S∩Gn(S)

)= S∩G−n(S). In order to locate those points inS that will stay inS

forever and those that originated in S in the past, the intersection of the horizontaland vertical strips must be considered. Figure 4.30 depicts this intersection for two

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164 Chapter 4

The Invariant Set in the Horseshoe MapFigure 4.30

forward and backward iterations. For n → ∞, the rectangles in Figure 4.30 shrinkto points. The emerging set of points is an example of a so-called Cantor set.54

Formally, the set of points in Figure 4.30 for n→ ∞ is given by

I =

( ∞⋂n=0

Gn(S)) ⋂ ( ∞⋂

n=0

G−n(S)).

The set I represents the invariant set of the square S for the map G. Points in Ioriginated in I and will stay in I for n→ ∞.

Starting at an arbitrary point in the invariant set I, successive iterations of G cancarry the initial point eventually back to itself. It is also possible that the motion ofa point in the invariant set is completely aperiodic. Smale (1963) proved with thehelp of symbolic dynamics55 that the invariant set in the horseshoe map

• contains a countable set of periodic orbits,• contains an uncountable set of bounded nonperiodic motions,• contains a dense orbit, i.e., there is at least one point in I whose orbit comes

arbitrarily close to every other point in I.

54 Compare also Section 6.2.2. and Figure 6.5 for the construction of a Cantor set. Figures4.27.b and 4.27.c contain horizontal distance lines that correspond to the first lines inFigure 6.5.

55 Symbolic dynamics describe the evolution of a point from n = −∞ to n = ∞ by asequence of symbols like 0 and 1. For example, the symbol 0 may be assigned to pointsin the lower half of Figure 4.30 and the symbol 1 to points in the upper half. A sequence. . . 1001 . . . then means that a point in the upper half is mapped to the lower half inthe first two iterations and returns to the upper half after the third iteration. For thehorseshoe map there exists such a sequence with positions from −∞ to ∞ for everypoint in I, and, vice versa, for every sequence there is exactly one point in I.

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4.3. Complex Transients in Discrete-Time Dynamical Systems 165

The corresponding motion in the original flow is then characterized by the presenceof motion on tori and of irregularly wandering trajectories.

As was mentioned above, Smale’s original horseshoe map does not possess anattracting invariant set. Almost all points in S eventually leave the square. The mapG can then be viewed as a tool for describing transient chaos when the invariant setaffects the behavior of an arbitrary point before it finally leaves the square.

Most importantly, it can be shown that the specific features of the invariant set ofthe horseshoe map arise when so-called transversal homoclinic orbits exist in the map.Consider a saddle-type fixed point x∗ with stable and unstable manifolds W s(x∗)andWu(x∗), respectively. If the stable and unstable manifold intersect transversely(i.e., non-tangential) at another point p, this point is said to be a homoclinic point. Theforward and backward orbit of p is then called a transversal homoclinic orbit. Whena homoclinic point exists, then there are also infinitely many other homoclinicpoints: p lies on W s(x∗), implying that all iterates of p lie on W s(x∗) as well. Butp and its iterates also lie on Wu(x∗). Thus, every iterate of p lies both on W s(x∗)and Wu(x∗), i.e., every iterate of p is a homoclinic point.

A Transversal Homoclinic Orbit for a Map in R2

Figure 4.31

When the fixed point x∗ is approached on the stable manifold, more and moreintersections with the unstable manifold occur. This implies that the unstable man-ifold winds in a wild manner around W s(x∗). Equivalently, W s(x∗) winds wildlyaround the unstable manifold when x∗ is approached on the backward orbit (cf.Figure 4.31).

When transversal homoclinic orbits exist, the behavior of initial points that arenot located in this orbit can be extremely complex. It follows from the Smale-Birkhoff homoclinic theorem56 that, when such orbits are present in a map g, theinvariant set of g is topologically equivalent to the invariant set in the horseshoemap, i.e., the above mentioned properties of the horseshoe map apply to the mapg as well.

56 Cf. Smale (1967), p. 29.

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166 Chapter 4

Y

Time

The Transient Motion in the Kaldorian Model (4.2.5); α = 21.0Figure 4.32

It is difficult to locate homoclinic orbits in specific dynamical systems. Thehorseshoe map is nevertheless important, because it uncovers that a possibly com-plex transient motion cannot be excluded per se in discrete-time, two-dimensionalsystems. Figure 4.32 shows the result of a numerical simulation of the Kaldor-typemodel described in Section 4.2.2. The time series has been obtained from the sameparameter set, but the adjustment coefficient α is lower than before (α = 21.0).The attractor of the system for this parameter set is a regular period-80 attractor.For a relatively long time span the transient motion is remarkably complex. Whenthe largest Lyapunov exponent (cf. Section 6.2.4) is calculated for the first 350 pe-riods it turns out that it is positive. Thus, the transient time series behaves likea chaotic time series although the trajectory eventually settles down on a regularattractor (with a negative largest exponent).

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Chapter 5

Chaotic Dynamics in Continuous-Time

Economic Models

Most existing economic models dealing with the chaos property are discrete-time models and can be reduced to a one-dimensional dynamical system.

The main reason for this concentration on one-dimensional systems can probablybe found in the relative ease with which chaotic motion can be established in thesesystems and because the two-dimensional case is already much more difficult tohandle. However, chaos does not occur only in discrete-time models, but may be aproperty of continuous-time models as well.

5.1. Basic Ideas

In the one-dimensional, discrete-time case, chaos according to the Li/Yorke defini-tion is characterized by the simultaneous presence of multiple periodic and aperi-odic orbits. The sequence of points may jump irregularly in the appropriate intervalon which the map is defined. Equivalently, in higher-dimensional, discrete-time sys-tems the sequence of points can jump irregularly in the space of the variables. As thetime step underlying the motion of the state variables in differential equations is in-finitely small, the evolution is smooth and the jumps typical for discrete-time systemsusually do not occur.1 While chaotic motion in discrete-time systems is described by

1 Compare, however, Section 2.5 and Chapter 7 dealing with relaxation oscillations andcatastrophe-theoretic models, respectively, for approaches that attempt to illustrate the pos-sibility of discontinuous jumps in continuous-time systems.

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a sequence of seemingly arbitrarily jumping points, chaos in a continuous-time dy-namical system appears as the irregular wandering of the entire trajectory in phasespace.

One of the most prominent, chaotic, continuous-time, dynamical systems is theLorenz system, named after the meteorologist E.N. Lorenz who investigated thethree-dimensional, continuous-time system

x = s(−x+ y),y = rx− y − xz,

z = −bz + xy,

s, r, b > 0, (5.1.1)

emerging in the study of turbulences in fluids. For r above the critical value r =28.0,the trajectories of (5.1.1) evolve in a rather unexpected way. Suppose a trajectorystarts at an initial value in the center of the left wing in Figure 5.1. For some timethe trajectory regularly spirals toward the outer region of that wing. However, thetrajectory eventually leaves the left wing, wanders to the center of the right wing,and starts spiraling outwards again. When the trajectory has reached a region far

The Lorenz Attractor; s = 10, r = 28, b = 2.66Figure 5.1

enough away from the center, it again wanders toward the left wing and the storyrepeats. As the trajectory does not necessarily have to pass the initial starting point,the trajectory in this second round can differ completely from that in the first round:the trajectory may wander through different points in phase space and may need alonger time before it turns toward the second wing. When the time horizon is longenough, both wings will densely be filled by the trajectory.

Note however, that the two wings do not exist isolated from the motion itself.While regular objects like fixed points or limit cycles are defined independent of

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5.1. Basic Ideas 169

the specific trajectories which converge toward these objects from different initialpoints, the precise location of the trajectory in Figure 5.1 depends on the choseninitial point. It is the evolution of the system that generates the geometric shapeshown in the figure.2 Different initial points therefore imply different trajectoriesbut the shape of the object in Figure 5.1 remains unchanged. As trajectories startingat different initial values in a neighborhood of the object all converge to and remainin the region with the two wings, the region is an attractor . It is a strange attractor be-cause it is neither a point nor a closed curve (including complicated closed curves).The notion of a strange attractor was introduced by Ruelle/Takens in 1971.

The geometric shape of Figure 5.1 is astonishing because the nonlinearities in(5.1.1) are relatively weak as compared with other quadratic or higher-order firstderivatives. A strange attractor with even weaker nonlinearities is the Rossler attractorshown in Figure 5.2, the underlying differential equation system of which is

x = −(y + z),y = x+ ay,

z = b+ z(x− c),a, b, c > 0. (5.1.2)

The Rossler Attractor; a = 0.2, b = 0.2, c = 5.7Figure 5.2

While there is no common agreement on the “strangeness” of a strange attractor,the following definition summarizes the verbal description given above:

2 The notion of the trajectory’s wandering on a wing is therefore only used for illustrativepurposes.

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Definition 5.1 (Ruelle (1979)): Consider the n-dimensional dynam-ical system

x = f(x, µ), x ∈ Rn, µ ∈ R (5.1.3)

with µ as a parameter. A bounded set A ⊂ Rn is a strange attractor for(5.1.3) if there is a set U with the following properties:i) U is an n-dimensional neighborhood of A.ii) A is an attracting set in the sense of Definition 2.1.iii) There is a sensitive dependence on initial conditions when x(0) is

in U, i.e., small variations in the initial value x(0) lead to essentiallydifferent time paths of the system after a short time.

iv) The attractor is indecomposable, i.e., it cannot be split into two ormore separate pieces.

In the following, chaos in continuous-time dynamical systems will be identified withthe existence of a strange attractor:3

Definition 5.2: A dynamical system (5.1.3) is chaotic if it possesses astrange attractor in the sense of Definition 5.1.

When a continuous-time dynamical system possesses a strange attractor and gener-ates chaotic motion it has to be kept in mind, however, that for a sufficiently shorttime interval a chaotic trajectory in a continuous-time system seems to behave reg-ularly with a smooth evolution of the variables over time. The irregularity in thesesystems appears in the emergence of a sequence of cycles with different amplitudesand frequencies.

While chaotic dynamics in discrete-time systems can already occur in one-dimen-sional systems like the logistic equation, the equivalent phenomenon in continuoustime can emerge only in at least three-dimensional systems. Canonically, chaos can-not occur in two-dimensional, continuous-time systems because a trajectory cannotintersect itself. The most complex type of motion that can arise in two-dimensionalsystems is a motion in a closed orbit, a homoclinic orbit, or the convergence of thetrajectory toward these limit sets.

A very useful concept in descriptions of the dynamic behavior of continuous-time dynamical systems are so-called Poincare sections and maps. These maps canalso be used for an illustration why chaotic motion cannot occur in two-dimensionalsystems.

Consider first the trajectory of a planar continuous-time system and suppose thatthe system converges toward a closed orbit as shown in Figure 5.3.a. Draw a straightline, Σ, through the trajectory and mark the points of intersection of the line withthe trajectory every time the trajectory crosses the line in the same direction. Theset of all marked points is called the Poincare section. Denoting the first point of

3 The expressions “strange attractor” and “chaotic attractor” are thus treated synonymous-ly. Compare, however, Grebogi/Pelikan/Ott/Yorke (1984) for examples where adistinction of the two concepts is appropriate.

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5.1. Basic Ideas 171

intersection as y1, the next as y2, etc., a sequence of points {yi} is constructeddepending on the motion in the differential equation system: given a certain yi, thepoint yi+1 is determined as well, provided the solution of the differential equationis known. As the points yi ∈ R2 are all located on the (one-dimensional) line Σ,they can be described by points Yi ∈ R. The sequence {yi}mi=1 can therefore bedescribed by a 1D map P : R → R, which maps Yi to Yi+1 according to the motionin the vector field. The map P is the Poincare map of the continuous-time dynamicalsystem.

5.3.a: n = 2 5.3.b: n = 3Poincare Sections of a Continuous-Time Dynamical System

Figure 5.3

The interesting property of this Poincare map consists in the fact that the mapprovides complete information on the qualitative behavior of the original differen-tial equation though the map has a dimension of only n − 1. In Figure 5.3.a, theconvergence toward a limit cycle in the differential equation system is representedby a converging sequence of points toward a fixed point in the Poincare map. Equiv-alently, if the trajectory of the differential equation system describes a closed orbit,the Poincare map consists of a single point which is not the stationary equilibrium.Obviously, a planar dynamical system can only have Poincare maps exhibiting sta-tionary fixed points or monotonically increasing or decreasing sequences of points{Yi}. Suppose that the sequence of points belonging to the Poincare section of aplanar differential equation system are located on a line like the one in Figure 5.4.The dynamic behavior of this map can be analyzed with the help of the method em-ployed in Chapter 4, i.e., the 450 line can be used to demonstrate the evolution ofYi. As the graph in Figure 5.4 is (necessarily monotonically) increasing with a slopeless than 1 at the fixed point, the sequence {Yi} converges toward this fixed pointregardless of the initial value of Y . According to the principles in constructing themap, the differential equation system is therefore characterized by convergence to-ward a stable limit cycle. Alternatively, a diverging sequence in the Poincare mapcorresponds to an unstable cycle.

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172 Chapter 5

The Dynamic Behavior in a One-Dimensional Poincare MapFigure 5.4

-30 -20 -10 0 10 20 30

30

20

10

-10

-20

-30

y

xC

C ′

The Two-Dimensional Poincare Map of the Lorenz Attractor,z = constant. Source: Berge et al. (1986), p. 126 (Re-drawn from the Original).

Figure 5.5

Consider next a three-dimensional system whose trajectory forms the spiralingcurve in Figure 5.3.b. The Poincare section is generated by laying a two-dimensionalplane Σ through the trajectory. The sequence of the three-dimensional points ofintersection generates a two-dimensional mapping in a way similar to the abovementioned procedure. For the case of the Lorenz attractor, this two-dimensionalPoincare map is illustrated in Figure 5.5. The map seems to consist of two separatesegments, each corresponding to a separate spiraling motion around one of the two(unstable) fixed points C and C′. While this view of the two-dimensional Poincaremap does not provide essential new insights into the character of the underlyingdifferential equation system, the inspection of only one variable in the Poincaremap indicates the presence of complex behavior in the three-dimensional Lorenzsystem. The first return map is defined as the sequence {xji}mi=1 of a single variablexj , j = 1, 2, 3 on a Poincare section.

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5.1. Basic Ideas 173

zmaxi+1

zmaxi

A First Return Map of the Lorenz Attractor; z = 0Figure 5.6

It is useful to study the first return map on the Poincare section generated bythe surface on which one of the variables does not change, i.e., where it reaches alocal extremum. The first return map for the variable z of the Lorenz equations isshown in Figure 5.6. The Poincare section was created by the surface with z = 0. Asz changes its direction on this surface, the map shown in Figure 5.6 thus describesthe successive extremal values (the maximal values actually) of the coordinate z inthe attractor: let zi be the maximum value of z the first time the attractor performsa cyclical round, then zi+1 will be the maximal z value in the next round.

The first return map in Figure 5.6 is a noninvertible, one-dimensional map andcan therefore be studied by means of the techniques presented in Section 4.1.1.The slope of the fictitious curve on which the observed pairs (zi, zi+1) are located isabsolutely larger than one at the point of intersection with the 450 line, indicatingpossibly complex behavior. As all realized points of the first return map nearly forma continuous curve, it is likely that chaos is present in this map. When chaos prevailsin this first return map, then the behavior of the original flow is also characterizedby irregular motion, i.e., the orbits in the flow erratically change their diameter inthe z direction.

It should be noted, however, that chaos in continuous-time dynamical systemscannot be established via general and simultaneously simple characteristics of thesesystems like, e.g., the Li/Yorke criterion in one-dimensional, discrete-time equa-tions. During the last decades, a variety of higher-dimensional systems belonging todifferent families has been investigated proving the presence of a strange attractor,4

but it is not always clear whether the diverse examples possess common (possibly

4 A summary of known chaotic dynamical systems can be found in Garrido/Simo (1983).

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174 Chapter 5

hidden) structural properties. In the following sections, two classes of dynamicalsystems will be presented together with economic applications which are fairly wellunderstood in the dynamical systems literature, namely coupled oscillator systemsand forced oscillators. The last section deals with the Shil’nikov scenario and thepresence of horseshoes in the Poincare maps of the underlying continuous-timesystem. A modified version of Metzler’s business-cycle model with inventories ispresented as an economic example of spiral-type attractors.

5.2. The Coupling of Oscillators

Nonlinear oscillators have already been discussed in Chapter 2. The economicrelevance of these oscillators was illustrated with examples from macroeconomicsor price and quantity adjustments in a single market. A common property of theseexamples can be found in the implicit assumption that the motion of the relevantvariables does not depend on exogenous influences in an essential way. It is, ofcourse, possible to study the effects of varying parameters but the influence ofpermanently changing exogenous variables (i.e., variables which do not belong tothe considered system) has not been taken into account yet. These influencescan be investigated by modelling the remaining, previously exogenous parts of theeconomy in an explicit manner and by emphasizing the links between the differentparts. Alternatively, a general system in which all possible variables are treatedsimultaneously can be split into sub-systems linked together by coupling terms.

A dynamical system can thus be understood as a set of sub-systems. The partial-analytic view dominating the examples in the previous sections results when nodynamic coupling of the sub-systems takes place. Interdependencies between thesub-systems are then interpreted as coupling effects. A system of coupled oscillatorsemerges when the sub-systems generate endogenous fluctuations in the absence ofcoupling effects.

5.2.1. Toroidal Motion

Consider a set of two independent, two-dimensional, nonlinear oscillators, i.e., dy-namical systems generating endogenous fluctuations:

x = f1(x),

y = f2(y),x, y ∈ R

2, (5.2.1)

and let both oscillators represent dissipative systems.5

In (5.2.1) the motion in each of the two oscillators depends entirely on the valueof the variables xi, i = 1, 2, and yi, i = 1, 2, respectively, in the isolated oscillators.

5 The alternative consideration of conservative dynamical systems (cf. Section 2.4.1) willbe neglected in the rest of the book because those systems do not seem to be genericin economics.

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5.2. The Coupling of Oscillators 175

The limit cycles generated by these oscillators are one-dimensional geometric ob-jects in the plane. Assume that these limit cycles are equivalent to the unit circleS1, i.e., a circle with radius r = 1.

Although both oscillators are independent it is useful to consider the joint mo-tion of the variables.6 This motion of the four variables (x1, x2) and (y1, y2) in(5.2.1) takes place on an object which is the product of the two limit cycles, namelyS1×S1. This geometric object in four-dimensional space is called a two-dimensionaltorus. Heuristically, a motion on a torus may be characterized by an oscillation in thehorizontal direction and another one in the vertical direction. As (for obvious rea-sons) it is difficult to present objects in four-dimensional space, Figure 5.7.a depictsa two-dimensional torus with different radii of the motion in three-dimensionalspace. Figure 5.7.b illustrates the two basic directions of the motion. The figuredemonstrates that a “horizontal” and a “vertical” cyclical component are involved inthe motion on the torus. The torus in Figure 5.7.a is a two-dimensional object be-cause it can be constructed from a two-dimensional plane by appropriate bendingand gluing.

5.7.a: The Motion on the Surface 5.7.b: The Directions of the MotionA Two-Dimensional Torus with its Cyclical Components

Figure 5.7

A trajectory on a torus may form simple closed curves as well as rather compli-cated ones. Assume a system like (5.2.1) and consider an initial point located on thesurface of a two-dimensional torus. Denote the frequencies involved in the motionof each of the two independent oscillators as ωi, i = 1, 2. The following types ofoscillatory motion on the torus can be distinguished:

• Both separate motions describe a closed curve within the same time interval,i.e., when the two cycles are completed the system has reached the initial pointagain. The frequencies ω1 and ω2 are identical.

• One of the oscillators describes a closed curve faster than the other but the ratioof the involved frequencies is a rational number, for example ω1/ω2 = 2. In that

6 It may be argued that the geometric complexity is of no economic interest since theoscillators are independent. However, if the two oscillators describe the actual valuesof variables like, for example, unemployment and prices, both variables will surely beconsidered simultaneously because of several macroeconomic reasons.

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176 Chapter 5

case the first oscillator generates two complete cycles while a single closed orbitis described in the second oscillator. The system passes the initial point on thetorus after two complete cycles of the first oscillator.

• Again, one of the oscillators generates a faster motion, but the ratio of the in-volved frequencies is irrational, e.g., ω1/ω2 = π. In that case the trajectory onthe torus will never meet its initial point again. Instead, the entire surface of thetorus will eventually be covered by the trajectory. The motion is then said to bequasi-periodic.

5.8.a.: Rational Frequency Ratio 5.8.b.: Irrational Frequency RatioTrajectories on a Two-Dimensional Torus (Projections)

Figure 5.8

Figures 5.8.a and 5.8.b contain two-dimensional illustrations of the motion on atwo-dimensional torus for a set of parameter values such that the ratio of the fre-quencies is rational (cf. 5.8.a) and irrational (cf. 5.8.b). The figures contain planarrepresentations of a two-dimensional torus. The torus is constructed from the planeby rolling the planar sheet and gluing together the upper and lower edges. The leftand right edges of the resulting tube are then connected in a similar way. Accord-ing to this construction, a trajectory reaching, for example, the upper edge of theplane reappears on the lower edge. While a trajectory returns to the starting pointafter one or more orbits in the rational case, the trajectory in Figure 5.8.b never lieson a closed curve.7

7 Cf. Haken (1983b), pp. 28f., for details. Note that the planes in Figure 5.7 are stylizedpictures. The statement is true (for the figure) when the trajectory starting at the origindoes not return to this point.

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5.2. The Coupling of Oscillators 177

A numerically precise two-dimensional torus is shown in Figures 5.9.a and 5.9.b.8

Figure 5.9.a shows the trajectory after 5000 Iterations. When a longer time hori-zon with 20000 iterations is considered the torus will more densely be covered bythe trajectory (cf. Figure 5.9.b). In spite of its complexity the trajectory on thetorus behaves regularly in the sense that there is no sensitive dependence on initialconditions.

A Quasiperiodic Motion on a Two-Dimensional Torus; 5000 Iterations; ∆t = 0.1Figure 5.9.a

A Quasiperiodic Motion on a Two-Dimensional Torus; 20000 Iterations; ∆t = 0.1Figure 5.9.b

8 The underlying three-dimensional system consists of the differential equations

x = (a− b)x− cy + x(z + d(1.0 − z2)

)y = cx + (a− b)y + y

(z + d(1.0 − z2)

)z = az − (x2 + y2 + z2).

The parameter values are a = 2.105, b = 3.0, c = 0.25, and d = 0.2. Cf. Langford(1985) for details on this dynamical system. The time step in the numerical simulationis 0.1 time units. For the present system this relatively large value is not problematic(the motion is very slow) but necessary in order to generate a sufficiently long timeseries.

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178 Chapter 5

Suppose now that the motion of the variables in both oscillators depends alsoon the motion in the other oscillator, i.e.,

x = f1(x, y),

y = f2(x, y),, x, y ∈ R

2, (5.2.2)

or, in other words, that the two two-dimensional oscillators are coupled. The generalcase of m coupled, -dimensional oscillators can be written as

x1 = f1(x1, x2, . . . , xh, . . . , xm, µ),...

xh = fh(x1, x2, . . . , xh, . . . , xm, µ),...

xm = fm(x1, x2, . . . , xh, . . . , xm, µ),

xh ∈ R�, µ ∈ R, (5.2.3)

with µ as a parameter.9 Assume that (5.2.3) possesses complex conjugate eigenval-ues and that for low values of the parameter µ the attractor of the system is a fixedpoint. By increasing the parameter, a Hopf bifurcation10 may occur, i.e., a pairof complex conjugate eigenvalues becomes purely imaginary and a closed orbitemerges in a neighborhood of the fixed point.

A further increase in the parameter µ may generate a second Hopf bifurcation.In that case the former limit cycle bifurcates into a two-dimensional torus. Analyti-cally, this second bifurcation can be determined only in special cases: the first Hopfbifurcation makes use of the Jacobian evaluated at the fixed point, i.e., the entriesof the matrix are constants. However, in the case of a limit cycle, the entries ofthe Jacobian have to be evaluated along the cycle, i.e., the Jacobian becomes time-dependent. It must therefore be assured that another pair of eigenvalues becomespurely imaginary independent of the location of the system on the cycle. For thesake of simplicity, assume that such a second Hopf bifurcation indeed takes place.

Fixed Point =⇒ T1 =⇒ T2 =⇒ T3 =⇒ · · · =⇒ Tn

↑ ↑ ↑ ↑(1st Hopf) (2nd Hopf) (3rd Hopf) · · · (nth Hopf)

The Landau Scenario for the Onset of TurbulencesTable 5.1

9 Equation (5.2.2) is then the special case of (5.2.3) with � = 2 and m = 2.10 Cf. Section 3.2.2.

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Provided that the dimension of the dynamical system is large enough, furtherbifurcations may lead to the emergence of a three-dimensional torus, T3, a four-dimensional torus, T4, etc. Every bifurcation increases the complexity of the mo-tion. This scenario long served as the standard model for the onset of turbulences.A low-dimensional system can involve only a few Hopf bifurcations, and the com-plexity of the motion is limited. If a higher degree of complexity is to be modeled,more variables (degrees of freedom) must be included in the analysis so that morebifurcations can occur. In the limit, a system with an infinite number of variablesundergoing a large number of bifurcations resembles a random process which isconsidered to be the appropriate description of turbulence. Table 5.1 schematicallydescribes this so-called Landau scenario for the onset of turbulence.

A major drawback of this scenario is the fact that even after a large number ofbifurcations the motion is not sensitive to initial conditions. Initial points whichare close together will stay close together as time elapses. This regularity aspectobviously contradicts the intuitive notion of turbulence.11

Another possibility for the onset of turbulence was proposed by Ruelle/Takensin 1971. Instead of a very large number of bifurcations as a prerequisite for the onsetof turbulent behavior, the Ruelle/Takens scenario implies that already after threeHopf bifurcations the motion can become chaotic.

Theorem 5.1 (Newhouse/Ruelle/Takens(1978)):Let x = (x1, . . . , xm) be a constant vector field on the torus Tm.• If m = 3, in every C2 neighborhood of x there exists an open vector

field with a strange attractor.• If m ≥ 4, in every C∞ neighborhood of x there exists an open vector

field with a strange attractor.

When the dimension of the dynamical system is high enough and when the motiontakes place on an at least three-dimensional torus (for example, via three successiveHopf bifurcations) then there may exist a strange attractor in the neighborhood of

Fixed Point =⇒ T1 =⇒ T2 =⇒ Chaos↑ ↑ ↑

(1st Hopf) (2nd Hopf) (3rd Hopf)

The Ruelle/Takens ScenarioTable 5.2

11 In fact, the Landau scenario could not be observed experimentally in the natural sci-ences. The successive emergence of higher-dimensional tori would imply the emergenceof an increasing number of incommensurate frequencies in the associated power spec-tra (cf. Section 6.1). However, only a few dominant frequencies together with linearcombinations could be observed in, e.g., fluid dynamics laboratory experiments. Cf.Berge et al. (1986), pp. 165ff., for details.

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180 Chapter 5

the torus. If the system is slightly perturbed, it may not move on the torus anymore,but initial points may instead be attracted by a strange attractor, i.e., the motionbecomes chaotic. This Ruelle/Takens scenario is schematically described in Table5.2.

5.2.2. International Trade as the Coupling of Oscillators

Consider the following simple dynamic IS - LM model as the starting point for anexample of a coupled oscillator system.12 Let Y denote income, r is the interestrate, andM describes the (constant) nominal money supply. Assume that the pricelevel, p, is fixed during the relevant time interval. Suppose that gross investment, I,depends on income in the sigmoid Kaldorian fashion and on the interest rate, i.e.,

I = I(Y, r), IY > 0, Ir < 0.

Savings depends on income and the interest rate:

S = S(Y, r), 0 < SY ≤ 1, Sr > 0.

Income adjusts when a positive or negative excess demand prevails in the goodsmarket, i.e.,

Y = α(I − S), α > 0. (5.2.4)

The liquidity preference, L(Y, r), depends on Y and r in the usual way, i.e.,LY > 0, Lr < 0. Assume that the interest rate adjusts according to

r = β(L(Y, r) −M/p), β > 0. (5.2.5)

Let (Y ∗, r∗) be the unique fixed point of the system and assume that it is unstable.Assume further that equations (5.2.4) and (5.2.5) constitute a nonlinear oscillatorsuch that the model generates endogenous fluctuations.13

12 The following example is adopted from Lorenz (1987a) and relies on a model origi-nally studied by Torre (1977) in the context of bifurcation theory. Another economicexample of coupled oscillator systems in the context of international trade can be foundin Puu (1987). Note that the adjustment equation for the interest rate is not unprob-lematic. The interest rate is determined on the bonds market, and the assumed formof its adjustment equation implies that the excess supply of bonds equals the excessdemand for money. However, it remains unclear how possible excess demands in thegoods market are financed. In a properly specified model this excess demand shouldinfluence the excess supply of bonds. Cf. Lorenz (1993b) for an example.

13 As was demonstrated in Chapters 2 and 3, it is easy to specify the functions I, S, or Lsuch that the requirements of the Poincare/Bendixson theorem or the Hopf bifurcationtheorem are fulfilled.

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Consider now three economies, each of which is described by equations like(5.2.4)-(5.2.5) with possibly different numerical specifications of the functions, i.e.,

Yi = αi

(Ii(Yi, ri) − Si(Yi, ri)

),

ri = βi(Li(Yi, ri) −Mi/pi

),

i = 1, 2, 3. (5.2.6)

The equation system (5.2.6) constitutes a six-dimensional differential equation sys-tem which can also be written as a system of three independent, two-dimensionallimit-cycle oscillators. If all three economies are indeed oscillating, the overall mo-tion of system (5.2.6) constitutes a motion on a three-dimensional torus T3.

International trade is introduced by assuming standard export and import func-tions Exi = Exi(Yj , Yk), i �= j, k and Imi = Imi(Yi), respectively. In addition tothe change in the excess demand for goods, the onset of international trade canimply a change in the money stock in country i if Exi �= Imi. In the following itwill be assumed that a trade-balance disequilibrium leads to an immediate changein the money stock.14 The resulting nine-dimensional system is

Yi = αi

(Ii(Yi, ri) − Si(Yi, ri) + Exi(Yj , Yk) − Imi(Yi)

),

ri = βi(Li(Yi, ri) −Mi/pi

),

Mi = Exi(Yj , Yk) − Imi(Yi),

(5.2.7)

with i, j, k = 1, 2, 3, j, k �= i. Equation system (5.2.7) constitutes a system of couplednonlinear oscillators which can be understood as a perturbation of the motion of theautonomous economies on a three-dimensional torus. The Newhouse/Ruelle/Takens(1978) theorem therefore implies that the international trade system (5.2.7)may possess a strange attractor.

The result of a numerical simulation of the dynamic behavior of system (5.2.7)is depicted in Figure 5.10. The three plots contain projections of the system’s at-tractor on the Yi−ri spaces of the three economies. The motion is chaotic in thesense of a positive largest Lyapunov (cf. Section 6.2.4 for details). The parametersassumed in the simulation imply that the economies i = 1 and i = 2 possess stablefoci in the autarkic case. The economy i = 3 displays a limit cycle behavior in theautarkic case. This scenario can be considered as a hint that it may not be necessaryto encounter closed orbits in all uncoupled sub-systems before the coupling intro-duces a complexity to the system. Simulating a system like (5.2.7) also uncoversthat complex attractors emerge rather incidentally; most simulation runs result indeformed closed orbits.15

14 Alternatively, it can be assumed that the central bank attempts to hold the money stockconstant in the case Exi �= Imi. The only way to achieve this goal in this simple modelis to offer bonds in the bonds market. The trade-balance deficit/surplus, Exi − Imi,then has to be considered in the interest rate adjustment equation.

15 This is partly due to the specification of (5.2.7). The discussion of the uniqueness oflimit cycles in Section 2.3 has shown that standard, two-dimensional oscillator systemslike the Lienard equation (which are usually assumed in discussions of coupled oscillator

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182 Chapter 5

r1 r2 r3

Y1 Y2 Y3

An Example of Chaotic Motion in the Coupled System (5.2.7) with 3 Countries;Ii = 8 arctan(0.025Yi) − 10ri, Si = 0.15Yi, Li = 0.1Yi − 15ri, Exi = Exij +

Exik, i �= j, k, Exij = 0.0001ξijY3j , i �= j, Imij = Exji, ξ12 = 3, ξ13 = 3,

ξ21 = 4, ξ23 = 3, ξ31 = 4, ξ32 = 3.Figure 5.10

This procedure of coupling autonomous oscillators can be applied to a varietyof different economic problems. A first investigation of the influence of dynamiccoupling was presented by Goodwin (1947) in a model describing the interde-pendence of markets. Larsen/Mosekilde/Rasmussen/Sterman (1988), Mose-kilde/Larsen/Sterman/Thomsen (1992) and Sterman/Mosekilde (1993) stud-ied frequency-locking behavior in a long-wave business cycle model. A multisector,Kaldorian-type business-cycle model with a structure essentially identical with theinternational trade model presented above was studied by Lorenz (1987b). If thecoupling between three different sectors of an economy takes place via the demandfor investment goods delivered from other sectors, and if the coupling is unidirec-tional, i.e., if a sector i receives goods from a sector j, but delivers goods only tosectors h �= j, which are closer to the final demand, then a strange attractor cannumerically be shown to exist.

5.3. The Forced Oscillator

In a series of papers, Cartwright/Littlewood (1945), Cartwright/Reuter(1987), Levinson (1943a,b, 1949), and Littlewood (1957a,b) demonstrated thatthe introduction of dynamic forcing in the van-der-Pol equation can involve a kindof dynamic behavior which at that time was assigned to stochastic dynamical systemsalone. In fact, these post-war studies laid the foundation for the introduction of thehorseshoe map by Smale (1963, 1967). Recent geometric methods in the study of non-linear dynamical systems have revived the interest in forced oscillator systems (e.g.,Abraham/Scott (1985), Levi (1981), Guckenheimer/Holmes (1983), Tomita(1986)).

systems) do not occur very often in economic dynamics. For example, the IS-LM system(5.2.6) possesses the structure of a Lienard equation only if ∂Li/∂ri = 0 or if all threefunctions I, S, and L are separable in their arguments.

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5.3. The Forced Oscillator 183

While forced oscillator systems naturally emerge in theoretical investigations ofseveral physical and technical devices, economic examples for this special familyof functions have only rarely been provided. The main reason for this deficiencymay lie in the fact that the necessary periodicity of the dynamic forcing may not beobvious in most economic applications.16

In the following, two economic examples of forced oscillator systems will be pre-sented. After a short and more or less heuristic introduction to the mathematics offorced and unforced oscillator systems it will be shown that Goodwin’s nonlinearaccelerator model with autonomous investment outlays as well as a simple macroe-conomic demand-stabilization model can imply the existence of a forced oscillatorsystem.

5.3.1. Forced Oscillator Systems and Chaotic Motion

Consider a nonlinear, autonomous, second-order differential equation

x+ f(x)x+ g(x) = 0. (5.3.1)

Recall from Section 2.3 that equation (5.3.1) is able to generate endogenous oscil-lations if the functions f(x) and g(x) fulfill certain requirements. For example, iff(x) is an even function with positive second derivative and two zero roots, and ifg(x) is an odd function with positive first derivative, the equation possesses a uniquelimit cycle.

The autonomous equation (5.3.1) is a special case of the more general form

x+ f(x)x+ g(x) = h(t), (5.3.2)

with h(t) as a periodic function, i.e., h(t) = h(t+∆t) ∀t. As time enters the equa-tion in an explicit manner, (5.3.2) is called a nonautonomous differential equation.Equation (5.3.2) is called a forced oscillator when f(x) and g(x) fulfill the require-ments of an oscillator. If the amplitude of the forcing term h(t) is small relative tothe dampening term f(x), (5.3.2) is called a weakly forced oscillator . Otherwise, theoscillator is called a strongly forced oscillator.

While the weakly forced oscillator does not add essentially new qualitative prop-erties to the dynamic behavior of (5.3.1) – in fact, the oscillator is still characterizedby a limit cycle behavior – the strongly forced oscillator may involve the emergenceof irregular dynamics. Consider the following special form of equation (5.3.2), i.e.,a dynamically forced equation of the van-der-Pol type:17

x− a(1 − x2)x+ x3 = a cosωt, (5.3.3)

16 Cf., however, Samuelson (1947), pp. 335ff., for an early discussion of the role of exoge-nous forcing in dynamic economic models.

17 In the original van-der-Pol equation the cubic term of (5.3.3) is replaced by g(x) = x.

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184 Chapter 5

x(t)

x(t)−4 4−8.1

8.2

The Motion in a Forced Oscillator System: x− 0.1(1 − x2)x + x3 = 10 cos(t)Figure 5.11

with a determining the amplitude of the forcing term and ω influencing its fre-quency.18

The result of a numerical simulation of (5.3.3) is shown in Figure 5.11. Theobject represents a chaotic attractor with a positive largest Lyapunov exponent (cf.Section 6.2.4 for details). Similar attractors can be generated when different oddterms g(x) are assumed in (5.3.2).

Forced oscillator systems do not only possess chaotic attractors but can generatecomplex transient behavior even if the attractor is a regular object. The follow-ing heuristic argument attempts to explain the reason for this complex transientmotion. When a is large, the dynamic behavior of (5.3.3) can be described by aone-dimensional geometric approximation of the involved Poincare map, whichwill be called the Levi-Poincare map in the following.19

18 System (5.3.3) can be interpreted as a three-dimensional system when t is considered astate variable with t = 1, i.e., the system (5.3.3) can be written as

x = y, y = a(1 − x2)y − x3 + a cosωt, t = 1.

Writing (5.3.3) in this three-dimensional form uncovers that the system can be a can-didate for chaotic dynamics although only two state variables seem to be involved in(5.3.3).

19 For details on the construction of the Levi-Poincare map see Levi (1981) and Gucken-heimer/Holmes (1983).

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5.3. The Forced Oscillator 185

The Levi-Poincare Map The Region B-C of the MapFigure 5.12 Figure 5.13

Figure 5.12 shows the Levi-Poincare map of equation (5.3.3). According to theconstruction of Poincare maps, a fixed point of the map corresponds to a closedorbit in the original flow. The four fixed points A through D in Figure 5.12 there-fore represent four closed orbits of equation (5.3.3). As the absolute slopes of thegraph of the Levi-Poincare map are smaller than 1 at the fixed points A and D, thecorresponding closed orbits in the flow are stable. Equivalently, the unstable fixedpoints B and C represent unstable closed orbits.

The existence of two stable closed orbits implies that the dynamic behavior of(5.3.3) depends on the initial conditions. If a trajectory starts at an initial point inphase space corresponding to a point to the left of B or to the right of C in the Levi-Poincare map, the trajectory will converge toward the closed orbit correspondingto points A or D, respectively. As is the case in all dynamical systems with more thanone limit cycle, the initial point therefore determines the final state of the system.

In contrast to dynamical systems exhibiting multiple limit cycles with alterna-tively stable and unstable orbits the forced oscillator allows for a more complicateddynamic behavior. When the initial point of the system is located to the right of Band to the left of C, a sequence of points in the Levi-Poincare map will obviouslyapproach neither B nor C because of their instability. In order to get an intuitiveunderstanding of the dynamic behavior, consider an enlargement of the region B-C(cf. Figure 5.13).

It is possible to find initial values in this region of the Levi-Poincare map whichgenerate a period-three cycle, namely

β3 < β2 < β1 < β4. (5.3.4)

As is well-known from the theory of one-dimensional maps, the existence of aperiod-three cycle implies the existence of chaotic motion in this map. The samequalitative property persists in the original Poincare map of which the Levi map isan approximation. As chaotic motion in a Poincare map implies irregular behaviorof the underlying flow as well, the essentially three-dimensional differential equa-

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186 Chapter 5

tion (5.3.3) is therefore characterized by chaotic motion as long as its trajectory islocated in a region corresponding to B-C in the Levi-Poincare map.

It is obvious from Figure 5.12 that the region B-C is not a trapping region. Thesystem can leave the region and will eventually converge to one of the two stablefixed points A or D. The possibly complex motion in the interval B-C is then anexample of transient chaos (cf. Section 4.3 for details) .

Whether a specific forced oscillator system possesses a strange attractor or a reg-ular attractor like a closed orbit with possibly complex transient motion (as demon-strated with the help of the Levi-Poincare map) depends on the exact algebraic andnumerical specification of the involved functions f(x) and g(x) and the forcingterm h(t). Mathematical results for the general equation (5.3.2) can be found inCartwright (1957b) and Cartwright/Reuter (1987).

In the following sections, two examples of how a forced oscillator system natu-rally emerges in standard economic modelling will be presented. Other economicexamples of forced oscillator systems can be found in Puu (1987, 1989) in mod-els of international trade. Haxholdt/Larsen/Tvede/Mosekilde (1991) stud-ied the complexity arising in the basin of attraction in another version of Good-win’s nonlinear accelerator model. Models of economic long waves are describedin Larsen/Morecroft/Thomsen/Mosekilde (1991), and Mosekilde/Larsen/Sterman/Thomsen (1992).

5.3.2. Goodwin’s Nonlinear Accelerator as a Forced Oscillator

Goodwin’s (1951) nonlinear accelerator model is usually quoted as a milestone inthe development of nonlinear business cycle theory because it represents an earlyalternative to the restrictive linear multiplier-accelerator models of the Samuelson-Hicks type. However, most textbooks deal only with Goodwin’s simplest case,namely that of a piecewisely defined accelerator in different stages of the businesscycle. For the purpose of this section, Goodwin’s final modification of his basicmodel deserves the greatest attention because it constitutes one of the very feweconomic examples of a forced oscillator system when specified appropriately.

By introducing lagged investment outlays, Goodwin (1951) finally obtained thesecond-order, nonautonomous differential equation

εθy +(ε+ (1 − α)θ

)y − ϕ(y) + (1 − α)y = Ia(t) (5.3.5)

with y as income, α as the marginal rate of consumption, ε as a constant expressinga lag in the dynamic multiplier process, θ as the lag between the decision to investand the corresponding outlays, ϕ(y) as induced investment, and Ia as the amountof autonomous outlays at t.

First consider the case in which Ia(t) = 0 ∀ t. Equation (5.3.5) is then anautonomous differential equation of the so-called Rayleigh type, which can easily betransformed into an equation of the van-der-Pol type. Differentiating (5.3.5) with

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5.3. The Forced Oscillator 187

respect to time and substituting x for y yields:

εθ˙y +(ε+ (1 − α)θ

)y − ϕ′(y)y + (1 − α)y = 0 (5.3.6)

or

x+A(x)x+B(x) = 0, (5.3.7)

with x = y, A(x) =[(ε+ (1 − α)θ

)− ϕ′(x)]/(εθ) and B(x) = (1 − α)x/(εθ), i.e.,

B is an odd function with respect to x = 0. It can be shown that (5.3.7) possesses aunique limit cycle if A(x) is an even function with A(0) < 0 and A′′(0) > 0.20

By means of graphical integration Goodwin illustrated that the transformedequation (5.3.6) or (5.3.7) possesses a unique limit cycle which shifts in phase spaceif the outlay Ia(t) is occasionally altered. If the shifting is irregular, the resultingtrajectories of income naturally deviate from harmonic motion.

Now consider the case of a time-dependent outlay function such that Ia(t) isτ -periodic over the business cycle, i.e., Ia(t + τ) = Ia(t); τ > 0. Let ia(t) =

Ia(t)/(εθ). If Ia(t) is a periodic function, ia(t) is periodic as well, and equation(5.3.5) turns into a forced oscillator. Suppose for simplicity that ia(t) has a sinu-soidal form, e.g., ia(t) = a sinωt, a > 0.

Under the assumptions regarding the functions A(x) and B(x) and the period-icity of exogenously determined outlays, (5.3.5) is then qualitatively identical withthe forced van-der-Pol equation. Goodwin’s nonlinear accelerator model with pe-riodic forcing can generate chaotic motion.

5.3.3. Keynesian Demand Policy as the Source of Chaotic Motion

It can be argued that one reason for the failure of Keynesian demand policy liesin the fact that in practice mainly discretionary, once-and-for-all policy measuresare performed which offset major economic variables to some degree but whichare not suited for neutralizing economic fluctuations entirely. It is therefore worth-while to investigate the dynamic effects of permanent hypothetical demand policieswhich are designed to be strictly anticyclic. In the following it will be demonstratedthat some Keynesian income policies can be ineffective when the perception of theunderlying economic dynamics as well as the proposed time path of policy inter-ventions are too simplistic. It will be shown that certain policy measures in a simpleKeynesian framework can lead to the formal presence of a strongly forced oscillatorsuch that the system behaves chaotically.

Consider the following thought experiment.21 Suppose that the dynamic behav-ior of an economy is precisely determined by the following standard laws of motion:

20 Compare Section 2.3 on the uniqueness of limit cycles.21 A longer version of the following model can be found in Lorenz (1987c).

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188 Chapter 5

net income reacts positively to excess demand in the goods market, i.e.,

Y = α(I(Y, r) − S(Y, r)

), α > 0, (5.3.8)

with α as an adjustment coefficient, I(Y, r) as net investment with Ir < 0, IY > 0and the Kaldorian sigmoid form of I(Y, ·), and S(Y, r) as savings with SY > 0 andSr > 0.

The interest rate depends negatively on the excess demand in the bonds marketwhich is assumed to be proportional to excess supply in the money market, i.e., 22

r = β(L(Y, r) −M/p

), β > 0, (5.3.9)

with r as the real interest rate, L(Y, r) as the liquidity preference with LY > 0 andLr < 0, M as the constant nominal money supply, and p as the price level.

Finally, assume that prices change according to a simple Phillips relation:

p = γ(Y − Y ∗), γ > 0, (5.3.10)

with Y ∗ as the natural level of income.Summarizing, equations (5.3.8)-(5.3.10) constitute the three-dimensional con-

tinuous-time system

Y = α(I(Y, r) − S(Y, r)

)r = β

(L(Y, r) −M/p

)p = γ(Y − Y ∗).

(5.3.11)

Suppose that the interest rate adjusts immediately to discrepancies between thedemand and supply of money such that

r = 0 = L(Y, r) −M/p ∀ t, (5.3.12)

and assume that (5.3.12) can implicitly be solved for r with

r = r(Y, p), rY > 0, rp > 0. (5.3.13)

Substitution for r in (5.3.8) and (5.3.9) leads to the two-dimensional continuous-time system

Y = α(I(Y, r(Y, p)) − S(Y, r(Y, p))

),

p = γ(Y − Y ∗).(5.3.14)

22 Compare the remarks on this assumption made in Section 5.2.2.

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5.3. The Forced Oscillator 189

Under certain assumptions the system (5.3.14) constitutes an oscillator, i.e., a dy-namical system which is able to endogenously generate fluctuations. Differentiatingthe income adjustment equation with respect to time yields

Y = α(IY Y + Ir(rY Y + rpp) − SY Y − Sr(rY Y + rpp)

). (5.3.15)

Rearranging terms and substituting for the price adjustment p leads to

Y − α(IY + IrrY − SY − SrrY

)Y − α

(Irrp − Srrp

)p = 0, (5.3.16)

Y − α(IY + IrrY − SY + SrrY

)Y − αγ

(Irrp − Srrp

)(Y − Y ∗) = 0.

Let A(Y ) = −α(IY + IrrY − SY − SrrY ) and B(Y ) = −αγ(Irrp − Srrp)(Y − Y ∗)and write (5.3.16) as

Y +A(Y )Y +B(Y ) = 0. (5.3.17)

In order to establish a result on the oscillation properties of (5.3.17), the followingsimplifying assumptions will be made:

Assumption 5.1: Ir, Sr, and rp are constant.

This assumption implies that B(Y ) is an even function with B(Y ) > ( < ) 0 ifY > (<) Y ∗. Furthermore, lim

Y→∞∫ Y

Y ∗ B(ξ)dξ = ∞.

Assumption 5.2: A(Y ) is an even function of Y with respect to Y ∗, andA(Y ) < 0 at Y ∗. Furthermore, ∃ Y > Y ∗ such that A(Y ) > 0 ∀ Y > Yand A(Y ) is nondecreasing ∀ Y > Y .

Assumption 5.2 implies that limY→∞

∫ Y

Y ∗A(ξ)dξ = ∞.

Assumptions 5.1 and 5.2 have the following consequence:

Proposition 5.1 If Assumptions 5.1 and 5.2 hold true, then (5.3.17)has exactly one limit cycle.

Proof : With the assumed properties, equation (5.3.17) is a generalized Lienardequation to which the Levinson/Smith theorem (cf. Section 2.3) on the uniquenessof limit cycles can be applied.

The uniqueness of the limit cycle depends crucially on the symmetry propertiesof the functions A(Y ) and B(Y ). Figure 5.14 illustrates one possible form of thefunction A(Y ), whose properties do not seem to allow a simple generalization ofthe proposition.

Equations (5.3.8)-(5.3.10) were postulated under the assumption that the gov-ernment does not intervene in the economic process. If the equations (5.3.8) –(5.3.10) indeed describe the evolution of the economy and if assumptions 5.1 and

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190 Chapter 5

A(Y ) as an Even FunctionFigure 5.14

5.2 hold true, i.e., if the economy is oscillating, the government may encounterthe task of stabilizing the fluctuating economy. This necessitates perception of thedynamic behavior of the economy. While it is certainly unreasonable to assumethat the government knows the exact model of the economy it can nevertheless beassumed that stylized facts such as turning points and lengths of the cycles can bedetected more or less exactly in time series analyses. Suppose that the governmentis approximating the observed time series in the past by a sinusoidal motion:

Assumption 5.3: The time series of income values Y (t) observed in thepast and generated by (5.3.17) are approximated by Y (t) = Y ∗+a sinωtwith a and ω chosen to fit the observed data.

The assumption implies that the government obviously considers the evolution ofincome as a process which can be perceived separately from those of other variables.It will therefore directly intervene in the goods market in an attempt to stabilize theeconomy by anticylic demand policies.

If the demand-stimulating policy follows a rule G(t) such that the impact on theeconomy is described by D(t) = bG(t), the excess demand in the model becomesI − S +D(t) and (5.3.8) turns into

Y = α(I(Y, r) − S(Y, r) +D(t)

). (5.3.18)

Obviously, the government has to determine an optimal date t0 for the beginning ofthe program. Without precise knowledge of the underlying structure of the econ-omy even this seemingly simple task may be difficult. Suppose, for example, thatt0 is chosen such that the (absolutely) maximum impetus occurs when the laissez-faire economy is at a turning point. At the turning points, the possibly observableexcess demands in the goods market equal zero. Therefore, the government mustbe positively convinced of the correctness of its policy because otherwise it may betempted to withdraw from intervention at the turning points.

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5.3. The Forced Oscillator 191

Suppose that the policy is introduced at a point in time t0: 23

Y (t0) = α(I(Y (t0), r(t0)) − S(Y (t0), r(t0)) +D(t0)

)= α

(I(Y (t0), r(t0)) − S(Y (t0), r(t0)) + a sinω(t0 − π)

).

(5.3.19)

For example, assume that the program is started when the economy is in a down-swing phase and takes on its natural level of income, i.e., t0 = π:

Y (t0) = α(I(Y (t0), r(t0)) − S(Y (t0), r(t0)) + a sinω(t0 − π)

)= α

(I(Y ∗, r(t0)) − S(Y ∗, r(t0)) + a sinω(t0 − t0)

),

(5.3.20)

with Y (π) < 0. Rescale the time axis by setting t′0 = 0 at t0 such that the incomeadjustment equation can be written as

Y = α(I(Y (t), r(t)) − S(Y (t), r(t)) + a sinωt

); t ≥ t′0 = 0, (5.3.21)

with Y (0) and r(0) determined appropriately.Correct timing of the introduction of the policy requires that, at t′0 = 0, output,

Y , and interest rate, r, are at their natural levels. Incorrect timing of the programcan therefore be considered by assuming other starting values of the variables at t′0,provided that sinωt′0 = 0, i.e., that the program is initiated at t′0.

The dynamics of the economy are described by the income adjustment equation(5.3.21), the adjustment equation of the price level (5.3.10), and equation (5.3.12)for the instantaneously adjusted interest rate. Differentiating (5.3.21) with respectto time and performing basically the same procedure as above leads to

Y +A(Y )Y +B(Y ) = αa cosωt, (5.3.22)

with A and B as defined above.

Assumption 5.4: i) The adjustment coefficient α in (5.3.22) is greaterthan 1, and furthermore, ii) the product of the coefficient a and theadjustment coefficient α is greater than 1.

If i) holds true, ii) can easily be justified because the amplitude-controlling param-eter is surely greater than 1 in order to speak of a relevant business cycle model.

Under assumptions 5.1-5.4, equation (5.3.22) is a strongly forced oscillator ofthe Lienard type. It follows that for appropriate parameter values the system canpossess a chaotic attractor or display complex transient behavior. A Keynesian pol-icy designed as a measure for completely neutralizing the cycle may instead lead toirregular oscillations. A political consequence of this result may consist in a suspen-sion of Keynesian ideas in this stylized model economy. Though the demand policyhas simply been superimposed upon the economy’s self-sustained evolution with nofeedback processes between the state of the economy and government expenditure,

23 Note that − sin t = sin(t− π).

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192 Chapter 5

it may appear to the government as if its policy is not well designed and that theperception of the economy’s laws of motion is incorrect. However, the failure ofthe demand management is not due to unexpected reactions of individuals but it isgenerated by the sensitive reaction of the superposition of two separate and regulartime series.

5.3.4. Conclusion

Forced oscillator systems constitute some of the most interesting dynamical systemswith respect to the possible complexity of the dynamic motion. Intuitively, thedynamics of these systems are rather simple. If a dynamical system like, e.g., a pen-dulum, is oscillating, and if this oscillatory motion is periodically influenced by anexogenous force, the outcome may be unpredictable. The effect of the forcing mayconsist in increased amplitudes, total dampening of the oscillation, or completelyirregular and permanent motion depending on the amplitude and frequency ofthe exogenous disturbance.

It was demonstrated above that it is relatively easy to find economic examplesof forced oscillator systems. However, most examples can be criticized because thenecessary assumptions seem to be artificial and ad hoc. The Goodwin model turnsinto a forced oscillator of the desired type only because of the additional assump-tion of periodic exogenous investment outlays. Actually, the nonlinear acceleratormodel therefore looses its character as an endogenous business cycle model. In theKeynesian stabilization model periodic forcing is obtained by an assumed (thoughnevertheless practically unavoidable) misperception of the actual cyclical behaviorof the economy. Similar arguments can probably be found in most other economicexamples of forced oscillator systems. Summarizing, forced oscillator systems ineconomics implying chaotic behavior usually do not represent generic economicmodels. The models may however be instructive from a pedagogical point of viewsince they uncover the possible complexity of higher-dimensional dynamical pro-cesses.

5.4. Homoclinic Orbits and Spiral-Type Attractors

As was pointed out above, no general criterion exists that allows to establish thepresence of a strange attractor in continuous-time dynamical systems. However, ithas turned out that homoclinic orbits play an important role in the emergence ofchaotic motion in many continuous-time systems and that complicated invariantsets exist in the associated Poincare maps.

The following section contains a brief presentation of the Shil’nikov scenarioand presents a result by Arneodo et al. (1981) which can be handled rather easily.This specific analytical example is demonstrated with a simple modification of astandard business cycle model in the second section.

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5.4. Homoclinic Orbits and Spiral-Type Attractors 193

5.4.1. The Shil’nikov Scenario

Section 4.3.2 contains a brief description of the horseshoe map and the complicateddynamics initiated by the presence of its invariant set. However, it is usually difficultto establish the existence of such an invariant set in a specific system. A theorem byShil’nikov (1965) provides sufficient conditions for the existence of horseshoes inthe Poincare map of a three-dimensional, continuous-time system.24 The theoremrests on the existence of homoclinic orbits (cf. Figure 5.15 for a standard homoclinicorbit in R2).

A Homoclinic Orbit for a Flow in R2

Figure 5.15

Consider the following specification of a three-dimensional system:

x1 = αx1 − βx2 + P1(x1, x2, x3),x2 = βx1 + αx2 + P2(x1, x2, x3),x3 = λx3 + P3(x1, x2, x3).

(5.4.1)

with Pi, i = 1, 2, 3, as Cr – functions (1 ≤ r ≤ ∞) vanishing together with their firstderivatives at the origin 0 = (0, 0, 0).

Theorem 5.2 (Shil’nikov (1965)):25 Assume that the vector field(5.4.1) has a fixed point x∗ such that

(i) the eigenvalues at x∗ are α± iβ and λ with |α| < |λ| and β �= 0;

(ii) there is a homoclinic orbit Γ for x∗.

24 Cf. Section 4.3.2. Details are described in Guckenheimer/Holmes (1983), pp. 319ff.,and Arneodo/Coullet/Tresser (1981), p. 574.

25 Cf. Guckenheimer/Holmes (1983), p. 319, for details.

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194 Chapter 5

Then there is a perturbation of (5.4.1) such that the perturbed systemhas a homoclinic orbit Γ ′ near Γ and the Poincare map of Γ ′ for theperturbed system has a countable set of horseshoes.

A geometric illustration of a homoclinic orbit which is consistent with the Shil’nikovscenario is given in Figure 5.16. After leaving the equilibrium point on the unstablemanifold, a trajectory returns in an oscillating manner toward the equilibrium onthe stable manifold.

A Shil’nikov-Type Homoclinic OrbitFigure 5.16

While the fulfillment of the local conditions (i) of this theorem can easily beexamined, the required presence of a homoclinic orbit constitutes a major obstaclewhich in most cases prevents a direct application of the theorem.26 Fortunately,some specific dynamical systems are known which possess a homoclinic orbit andallow the fulfillment of the local stability properties of the Shil’nikov theorem to beeasily verified.

In a series of papers, Coullet/Tresser/Arneodo (1979), Arneodo/Coul-let/Tresser (1981, 1982), and Tresser (1982)27 demonstrated that the dynamicalsystem

x+ ax+ x = z,

z = fµ(x),(5.4.2)

or, written as a third-order differential equation,

˙x+ ax+ x = fµ(x), (5.4.3)

26 A numerical algorithm for the detection of homoclinic orbits is described in Beyn(1990).

27 Compare also Glendinning/Sparrow (1984).

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5.4. Homoclinic Orbits and Spiral-Type Attractors 195

with a as a constant, exhibits chaotic behavior for appropriate forms of the one-parameter family of functions fµ(x). For example, the specification fµ = µx(1−x),i.e., a logistic function, yields geometric objects that resemble the diverse Rosslerattractors. The motion is characterized by a screw-type or spiral-type structure de-pending on the magnitude of the parameter µ.28 Other forms of the function fµwith similar non-invertibilities lead to comparable results.29

5.4.2. Spiral-Type Chaos in a Business-Cycle Model with Inventories

As a very simple economic example of the emergence of a chaotic motion in three-dimensional systems analogous to the cases studied by Arneodo et al. consider thefollowing modification of a macroeconomic business-cycle model with inventorieswhich in its discrete-time formulation was first discussed by Metzler (1941). Acontinuous-time version of the model is studied in Gandolfo (1983)30 and willthus only be outlined in the following.

Let Y denote the national product and assume that output adjusts according todiscrepancies between the desired and actual inventory stocks, i.e.,

Y = α(Bd(t) −B(t)

), α > 0, (5.4.4)

with Bd(t) as the desired and B(t) as the actual inventory stock at t. The actualinventory stock changes when disequilibria prevail on the goods market, i.e.,

B(t) = S(t) − I(t), (5.4.5)

with S and I as savings and investment, respectively. The desired inventory stock isassumed to depend linearily on the expected output, Y e(t), in t

Bd(t) = kY e(t), k > 0, (5.4.6)

implying that

Bd(t) = kY e(t). (5.4.7)

28 A geometric description of the dynamical behavior in these spiral-type attractors can befound in Berge/Pomeau/Vidal (1986), pp. 119f.

29 For example, Arneodo et al. (1982) studied equation (5.4.3) with the piecewise-lineartent function

fµ(x) =

{1 + ax if x < 0,

1 − µx if x ≥ 0,

with a > 0 and µ > 0 as parameters.30 Cf. Gandolfo (1983), pp. 259ff.

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196 Chapter 5

The expected output is determined according to a modified hypothesis of extrap-olative expectations which considers not only the rate of change of current outputbut which also includes the changes in this rate:

Y e(t) = Y + a1Y (t) + a2Y (t). (5.4.8)

Thus, expected output changes according to

Y e(t) = Y + a1Y (t) + a2 ˙Y (t). (5.4.9)

Differentiating (5.4.4) with respect to time and substituting for Bd(t) and B(t)yields the third-order differential equation

˙Y +αka1 − 1αka2

Y +1a2Y =

S(t) − I(t)ka2

, (5.4.10)

or, abbreviated,

˙Y +A1Y +A2Y = β(S(t) − I(t)

). (5.4.11)

Gandolfo (1983) demonstrated that (5.4.11) is unstable when savings is a linearfunction of output, e.g., S(t) = (1 − c)Y (t) − S0, 1 ≥ c > 0, when investment isautonomous, i.e., I(t) = I0, I0 > 0, and when A1 < 0.31

Savings and Investment in a Modified Metzler ModelFigure 5.17

31 Theoretically, A1 can be positive or negative depending on the relative magnitudes ofα, k, and a1. However, negativity seems to be more convincing when the adjustmentcoefficient α is low.

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5.4. Homoclinic Orbits and Spiral-Type Attractors 197

Y

Y

The Spiral-Type Attractor of (5.4.12); Y versus YFigure 5.18

Y

Y

The Spiral-Type Attractor of (5.4.12); Y versus YFigure 5.19

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198 Chapter 5

The linearity of the savings and investment functions in the Metzlerian modelhas been assumed in this model for technical convenience. However, there is noconvincing reason why these linear functions should constitute the only economi-cally relevant forms. Once the linearity assumption is abandoned, it can be shownthat the modified Metzler model has the form of (5.4.3) with a noninvertible func-tion fµ(·).

Define Y ∗, Bd∗, B∗, S∗, and I∗ as the equilibrium values of output, desiredand actual inventory stock, savings, and investment, respectively, and consider thedeviations from these equilibrium values, i.e., y = Y − Y ∗, bd = Bd − Bd∗, b =B −B∗, s = S − S∗, and i = I − I∗. Equation (5.4.11) then becomes

˙y +A1y +A2y = β(s(t) − i(t)

). (5.4.12)

Assume that both savings and investment are nonlinear functions of output. Possi-ble shapes of the functions are illustrated in Figure 5.17 where it has been assumedthat two points of intersection of the savings and investment functions exist. Thedifference

(s(y) − i(y)

)therefore describes a one-humped curve similar to the lo-

gistic function fµ(x) = µx(d − x) used by Arneodo et al. (1981) for the case ofd = 1.

Assumption 5.5: Equation (5.4.12) is characterized by the followingproperties:

(i) A1 > 0 and A2 close to unity.(ii) β

(sµ(y)−iµ(y)

)is a one-humped function fµ(y) with a critical value

yc > 0, the slope of which can be controlled by a single parameter µ.

Under Assumption 5.5, the Metzlerian model (5.4.12) is nearly identical with equa-tion (5.4.3). The Lie derivative (the divergence) of (5.4.12) is negative because ofA1 > 0.32 The system is therefore volume contracting and possesses an attractinginvariant set. Figures 5.18 and 5.19 show the results of a numerical investigationof (5.4.12) in (Y − Y ) – space and in (Y − Y ) – space. The dynamic behaviorof (5.4.12) is not essentially different from that of (5.4.3) and it can be seen that(5.4.12) possesses a Shil’nikov-type structure for the assumed values of A1, β, andthe slope of the excess supply function. In contrast to logistic, one-dimensionaldifference equations, rather flat shapes of the one-humped curve are sufficient toencounter chaotic motion.33

The basin of attraction of the attractor is depicted in Figures 5.20 and 5.21.White areas represent the basin of attraction; the grey-shaded areas constitute the

32 The numerical calculation of Lyapunov exponents (cf. Section 6.2.4.) for the assumedparameters yields a positive and a negative exponent in addition to the zero exponent.Cf. Lorenz (1992c) for details.

33 In the numerical investigation of equation (5.4.3), Arneodo et al. (1982) detect theShil’nikov attractor for a = 0.4 and µ < 1. Larger values of µ lead to the appearanceof regular periodic or double-periodic attractors.

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5.4. Homoclinic Orbits and Spiral-Type Attractors 199

Y

Y

Figure 5.20: Basin of Attraction of (5.4.12), (Y − Y space)

Y

Y

Figure 5.21: Basin of Attraction of (5.4.12), (Y − Y space)

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200 Chapter 5

basin of infinity.34 It can be seen that the basin is formed by a relatively narrowregion. Initial points that are not very close to the attractor will diverge from it.It is also possible to encounter other types of behavior in system (5.4.12). For ex-ample, slightly different parameter values imply the existence of multiple compli-cated attractors, the basins of attraction of which constitute fractal sets (cf. Lorenz(1993a)).

The simple example presented in the above section demonstrated that it is in-deed possible to provide an economic application which is consistent with the re-quirements of the Shil’nikov scenario. It can be expected that several other modifi-cations of the model are possible which still imply the emergence of a Shil’nikov-typeattractor when the excess supply function is noninvertible. However, as was pointedout by Gandolfo (1983), economic models which can be reduced to a third-orderdifferential equation are really rare in standard dynamical economics, implying thatfurther applications of the Shil’nikov theorem will probably be complicated.

34 The overlapping of the basin boundaries and the attractors in Figures 5.20 and 5.21 is aresult of different projections: for example, in Figure 5.20 the attractor is a projection ofthe three-dimensional state space to the Y –Y plane (with a Y = 0 coordinate) while Yis, of course, changing during the motion; the basin has been calculated for a constantinitial value of Y .

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Chapter 6

Numerical Tools

The theoretical results presented above allow to establish the existence of chaotictrajectories in several dynamical systems, which fulfill the assumptions of the

appropriate theorems. For example, when the difference equation is unimodal, itis possible to apply the Li/Yorke theorem or Sarkovskii’s theorem and to establishthe existence of chaos (defined in the sense of one of the definitions provided inthe previous chapters). However, in many cases it may be difficult or analyticallyimpossible to detect a period-three cycle, and for most differential equation systemsthere are no theoretical results at all. Experiments show that even for cycles of arelatively low period it may be impossible to distinguish regular time series fromcompletely chaotic series by simple visual inspection.

It is therefore necessary to introduce more sophisticated methods of time seriesanalysis into the investigation of irregular motion, and the question arises whetherit is possible to apply numerical techniques evidencing chaotic dynamics in

• statistical time series for which the underlying dynamical system (if it exists) isnot known, and in

• given dynamical systems which do not fulfill the assumptions of the standardtheorems but which appear as good candidates for chaotic systems.

The following tools can be useful in deciding whether an actual statistical time seriesor a time series generated by a simulation of a known dynamical system is regular,chaotic, or stochastic.1

1 However, it must be stressed at the beginning that (abstracting from spectral analysis)the usage of these tools is in very early stages and that the progress in this field is rapid.The following survey is therefore neither complete nor very in depth.

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202 Chapter 6

6.1. Spectral Analysis

If a deterministic dynamical system is given whose behavior cannot be investigatedfurther by applying the standard geometric or analytical methods, numerical simu-lations are appropriate. The generated time series in such a simulation may exhibitsimple patterns like monotonic convergencies or harmonic oscillations. However,the series may also appear to be random due either to

• periodic behavior with a long period,

• quasi-periodic behavior with many incommensurate frequencies,

• deterministic chaos,

• noise generated by the use of specific algorithms during the simulation, or to

• the design of digital computers implying specific problems in the representationof numbers.

Spectral analysis has proven to be particularly useful in attempts to distinguish pe-riodic and quasi-periodic time series with few frequencies from random behavior(either chaotic or true random behavior).2

The aim of spectral analysis is dividing a given time series into different harmonicseries with different frequencies. For example, if a time series consists of two over-lapping harmonic series, spectral analysis attempts to isolate these two harmonicseries and to calculate the involved frequencies. Furthermore, spectral analysisprovides information on the contribution of each harmonic series to the overallmotion, i.e., whether there are dominating frequencies.

In the following, only an outline of the essential ingredients of spectral analysiswill be given.3 Assume that a time series xj , j = 1, . . . , n of a single variable hasbeen observed at equi-distant points in time. The Fourier transform of the series xjis defined as

xk =1√n

n∑j=1

xje−(i2πjk/n), k = 1, . . . , n, (6.1.1)

with i =√−1. The inverse Fourier transform maps the xk back to xj with the differ-

ence that xj is now periodic, i.e., xj = xj+n:

xj =1√n

n∑k=1

xke−(i2πjk/n), k = 1, . . . , n. (6.1.2)

2 See, for example, the pioneering work of Granger/Hatanaka (1964) for an elaboratedintroduction. Cf. Dale (1984) for applications in business cycle theory.

3 Cf. Berge et al. (1986), pp. 43ff., and Medio (1993), pp. 101-114, for comprehensivesurveys of Fourier transforms and power spectral analysis.

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6.1. Spectral Analysis 203

Consider next the autocorrelation function defined as

ψm =1n

n∑j=1

xjxj+m, (6.1.3)

with m as the lag between the correlated values. By applying the Fourier transformto (6.1.3) and substituting for xj , (6.1.3) becomes

ψm =1n

n∑k=1

|xk|2 cos(

2πmkn

). (6.1.4)

The inverse of (6.1.4) is4

|xk|2 =1n

n∑m=1

ψm cos(

2πmkn

). (6.1.5)

The function |xk|2 is thus proportional to the Fourier transform of the autocorre-lation function. The graph obtained by plotting |xk|2 versus the frequency f =(2π)/n is called the power spectrum.5

A power spectrum can loosely be defined as each frequency’s contribution to theoverall motion of the time series. For example, if there is no periodic component ina given series, the power spectrum will be a smooth monotonic curve with a peak atthe origin.6 If there are frequencies for which the associated |xk|2 are significantlyhigher than for others, spectral analysis indicates the existence of periodic behavior.The interpretation of the peaks depends on the underlying time concept.

When the basic dynamical system is formulated in continuous time, a single peakin the power spectrum is equivalent to the existence of a single closed orbit withthe associated frequency. Power spectra with several distinguishable peaks indicatethe presence of quasi-periodic behavior. The dominating peaks represent the basicincommensurable frequencies of the motion, while minor peaks can be explainedas linear combinations of the basic frequencies. If the underlying system is discrete,a single peak corresponds to a period-2 cycle, the emergence of two additionalpeaks to the left and to the right sides of the first peak, respectively, correspond to aperiod-4 cycle, 7 peaks correspond to a period-8 cycle, etc. If a continuum of peaksemerges7, the power spectrum is said to reflect broad band noise. The motion is theneither purely random or chaotic for both underlying time concepts.

4 Cf. Berge et al. (1986), p. 47, for details.5 In practical numerical work the Fourier transform is usually replaced by the Fast Fourier

transform, which (as the name suggests) is a much faster algorithm than the originaltransformation.

6 Depending on particular statistical procedures like detrending and tapering the slopeof the curve can be different.

7 It may be difficult to decide whether a continuum indeed prevails because subharmonicsmay add an unknown number of peaks to the spectrum.

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204 Chapter 6

Figures 6.1.a to 6.1.d represent power spectra of the logistic equation for dif-ferent values of the bifurcation parameter µ. Figure 6.1.a illustrates the powerspectrum for µ = 2.5. The fixed point of the map is stable, and the power spectrumdisplays no peaks at positive frequencies. Figure 6.1.b depicts the case of a stableperiod-2 cycle (µ = 3.2). The power spectrum possesses a peak at a frequency of0.5. The additional peak in Figure 6.1.c indicates the existence of a period-4 cycle(µ = 3.5). Cycles of higher order would generate additional peaks to the left and tothe right of the single peak in the figure. Figure 6.1.d contains the power spectrumfor a value of µ in the chaotic regime. There does not exist a peak that clearlydominates all other peaks.

PS PS

Frequency Frequency

PS PS

Frequency Frequency

Power Spectra for the Logistic Equationµ = 2.5 (upper left); µ = 3.2 (upper right); µ = 3.5 (lower left); µ = 4.0 (lower right)

500 Iterations; First-Degree Polynomial DetrendingFigure 6.1

While power spectra are thus particularly useful in investigating the periodic be-havior with few frequencies of higher-dimensional dynamical systems, chaotic andrandom behavior cannot be discriminated with this method. It might even beimpossible to discriminate between chaotic and quasi-periodic behavior. The fol-lowing section presents some concepts which can provide more definite answersto the question of which type of behavior prevails in a dynamical system or a timeseries.

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6.2. Dimension, Entropy, and Lyapunov Exponents 205

6.2. Dimension, Entropy, and Lyapunov Exponents

The short presentation of spectral analysis has shown that traditional statistical tech-niques fail to provide a definite answer to the question of whether a given complextime series is generated by a random process or by deterministic laws of motion.Appropriate concepts for distinguishing between these two sources of complex andirregular behavior have emerged only recently, and the development of new tech-niques is still in progress. The following section which outlines some of these rel-atively new methods and concepts will therefore be preliminary. In addition tothe empirical motivation for dealing with those concepts, their discussion will beuseful because new insights into the nature of deterministic chaotic systems can beprovided.8

6.2.1. Phase Space Embedding

Of central importance to the numerical investigation of complex dynamical sys-tems is the notion of the embedding dimension. Suppose that a dynamical process isgenerated by a deterministic set of equations like9

xit+1 = gi(xt), x ∈ Rn, i = 1, . . . , n, (6.2.1)

and let a certain xj be the variable which attracts the attention of an observer. Theobserver neither knows the structural form of (6.2.1) and its dimension n (andtherefore the values of all relevant other entries xit, i �= j, in (6.2.1) ), nor can hebe sure that his measurement of the quantity xjt is correct. Denote the observedvalue of the variable xj at t as xjt and let

xjt = h(xt), (6.2.2)

i.e., the observed variable depends on the “true” values xit, i = 1, . . . , n, but themeasurement of the variable10 may imply differences between xjt and xjt .

The measurement procedure over time generates a time series {xjt}Tt=1. Anembedding is an artificial dynamical system which is constructed from the one-dimensional time series in the following way: consider the last element xjT in

8 Surveys of the following topics can be found in Berge et al. (1986), pp. 144ff. and279ff., and with an overview of economic applications, in Frank/Stengos (1988b) andPeters (1991). Concise survey are provided in Brock (1990) and Sayers (1991). Themore technically interested reader should consult Eckmann/Ruelle (1985), Barnett/Chen (1988a), Brock (1986, 1987b, 1988a), Brock/Sayers (1988), Medio (1993), Ch.6 and 7, and Scheinkman (1990).

9 The continuous-time case can be treated analogously. Details are described in Guck-enheimer/Holmes (1983), pp. 280 ff., and Takens (1981).

10 Brock (1986), p. 170, calls the function h a measuring apparatus.

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206 Chapter 6

the observed time series and combine it with its m predecessors into a vectorxmT = (xjT , x

jT−1, . . . , x

jT−m+1). Perform this grouping for every element xjt in

the descending order t = T, . . . , t0. The m − 1 elements xjτ , τ = 1, . . . , t0 do nothave enough predecessors in the time series. It follows that only T −m vectors xm

t

can be generated. In this way, the scalar entries in the original time series havebeen rearranged into the m-dimensional vectors xm

t :

xmT = (xjT , x

jT−1, . . . , x

jT−m+1)

xmT−1 = (xjT−1, x

jT−2, . . . , x

jT−m)

...xmt0

= (xjt0, xjt0−1, . . . , x

jt0−m+1)

(6.2.3)

with t0 = m. The m-dimensional vector xmT is called the m-history11 of the observa-

tion xjT . Since the first elements do not possess a sufficient number of predecessors,the sequence of the vectors {xm

t }Tt=t0is shorter than the original time series and

varies with the value of m. The number m is called the embedding dimension.Each m-history describes a point in an m-dimensional space, the coordinates of

which are the delayed observed values in the vector xmt . The sequence {xm

t }Tt=t0

of points will therefore form a geometric object in this space. It was proven byTakens (1981) that this object is topologically equivalent to the appropriate objectgenerated by the true dynamical system (6.2.1) if12

i) the variables xi of the true dynamical system are located on an attractor, i.e.,transients have been excluded,

ii) the functions gi(x) in the true dynamical system and the observation functionh(x) are smooth functions, and

iii) m > 2n− 1.

If conditions i) - iii) are fulfilled, it is thus theoretically possible to reconstructthe behavior of the (unknown) true dynamical system from a single observed timeseries.13 However, as n is not known for an arbitrary, observed time series, thechoice of m is vague.14

11 The relevant literature actually considers what might be called the m-future of an obser-vation, namely xmt = (xt, xt+1, . . . , xt+m−1). The qualitative properties are the samefor both orientations.

12 For a precise formal description of the Takens theorem compare Brock (1986).13 In certain cases of low-dimensional dynamical systems this result is obvious. For exam-

ple, consider the generalized Lienard equation (2.3.2) of Section 2.3.1, in which thevariable x is defined as y. The (x− y) – space is therefore equivalent to the (y − y) –space. Berge et al. (1986), p. 77, provide the exact transformation between (x, y, z)-coordinates and (x, x, x) coordinates for the Rossler attractor.

14 If the underlying true dynamical system is purely random, n can be thought of as beinginfinitely large. In that case, no m-history of observed values can therefore be foundwhich mimics the true system.

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6.2. Dimension, Entropy, and Lyapunov Exponents 207

y

x

An Attractor of a Two-Dimensional, Discrete-Time SystemFigure 6.2

y1

y2

A Projection of the m-History of {yt}T1 onto the y2 − y1 Space; m = 10Figure 6.3

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208 Chapter 6

The power of this method can be illustrated with a simple numerical example.Figure 6.2 shows the attractor of a two-dimensional discrete-time system.15 Thegeometric object obtained by embedding the series {yt}T1 inm-dimensional vectorsyt with m = 10 is displayed in Figure 6.3. Though the two geometric objects arenot identical they are strikingly similar. If the object in Figure 6.3 consisted of aplasticine-like material it would be possible to transform its shape into that of theattractor in Figure 6.2 by an appropriate kneading. The topological properties ofthe object can survive in this kneading procedure.

The embedding procedure described above does not necessarily generate geo-metrically similar objects. When the series of xt values is lagged instead of the ytvalues, no object can be generated which resembles the original attractor. Further-more, most other projections of the ten-dimensional lagged object in the examplediffer drastically from the original attractor.16

The described m-histories of an empirically observed or numerically calculatedtime series are important in attempts to calculate the so-called correlation dimensionand the Lyapunov exponents from these series. Before these concepts can be pre-sented, another basic concept, namely that of fractal dimension, must be describedbriefly.

6.2.2. Fractal Dimensions

Intuitively, the dimension of a geometric object is connected with an integer value.For example, a point has dimension 0, a line has dimension 1, a plane has dimension2, etc., and it is difficult to imagine an object whose dimension is a nonintegernumber, say 1.5. In fact, the definition of the dimension used in these examplesis that of the Euclidian dimension which is always an integer. In addition to thisdefinition, other kinds of dimensions exist which permit not only integers andwhich allow an interesting insight into the nature of strange attractors.

Though the following concept of a dimension is interesting mainly for purelymathematical purposes, it is very useful in understanding different notions of di-

15 The simulated system is

xt+1 = 0.259(−0.1(xt − 10.0)3 + 2.0(xt − 10.0) + 80.0 − yt

)+ xt

yt+1 = 2.0(xt − 0.051yt − 5.0

)+ yt.

The system represents a numerical specification of a simultaneous price-quantity adjust-ment process, cf. Lorenz (1992a) for a discussion.

16 When the dynamical system under consideration is modeled in continuous time, an-other difficulty usually arises: the time step in the simulation of the system is an arbi-trarily fixed quantity (or a variable quantity, depending on the underlying algorithm)that cannot be compared with the fixed time step in the discrete-time case. The time lagis usually determined by practical considerations in these systems. A time lag of roughly20% of a full orbit usually delivers sufficient results. For example, when the simulationof a system generates an approximate orbit in 50 integration steps, the consideration ofevery 10th value in the generated time series can deliver good results.

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6.2. Dimension, Entropy, and Lyapunov Exponents 209

6.4.a. 6.4.b. 6.4.c.Illustration of the Hausdorff Dimension for the Plane

Figure 6.4

mensions. First, consider a two-dimensional space with a single point (cf. Figure6.4.a) and construct a two-dimensional square with length ε. The number N(ε) ofsuch squares needed to cover this single point is obviously

N(ε) = 1,

which is independent of the length ε. Next, consider a set of points located on aline with length L (cf. Figure 6.4.b). For a given ε, the minimal number of squaresto cover the line entirely is

N(ε) =L

ε.

As a final example, consider a set of points located in a rectangle ABCD whichcovers a surface S (cf. Figure 6.4.c). For a given ε, the minimal number of squaresnecessary to cover the rectangle is

N(ε) =S

ε2 .

The Hausdorff dimension DH is defined as17

DH = limε→0

lnN(ε)ln(1/ε)

, (6.2.4)

where the square used above for illustrative purposes can be replaced by hypercubesof length ε. Applying this definition to the three examples in Figures 6.4.a-c yields

17 Actually, this is the so-called Kolmogorov capacity, but the designation Hausdorff dimen-sion has become common in the dynamical systems literature.

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210 Chapter 6

The Construction of a Cantor Middle-Third SetFigure 6.5

DH = 0 in the case of the single point, DH = 1 for the line, and DH = 2 forthe surface S. The Hausdorff dimension is therefore identical with the Euclidiandimension in the case of simple geometric objects.

Consider however another object which is of central importance in the geo-metric descriptions of many strange attractors, namely a Cantor set. The so-calledCantor middle-third set (cf. Figure 6.5) is constructed in the following way: take astraight line of length L = 1, divide it into three equal parts and cut off the middlepart. The set now consists of two separate pieces. In the next step, perform thisprocedure with each of the two remaining lines, such that the two lines split intofour pieces, etc.

For the different steps in the construction of the Cantor set the number N(ε)of the minimal number of lines (i.e., one-dimensional Euclidian “cubes”) necessaryto cover the set is obviously

ε = 1 =⇒ N(ε) = 1ε = 1/3 =⇒ N(ε) = 2ε = 1/9 =⇒ N(ε) = 4...ε = (1/3)m =⇒ N(ε) = 2m.

(6.2.5)

For increasing m, i.e., decreasing ε, the Hausdorff dimension is then given as

DH = limm→∞

ln 2m

ln(1/(1/3)m)= lim

m→∞ln 2m

ln 3m=

ln 2ln 3

� 0.63, (6.2.6)

i.e., a noninteger number. If the dimension of an object is a noninteger number,the object is said to have a fractal dimension. If the attractor of a dynamical systempossesses a fractal dimension and if this number is small, there is evidence that theattractor is strange.

However, a fractal dimension is neither sufficient nor necessary for the existenceof a strange attractor in the sense of Definition 5.1.18 There exist attractors with

18 Other definitions of strange attractors, e.g., purely geometric definitions, may dissolvethis ambiguity.

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6.2. Dimension, Entropy, and Lyapunov Exponents 211

fractal dimension that are not strange. On the other hand, an attractor may bestrange though its Hausdorff dimension is an integer.19 While the concept of theHausdorff dimension can be useful in illustrations of the idea of fractal dimen-sion, other concepts are more fruitful in practical studies mainly with respect tooperationality, i.e., implementation of appropriate algorithms and their computingspeed.

6.2.3. Correlation Dimension

An alternative to the concept of Hausdorff dimensions (that saves a lot of computingtime in numerical studies) is the concept of correlation dimensions introducedby Grassberger/Procaccia (1983). Let {xit}Tt=1 be an observed time series of asingle variable and consider itsm-histories as defined in (6.2.3). Them-dimensionalvectors xm

t can be plotted in an m-dimensional phase space. If the requirementsof the Takens theorem mentioned above are fulfilled, the generated geometricobject will be topologically equivalent to the genuine attractor of the true dynamicalsystem.

Suppose that the attractor is chaotic and consider two points on this attractorwhich are far apart in time. Due to the sensitive dependence on initial conditions,these points are dynamically uncorrelated since arbitrarily small measurement er-rors in the determination of the initial point can lead to drastically different loca-tions of the second point. However, as both points are located on an attractor, theymay come close together in phase space, i.e., they may be spatially correlated.

The two points xmi and xmj are said to be spatially correlated if the Euclidiandistance is less than a given radius r of an m-dimensional ball centered at one ofthe two points, i.e., ‖xmi − xmj‖ < r. The spatial correlation between all pointson the attractor for a given r is determined by counting the number of these pairslocated in a ball around every point:

C(r,m) = limTm→∞

1T 2m

× [number of pairs i, j with

a distance ‖xmi − xm

j ‖ < r],(6.2.7)

or20

C(r,m) = limTm→∞

1T 2m

Tm∑i,j=1

H(r − ‖xmi − xm

j ‖), (6.2.8)

19 For details compare Grebogi et al. (1984).

20 Cf. Berge et al. (1986), p. 151.

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212 Chapter 6

with Tm as the length of the series of constructedm-histories xmt , ‖·‖ as the Euclidian

norm, and H as the Heaviside function

H(y) =

{1 if y > 0 ,0 otherwise.

The function C(r,m) is called the correlation integral. The correlation dimension isdefined as

DC(m) = limr→0

lnC(r,m)ln r

. (6.2.9)

The calculated values of the correlation dimension are close to the Hausdorff di-mension and do not exceed it, i.e., 21

DC ≤ DH . (6.2.10)

Obviously, the correlation dimension can be computed more easily than the Haus-dorff dimension since counting is the essential ingredient in calculating the corre-lation dimension: fix a small r and count the number of points N(r) lying in a ballcentered at a xmi. Perform this procedure for every xmi and calculate C(r,m) andDC(m).

Stylized Correlation Integral C(r,m) versus the Radius rFigure 6.6

The correlation integral C(r,m) depends on xm and thus on the length m ofthe m-history vectors. The question of whether and how the correlation dimensionvaries with changes in m thus arises. From (6.2.9) it follows that

lnC(r,m) ≈ DC(m) ln r, (6.2.11)

21 In fact, both concepts lead to nearly identical numerical values in the standard examplesof chaotic dynamical systems. Cf. Berge et al. (1984), p. 149.

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6.2. Dimension, Entropy, and Lyapunov Exponents 213

i.e., the correlation integral C(r,m) is proportional to rDC

. For a given m the rela-tion between lnC(r,m) and ln r can be illustrated graphically with the correlationdimension as the slope of the graph. Figure 6.6 is called a Grassberger/Procacciaplot in the following.

The most important point consists of the fact that the slope, i.e., the correlationdimension, settles to a stationary value for increasing length m of the m-historyvectors xm when the dynamical system is deterministic, whereas the slope perma-nently increases in the case of a stochastic process, i.e., a process with an infinitenumber of degrees of freedom. In other words, if the dimension continues to growwith the embedding dimension m, the process will be stochastic. If DC becomesindependent of m, the process will be deterministic. The correlation dimensiontherefore seems to constitute a powerful tool for distinguishing between randomand deterministic noise in an observed time series.

6.2.4. Lyapunov Exponents

Strange attractors are geometrically characterized by the simultaneous presenceof stretching and folding, implying that two initially close points will be projected todifferent locations in phase space.22 The presence and interaction of stretching andfolding in a certain dynamical system can be described via the so-called Lyapunovexponents. As this section is concerned only with attractors, it is assumed in thefollowing that the system is dissipative, i.e., that it contracts volume in phase spacewith time.23

In order to get an intuitive idea of the meaning of Lyapunov exponents, considera set of initial points located inside a circle in the plane and denote its radius byr0 (cf. Figure 6.7.a). When the dynamical system is dissipative, it will project theinitial points in the circle into an object with a smaller area, but possibly differentshape. Let the new shape be the ellipse in Figure 6.7.b, where the former radius r0

has been stretched into one direction and contracted into the other one. Denotethe major and minor axes of the ellipse as r1 and r2, respectively, with r1 = µ1r

0

and r2 = µ2r0, or

µi =ri

r0 , i = 1, 2. (6.2.12)

After N steps, the radii ri will become ri = µiNr0, or, written as logs

log2 µi =1N

log2ri

r0 . (6.2.13)

22 Compare also Section 4.3.2 for a demonstration of stretching, contracting, and foldingin the horseshoe map.

23 In the case of continuous-time dynamical systems, the Lie derivative (cf. Section 2.4.1)must therefore be negative.

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214 Chapter 6

Suppose that the limit

log2 µi = limN→∞

1N

log2ri

r0 (6.2.14)

exists for the number of iterations (or time t in the continuous-time case) approach-ing infinity. The numbers µi in (6.2.14) are called Lyapunov numbers, while the logsof the µi’s are called Lyapunov exponents denoted by λi = log2 µi. Obviously, thereare as many Lyapunov exponents in a dynamical system as there are phase spacecoordinates, i.e., i = n. The set of all Lyapunov exponents λi, i = 1, . . . , n, is calledthe Lyapunov spectrum of a dynamical system.

The signs of the Lyapunov exponents determine whether stretching and con-tracting occur in a dynamical system. If the two exponents r1 and r2 mentionedabove have opposite signs, the ellipsoid will be infinitely stretched for N → ∞.However, as the scenario takes place on an attractor, the ellipsoid cannot always bestretched in the same direction, but must be folded such that it is located in theneighborhood of the original circle (cf. the folding in the horseshoe map in Figure4.26.b).

The stretching implies that two initial points close together in the original circlewill diverge exponentially on the attractor. The Lyapunov exponents thereforeconstitute a quantity for characterizing the rate of divergence of two initial points.Note that this divergence on the attractor is a dynamical property. The foldingpresent in strange attractors may occasionally lead to geometrically close contactsbetween two points on different trajectories.

6.7.a. 6.7.b.Stretching and Contracting in a Dynamical System

Figure 6.7

It remains to formalize the development of the ratios ri/r0 during the dynamicalprocess. Consider first the discrete-time case with an n-dimensional mapping

xt+1 = f(xt), x ∈ Rn, (6.2.15)

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6.2. Dimension, Entropy, and Lyapunov Exponents 215

and two initial points x0 and x′0. Let the difference δx0 = ‖x0 − x′

0‖ be small. Afterthe first iteration, the difference between the two points x1 and x′

1 will be

‖x1 − x′1‖ = ‖f(1)(x0) − f(1)(x′

0)‖. (6.2.16)

A linear approximation of the difference yields

‖x1 − x′1‖ ≈ df(1)(x0)

dxδx0, (6.2.17)

where df(1)(x0)dx is the Jacobian matrix J:

J =

∂f (1)1

∂x1. . .

∂f (1)1

∂xn

.... . .

...

∂f (1)n

∂x1. . .

∂f (1)n

∂xn

. (6.2.18)

After N iterations the difference between the corresponding points will be

‖xN − x′N‖ = ‖f(N)(x0) − f(N)(x′

0)‖, (6.2.19)

with f(N)(x0) as the N th iterative. Linearization yields

xN − x′N ≈ df(N)(x0)

dxδx0, (6.2.20)

where, by the chain rule,(df(N)(x0)

)/(dx) = J(N) equals the product of the N

Jacobian matrices J in (6.2.18) evaluated along the orbit.As J(N) is an n×nmatrix, it also possesses n eigenvalues. Denote the eigenvalues

of this matrix as ΛNi and rearrange them such that ΛN

1 ≥ ΛN2 ≥ . . . ≥ ΛN

n . TheLyapunov exponents λi, i = 1, . . . , n, are defined as24

λi = limN→∞

1N

log2 |ΛNi |. (6.2.21)

From the so-called multiplicative ergodic theorem25 it follows that this limit exists foralmost all x0.

As an example, consider again the one-dimensional logistic equation (4.1.2).The eigenvalue of J(1) is, of course, the first derivative, and the eigenvalue of J(N)

24 Cf. Farmer et al. (1983), Guckenheimer/Holmes (1983), pp. 283ff., Eckmann/Ru-elle (1985), Wolf et al. (1985). It is also possible to use natural logarithms.

25 Cf. Eckmann/Ruelle (1985), pp. 629ff.

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216 Chapter 6

is the product of the derivatives along the orbit {xt}Nt=1 (cf. (4.1.7) ). If the mappossesses a stable fixed point, e.g., x∗ = 0.6 for µ = 2.5, the product of the deriva-tives at the fixed point is

∏= −0.5N . The Lyapunov exponent is then given as

λ = log2(.5N)/N = −1, indicating the fact that the sequence {xt} is rapidly con-verging to the fixed point. Table 6.1 contains the results of a simple calculation ofthe Lyapunov exponent for the logistic map with µ = 4, i.e., for the chaotic case.For N large, the Lyapunov exponent is positive and rapidly converges to λ = +1.

t xt |f ′(xt)|∏N

t=1 |f ′(xt)| λ(N)

1 .600 0.799 0.799 -.3212 .960 3.680 2.944 .7783 .153 2.771 8.158 1.0094 .520 0.160 1.307 0.096...

......

......

21 .262 1.899 0.178 · 107 0.98922 .774 2.195 0.392 · 107 0.995

......

......

...99 .221 2.225 0.598 · 1030 0.999

Lyapunov Exponents of the Logistic Map; µ = 4Table 6.1

Figure 6.8 shows calculated values of the Lyapunov exponents for the logisticmap versus the parameter µ. As can be seen from the figure, the exponents arenegative for values of µ lower than the critical value µc ≈ 3.59. In the chaoticregime, the exponents are typically positive, but there are values of µ with negativeLyapunov exponents, indicating the presence of stable period points.

An analogous procedure for the continuous-time case leads to

λi = limT→∞

1T

log2

(ΛTi

)(6.2.22)

with T ∈ R, i.e., the time step between iterations tends to zero.The sum

∑i λi, i = 1, . . . ,m ≤ n can be interpreted as follows:26 The first

Lyapunov exponent measures the extent of the ellipsoid into the first direction, thesum λ1 + λ2 measures the extent of the area defined by the first two principal axes,the sum of the first three exponents measures the extent of the volume defined bythe first three principal axes, etc. As this section deals only with dissipative systems,the volume contracts under successive iterations. In systems with n ≥ 2, the sum ofall Lyapunov exponents must therefore always be negative.

26 Cf. Wolf (1986), p. 280.

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6.2. Dimension, Entropy, and Lyapunov Exponents 217

λ

µ

Lyapunov Exponents of the Logistic Map; 2000 Intervals on the µ lineFigure 6.8

Dimension Asymptotic Limit Cycle Torus ChaosStability (T1) (T2)

n = 1 (−)

n = 2 (−,−) (0,−)

n = 3 (−,−,−) (0,−,−) (0, 0,−) (+, 0,−)

Lyapunov Exponents and Dynamic Behavior in Continuous-Time SystemsTable 6.2

The meaning of the Lyapunov exponents can be interpreted as follows: when allLyapunov exponents are negative on an attractor, the attractor is an asymptoticallystable fixed point. When one or more Lyapunov exponents are non-negative, thenat least one exponent must vanish.27 A limit cycle must involve a λi = 0 and thuscannot occur in the one-dimensional case. A torus can emerge only in at leastthree-dimensional phase space. As two cyclical directions are involved in a 2-torus,two of its Lyapunov exponents are equal to zero (the third one must be negative in

27 Cf. Eckmann/Ruelle (1985), p. 632.

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218 Chapter 6

a dissipative system). If one of the exponents is positive, chaotic motion prevails.28

This can be stated explicitly in the alternative definition of chaotic motion:

Definition 6.1: A dissipative dynamical system is chaotic if the largestLyapunov exponent is positive.

The characterization of the behavior of low-dimensional continuous-time dynam-ical systems by means of their Lyapunov exponents is summarized in Table 6.2.Empty fields indicate the impossibility of the appropriate dynamic behavior if thedimension n is too low.

Recently, efficient algorithms have been constructed to estimate the entire Lya-punov spectrum or only the largest Lyapunov exponent. The algorithm by Wolfet al. (1985) has emerged as the standard and constitutes a relatively easy wayof calculating the largest exponent from a time series. Brock/Dechert (1987)have demonstrated that a theoretically ideal version of the algorithm indeed con-verges to the true exponents. A modification of the algorithm of Wolf et al.can be found in Kurths/Herzel (1987). The algorithm proposed by Benettin/Galgani/Strelcyn (1980) permits the calculation of the entire Lyapunov spec-trum. Dechert/Gencay (1990, 1992) and Gencay/Dechert (1992) describe thecalculation of all Lyapunov exponents with the help of network techniques. The ex-ponents can accurately be determined even if the number of observations is ratherlimited.

6.2.5. Kolmogorov Entropy

It was stressed several times before that a strange attractor is characterized by asensitivity to initial conditions, i.e., two initially close points may imply completelydifferent trajectories. Suppose that two initial points are so close together that theycannot be distinguished one from another by the measuring device. Provided thatthe motion takes place on a strange attractor, the trajectories diverge and eventuallybecome distinguishable as time elapses. In other words, while at the start of anexperiment information on possible differences in the initial states may not beaccessible, it will be produced as time passes.29

An index which reflects the amount of information produced on an attractor isthe so-called Kolmogorov entropy, which occasionally is also denoted as metric entropyor just entropy. Technically, the number is derived as follows: partition the phasespace into hypercubes with side lengths ε and denote the resulting n cubes by ci, i =1, . . . , n (cf. Figure 6.9). Consider an initial measurement x(t1) and suppose that

28 Note that chaos therefore cannot occur in a two-dimensional, continuous-time system:with λ1 > 0 and λ2 necessarily equal to zero, the system would possess a repeller insteadof an attractor. The minimum phase space dimension for a strange attractor is thusn = 3.

29 Cf. Grassberger (1986), pp. 292 ff., for a precise formulation of the required informa-tion to specify a trajectory.

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6.2. Dimension, Entropy, and Lyapunov Exponents 219

A Partition of the Phase Space with Hypercubes in the PlaneFigure 6.9

subsequent measurements take place at specific points in time (t1 +τ), (t1 +2τ), . . .(t2). Denote the joint probability30 that the trajectory starting at x(t1) will be in cubec1 at (t1 + τ), in cube c2 at (t1 + 2τ), . . ., and in cube cn at the final point (t2) byρc1, . . . , cn . The Kolmogorov entropy is then defined as

K = − limε→0

limt2→∞ lim

τ→0

1t2τ

∑c

ρc1, . . . , cn log ρc1, . . . , cn . (6.2.23)

Equation (6.2.23) is numerically intractable when the joint probabilities ρc1, . . . , cnare not known. An approximation of the entropy K was proposed by Grass-berger/Procaccia (1983b), who related the entropy to the correlation integralpresented above. Let C(ε,m) be the correlation integral of a time series with em-bedding dimension m. It can be shown that the expression31

K2 = limm→∞ lim

ε→0

logC(ε,m)

C(ε,m+ 1)(6.2.24)

estimates the Kolmogorov entropy very well (K2 ≤ K). It has the advantage that itcan be computed as easily as the correlation dimension.

As the correlation integral does not change in case of a regular attractor like alimit cycle, i.e.,C(ε,m) = C(ε,m+1), the entropyK2 equals zero. If the dynamicalsystem is entirely random, the entropy is infinite. A chaotic system is characterizedby a finite entropy 0 < K2 < ∞, i.e., by increasing the embedding dimension theKolmogorov entropy approaches a finite and positive value.

30 Cf. Haken (1983a), pp. 26ff., for details.31 Cf. Grassberger/Procaccia (1983b), pp. 2591f., or Eckmann/Ruelle (1985), pp.

649f. The variable τ represents the time lag in the measurement procedure. In thecase of a time series generated by a differential equation, a value of, e.g., τ = 10 meansthat only every 10th value in the time series is considered in the calculation.

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220 Chapter 6

6.2.6. Summary

The different indices described above allow to distinguish regular, chaotic, and ran-dom behavior in a dynamical system or its reconstruction from a time series. Thecorrelation dimension provides information on the deterministic or random char-acter of a time series, whereas Lyapunov exponents and the Kolmogorov entropyare useful in discriminating chaotic and regular behavior.

In some cases, the relation between the three indices can be specified numer-ically, at least with respect to interacting bounds. As was mentioned above, thecorrelation dimension is a lower bound to the Hausdorff dimension, i.e.,

DC ≤ DH , (6.2.25)

and, in fact, both concepts provide nearly identical values in many cases.It has further been conjectured that the Hausdorff dimension (and thus implic-

itly the correlation dimension) are related to the Lyapunov exponents. For exam-ple, in the case of a two-dimensional map with Lyapunov exponents λ2 < 0 < λ1the conjecture reads32

DH = 1 +λ1

|λ2| . (6.2.26)

The r.h.s. of (6.2.26) is also referred to as the Lyapunov dimension.33 In some cases,the Lyapunov dimension approximates the Hausdorff dimension fairly well.

As positive Lyapunov exponents indicate the stretching of an initial set on anattractor in a single direction and as the Kolmogorov entropy measures the averagerate of simultaneous stretching in all directions, both indices can be related by

K ≤∑i

positive λi. (6.2.27)

In some cases, ≤ can be replaced by the equation sign; equation (6.2.27) is thencalled Pesin’s identity.

Table 6.3 contains the calculated values of the correlation dimension, Lyapunovexponents, and Kolmogorov entropy for some prototype examples of chaotic dy-namical systems mentioned in the text.34

32 Cf. Ott (1981), p. 662, or Wolf et al. (1985), p. 289.

33 The general definition of the Lyapunov dimension is DL = j+

∑ji=1λi

|λj+1|with j fulfilling

the condition that∑j

i=1 λi > 0 and∑j+1

i=1 λi < 0.34 Calculations for other dynamical systems in different fields can be found, for example,

in Wolf et al. (1985), p. 289. Own calculations should be considered preliminarybecause the excessive time consumption allowed only limited data sets.

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6.2. Dimension, Entropy, and Lyapunov Exponents 221

Equation Correlation Lyapunov KolmogorovType Dimension Exponents Entropy

Lorenz † 2.05 ± .01 λ1 = 2.16 ≈ 0.13 (∗)

λ2 = .0λ3 = −32.4

Rossler ‡ 2.29 ± 0.06 (∗) λ1 = 0.13 ≈ 0.011 (∗)

λ2 = .0λ3 = −14.1

Henon § 1.21 ± .01 λ1 = .603 0.35 ± 0.02λ2 = −2.34

† Cf. eq. (4.2.1); s = 16.; r = 45.92; b = 4.‡ Cf. eq. (4.2.2); a = 0.15; b = 0.2; c = 10.§ xt+1 = 1 − 1.4x2

t + 0.3yt; yt+1 = xt

Statistical Properties of Prototype Strange Attractors. Sources: Grassberger/Procaccia (1983a,b), Vastano/Kostelich (1986), Wolf et al. (1985), owncalculations (∗).

Table 6.3

In applying these tools in empirical investigations of the possible presence ofchaotic motion in an actual time series, the following two-step procedure seems tobe appropriate:

Step 1: Calculate the correlation dimension. If DC is very high, the system isdominated by random influences and the hypothesis of the presence ofchaos should be rejected.

Step 2: If DC is low, calculate the largest Lyapunov exponent and the K2 approx-imation of the Kolmogorov entropy. If a positive Lyapunov exponent canbe detected and if K2 converges to a finite positive value, it can be con-cluded that chaos is present.

In addition, other tests may be necessary to confidently establish chaotic dynam-ics in a time series and they will be outlined in the following section on economicapplications of the concepts described above. These tests may become necessary be-cause all of these concepts involve numerically vague statements. As, for example,the sample size of the time series, the size of the embedding dimension, the radiusr in the correlation dimension or its sufficiently low value are not precisely deter-mined, room for subjective interpretation of the results remains in most empiricalapplications.

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222 Chapter 6

6.3. Are Economic Time Series Chaotic?

Before some recent results on possibly chaotic empirical time series are presented, itmay be appropriate to discuss whether the distinction between chaotic and randomsystems is relevant to economics.

The aim of business cycle theory over the decades was to model the basic un-derlying dynamics of an economy, implying regular fluctuations. Much in the spiritof the mechanistic worldview, the general tendency in reflections on the cyclicalbehavior of major economic time series was characterized by the attempt at isolat-ing the regular forces in oscillating time series and explaining them by appropriateassumptions concerning the structure of the economy. Though nobody could re-ally claim that the regularly oscillating linear economic systems like, for example,the multiplier-accelerator systems of the Samuelson-Hicks type could satisfactorilypicture actual time series, it was nevertheless believed that these models could pro-vide an example of the basic underlying economic dynamics. As actual time series areobviously characterized by a much more irregular behavior not only with respectto the monotonicity of cycles (i.e., they are reminiscent of noise) but also with re-spect to occasional interruptions in the amplitude and the frequency, the idea wasexpressed that actual business cycles may theoretically (i.e., abstractly) be describedby regular linear systems, but that it is necessary to include stochastic exogenousinfluences in order to provide a more realistic picture of the cycle.

It was impressively demonstrated by Slutzky (1937) and Kalecki (1954) that itmay be impossible to distinguish between time series generated by stochastic pro-cesses and actually observed historical time series. Furthermore, with some effortit is almost always possible to create hypothetical time series by means of appropri-ately chosen stochastic processes superimposed on linear dynamical systems whichdiverge only minimally from actual time series. Though this is a procedure whichcan only be executed ex post, the recent Rational Expectations literature on businesscycles has been dominated by the idea that linear difference or differential equa-tions with their implicit regularity constitute a good starting point for describingactual cycles when stochastic exogenous influences are included which offset theregular cycles permanently or from time to time.

The presentation of chaotic, nonlinear dynamical systems in Chapters 4 and 5attempted to outline a possible alternative to this stochastic linear approach. Whilestochastic influences can certainly not be completely ignored in satisfactory non-linear approaches to real-life phenomena, nonlinear economic dynamics is mainlyinterested in explaining most of the irregularity in actual time series with the help ofa deterministic approach. Recent work on empirical chaos in economics has there-fore concentrated on the question whether an arbitrary time series is generatedby a stochastic linear process or by a nonlinear process having the chaos prop-erty. Economically, the problem can be relevant because an agent who is aware ofthe deterministic character of a process and who has sufficient information on thestructure of the economy might be able to calculate the future development of theeconomy to some degree while another, stochastically oriented agent may resignin face of the seemingly too complex behavior of the system. From the practicalpoint of view of an agent it may be rather irrelevant whether he is confronted with

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6.3. Are Economic Time Series Chaotic? 223

a stochastic or a chaotic system because both kinds of systems may prevent himfrom making precise predictions, but from the theoretical point of view it is highlyinteresting which kind of dynamic behavior prevails because it may be the task ofpolitical institutions to eliminate possible information barriers.

When the statistical tools described above are to be applied to economic timeseries, a serious problem arises. In contrast to signal measurement in laboratoryexperiments where several tens of thousands of data points can easily be obtainedin a reasonable time in many cases, the shortest time unit of measurement in eco-nomics is usually a single day. Remembering that the majority of economic timeseries consists of annual, quarterly, or monthly data (with some weekly or daily datain well-organized surroundings like financial markets) and that the history of re-liable empirical research encompasses a period of at best 80-100 years, the lengthof a standard time series is shorter than the maximal value of n ≈ 10, 000, andwill typically consist of a few hundred (or less) data points. The reliability of thecalculated indices is therefore obviously limited.

An example of the direct application of correlation dimensions and Lyapunovexponents to macroeconomic data with a small sample size is reported in Brock(1986) in a test for deterministic chaos in detrended quarterly US real GNP datafrom 1947-1985. The Grassberger/Procaccia correlation dimension is calculated asDC ≈ 3.0 to 4.0 for an embedding dimension of m = 20, and the largest Lyapunovexponent is slightly larger than zero. With some precaution it could therefore beconcluded that chaotic motion in the GNP data cannot be excluded.

In order to uphold these findings, additional tests are desirable. An attemptto support or to reject the results of the standard procedures described above isBrock’s residual diagnostic.35

Theorem 6.1 (Residual Diagnostics)(Brock (1986)

): Let {at}∞t=1 be a

deterministic chaotic time series. Fit a linear time series model with afinite number of lags to the series, i.e.,

at + γ1at−1 + · · · + γLat−L = ut, t = L+ 1, . . . ,

where ut is the residual at time t and γ1, . . . , γL are the estimated co-efficients. Then, generically, the correlation dimension and the largestLyapunov exponent of {at} and {ut} are the same.

Brock (1986) applied this residual test to the same detrended U.S. GNP data asabove. The autoregressive AR(2) model

xt = 1.36xt−1 − 0.42xt−2 + ut, (6.3.1)

with xt as detrended GNP, fits the data very well, and Theorem 6.1 implies that, e.g.,the correlation dimension of the residuals {ut} must equal the formerly calculated

35 The following presentation of Theorem 6.1 differs slightly from the original. Althoughthe residual test is theoretically valid only in the infinite-dimensional case, it can serveas a discriminating tool even in low-dimensional cases.

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224 Chapter 6

value for {xt}. However, the dimension nearly doubles for alternate values of thelength of the involved ε-cubes. It follows that the presence of chaos in the US GNPdata should be excluded.36

Another additional procedure was proposed by Scheinkman/LeBaron (1989b)in the form of the shuffle diagnostics. In contrast to a stochastic process, the (re)-constructed attractor of a nonlinear dynamical system via m-histories of observedvalues possesses a geometrically ordered form if the motion is regular or chaotic.Suppose now that the same data points are observed, but that the time indexes arechanged. This leads to different m-histories and therefore a different form of theattractor in phase space. If the interchange of the time indexes is arbitrary, it canbe expected that the attractor will no longer display an ordered form, and, conse-quently, the correlation dimension will increase. This shuffling of the data can thusbe used as a test for deterministic nonlinear dynamics versus stochastic processes: af-ter shuffling the data, a nonlinear system will have a (probably substantially) highercorrelation dimension, while a stochastic process will almost always imply the samehigh dimension before and after the shuffling.37

The numerical tools presented above have been applied to a variety of economicdata. The following list is only an excerpt of ongoing work.38 Business cycle theoryand economic policy mainly deal with GNP and employment as the two macroeco-nomic key variables. Therefore, it is important to know whether the observed timeseries of these variables behave randomly, or a nonlinear structure is present. WhileBrock’s results on GNP data already suggest to reject the hypothesis of chaotic dy-namics for US data, Frank/Stengos’ investigations of Canadian (Frank/Stengos(1988a) and international (Frank/Gencay/Stengos (1988)) GNP data supportthe above findings. For detrended Canadian data the authors calculate a correla-tion dimension of ≈ 2.4 to 4.0 for varying embedding dimensions up to m = 20.However, the residual test nearly doubles the dimension. Shuffling does not leadto higher dimensions, as would be the case in the presence of chaos. Instead,the dimensions of the shuffled residuals even decrease. The average dimensionof German, Italian, and U.K. data is between 6.0 and 7.0; and the residuals donot possess significantly higher dimensions. However, shuffling the residuals al-ters the dimensions only slightly. Japanese data have a lower dimension, which istripled by shuffling. In all countries, the largest Lyapunov exponents are slightlynegative. Summarizing, international GNP data do not seem to be chaotic, though

36 Brock (1986) points out that this phenomenon can arise in so-called unit root processes,i.e., processes with standard deviation of {xt} close to one: although the process isstochastic, ordered pairs (xt, xt−1) nearly form a line in R

2, suggesting some kind ofordering.

37 In addition to these two supplementary diagnostics, other procedures have been pro-posed. Brock/Dechert/Scheinkman (1987) introduced the W -statistics, which is afamily of procedures based on the correlation dimension. Cf. also Scheinkman/Le-Baron (1989a). Brock/Dechert/Scheinkman (1987) developed the BDS statistics,which is a collection of tests based on the correlation integral and which discriminatesbetween the null hypothesis of i.i.d. random variables and the hypothesis of determin-istic chaos. Cf. Granger (1991) and Westlund (1991) for discussions of this statistic.

38 Surveys of recent work can be found in Frank/Stengos (1988b) and Brock (1987b).

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6.3. Are Economic Time Series Chaotic? 225

Subjects Authors

Social Products Brock (1986), Brock/Sayers (1988),Frank/Stengos (1988a, 1988c),Scheinkman/LeBaron (1989a)Sayers (1989), Potter (1991)

Unemployment Sayers (1988a, 1988b, 1989)

Pig-Iron Production Sayers (1989)

Exchange Rates Bayo-Rubio et al. (1992),Hsieh (1988), Papell/Sayers (1990),Meese/Rose (1991)

Gold/Silver-Returns Frank/Stengos (1987)

Stock-Market Returns Eckmann et al. (1988)Scheinkman/LeBaron (1989b)

Monetary Aggregates Barnett/Chen (1988a, 1988b),Barnett/Choi (1988),Ramsey/Sayers/Rothman (1990)

Price-Quantity-Adjust- Schmidt/Stahlecker (1989)ments (Industrial Data)

Experimental Sterman (1988),Behavior Sterman/Mosekilde/Larsen (1988)

A Sample of Empirical Investigations of Chaotic Time SeriesTable 6.4

there is evidence of low-dimensional nonlinearities. Sayers (1988a,b) studied pos-sible nonlinearities in the unemployment rates indirectly via man-days idle to work-stoppages. Calculations of the correlation dimension and the Lyapunov exponentsand application of the residual diagnostics to the detrended data suggested to denythe presence of deterministic chaos but it seemed as if nonlinear structure prevailsin the series. The author arrives at the same conclusion in a study of business-cycleindicators, including GNP, pig-iron production and unemployment rates for theU.S. (cf. Sayers (1989)).

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226 Chapter 6

One of the very few studies that unambiguously established low-order determin-istic chaos in an economic time series is due to Barnett/Chen (1988a) and hasinitiated some criticism. W. Barnett has often stressed that the usual simple sumindex for monetary aggregates is “not even a first order approximation to the exact ag-gregation theoretic aggregate” (Barnett/Chen (1986)), and has proposed the use ofDivisa indices especially in empirical investigations of monetary aggregates. Theconstruction of Divisa indices relies on neoclassical macroeconomic theory andmeasures the flow of monetary services as perceived by the users of these services.39

Barnett/Chen (1988a) examined several monetary aggregates with sample sizesof > 800 observations for the presence of chaos.40 For example, the correlation di-mensions of the Divisa analogs of the monetary aggregates M2 and M3 lie between1.0 and 2.0 for embedding dimensions up to m = 6. Figures 6.10.a and 6.10.bcontain Grassberger-Procaccia plots of the correlation dimensions of M2 and M3,respectively. The largest Lyapunov exponents are reported to be slightly positive.Other indices like a simple sum index of M2 or supply-side analogs of the Divisa-M2index display more noise. No evidence for low-dimensional chaos can be found inthe simple sum and Divisa aggregates of M1. Ramsey/Sayers/Rothman (1990)have demonstrated that the same original data set used by Barnett/Chen (1988a)does not show evidence of chaos when the data is transformed to a stationary timeseries.

Macroeconomic time series therefore do not seem to be good candidates fordeterministic chaos. It may be argued that statistical procedures in generatingthe data can introduce such a great amount of noise that low-order deterministicchaos has to be rejected. On the other hand, it should not be excluded per se thatparticular procedures like the calculation of Divisa indices are able to generatestructure in basically stochastic time series.

The results on macroeconomic data suggest to study data on the microeconomiclevel instead. At first glimpse, financial data like foreign exchange rates, stock ex-change rates, etc. indeed appear to be potentially good candidates for chaotic timeseries. Scheinkman/LeBaron (1989b) studied time series based on a set of morethan 5000 daily stock return rates. The correlation dimension was found to be≈ 5.0 to 6.0 for m = 14. The dimension of the residuals are reported to be thesame as those of the original series. Shuffling the data significantly increases thedimension, implying that chaos should not be rejected. Frank/Stengos (1989)studied gold and silver rates of return based on London daily prices. The corre-lation dimension of the daily data lies between 6.0 and 7.0 for m = 25. Shufflingyields higher dimensions for all series. The K2 entropies of the series are in therange of 0.15 < K2 < 0.24, and thus indicate the presence of deterministic chaos.

39 Cf. Barnett/Hinich/Weber (1986) and Barnett/Chen (1988a) for details on Divisaindices. The growth rates of calculated Divisa monetary indexes diverge drastically fromofficial monetary growth rates. As aggregate monetary data are based on certainly reli-able counting procedures, this divergence may be interpreted as a failure of neoclassicaltheory in the face of empirical problems.

40 See also Barnett/Choi (1988).

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6.3. Are Economic Time Series Chaotic? 227

-13.00 -12.00 -11.00 -10.00 -9.00 -8.00 -7.00 -12.00 -11.00 -10.00 -9.00 -8.00 -7.00 -6.00

log2 ε log2 ε

6.00

7.50

9.00

10.5

012

.00

13.5

015

.00

8.00

9.00

10.0

011

.00

12.0

013

.00

14.0

015

.00

log 2

C∗ n(

ε)

log 2

C∗ n(

ε)

Case 1: DDM2 Case 2: DDM3

6.10.a 6.10.bCorrelation Dimensions of Divisa Monetary Aggregates

Source: Barnett/Chen (1988a) (redrawn from the original)Figure 6.10

The studies mentioned above deal with statistical economic time series. Em-pirical economics is, however, not exclusively concerned with anonymous numberslike GNP, M1, or exchange rates but also encloses experimental studies of humanbehavior. Sterman (1988, 1989) and Sterman/Mosekilde/Larsen (1988) per-formed the following laboratory experiment: Human beings (mainly economists)were confronted with a multiplier-accelerator model of the business cycle. Theirtask was to manage capital investment when the model economy was in disequi-librium. The (usually suboptimal) behavior could subsequently be described by aspecific decision rule. A final simulation of the decision rule with parameters esti-mated from the experiment showed that a large number of the participants (40%)produced unstable behavior including chaos as measured by a positive Lyapunovexponent. While such a long-term simulation of a decision rule ignores learningeffects and the experimental data includes transient behavior, the laboratory exper-iment indicates that human behavior is much more complex than microeconomictextbooks suggest.

Summarizing this recent empirical work on deterministic chaos in economictime series, the following conclusions can be drawn:

• Actual economic time series differ from their analogs in the natural sciencesalmost always with respect to the relatively small sample size.

• As the small sample size does not lead to reliable results, supplementary tests arenecessary in empirical economics. These additional tests can reject the chaos

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hypothesis in those cases in which the standard procedures indicate the presenceof deterministic chaos.

• Chaotic motion cannot be excluded in several micro- and macroeconomic timeseries. It does not seem that microeconomic data like, e.g., financial marketsdata, are per se better candidates for the presence of chaos. The presence of noisein official data, the generation of structure in constructing particular indexes,or statistical preliminaries like detrending play essential roles in the findings.

• Even when the presence of chaotic motion cannot be established, evidence oflow-dimensional nonlinearities exists in many economic time series.

6.4. Predictability in the Face of Chaotic Dynamics

At first glance, the presence of deterministic chaos seems to imply rather destructiveeffects on the predictability of an actual time series or the trajectories in a theoret-ical economic model: if a model has sensitive dependence on initial conditions,arbitrarily (but finitely) precise digital computers are conceptually unable to calcu-late the future evolution of the system. When prediction is impossible, economicsloses a major justification for its mere existence.41

Statements like the one above contrast chaotic dynamical systems with modelsconstructed in the classic deterministic tradition. Compared with the regular be-havior in linear or quasi-linear dynamical systems, chaotic systems display a wildand irregular behavior, a superficial inspection of which suggests that it does notseem to possess structure at all. When standard prediction techniques rely on apurely deterministic approach, it is easy to claim a general failure of forecastingprocedures.

However, chaotic dynamical systems should not be compared with regular deter-ministic systems but with purely random systems or linear systems on which stochas-tic influences are superimposed. The foregoing presentation of theoretical andempirical results on chaotic dynamics showed that the presence of structure is theessential property of chaotic dynamical systems as compared with random series.If structure prevails, it is possible (at least to some degree) to predict the evolu-tion of the system. Stochastic systems or time series can allow the future behaviorto be anticipated with a (hopefully) given probability, and it may be possible todetermine a corridor for a variable’s probable amplitude. In contrast, if a systemis purely deterministic and chaotic, trajectories in a higher-dimensional system di-verge exponentially, but for sufficiently small time horizons it is possible to predictthe system’s evolution with an acceptable preciseness. Farmer/Sidorowich (1987,1988a,b) proposed local prediction techniques for chaotic time series which seemto be promising for short-term economic forecasting. The approach relies on thereconstruction of the attractor with the Takens method and the search for the near-est neighbor of a given point on the attractor. The simplest method for predictingthe next realized value consists in assigning the succesor of this neighboring point

41 Compare Baumol (1987) for discussion of the predictability problem in econometrics.

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6.4. Predictability in the Face of Chaotic Dynamics 229

Observed Predicted

Xt−T

Xt

Actual and Predicted Phase Spaces, Prediction: T = 1 PeriodFigure 6.11

to the predicted value. Numerical applications of this technique to different pro-totype equations show surprisingly low prediction errors for short time intervals.

The following figures show the results of applying the nearest-neighbor tech-nique to the data obtained from simulating the two-dimensional, discrete-timeKaldor model of Section 4.2.2. Figure 6.11 contains the phase spaces Yt vs. Yt−1 ofthe original system and the predicted evolution.42

The two phase spaces were obtained in the following way. The time evolution ofY and K in the discrete-time Kaldor model was calculated in the standard fashionfor n = 10000 iterations. A transient motion of 1000 iteration has been excludedfrom the consideration. The sequence {(Yt,Kt)}nmax

n0=1 represents the true motionof the system. Suppose that the observer considers only income as the relevant vari-able. The observer’s (predictor’s) task consists in deriving information on income’sfuture evolution from an available data set. Assume that the predictor has accessto n0 past values of income. The number n0 of past values is called the numberof atlas points. For the purpose of demonstrating the potential power of the pre-diction technique the economically rather unrealistic number of 2000 initial atlaspoints has been assumed. When the observer predicts the time evolution of incomebased on the available information, the true system continues to evolve accordingto the underlying deterministic laws of motion. In the phase space in the left part ofFigure 6.11, the pairs (Yt, Yt−1) are shown for the iterations n = 2001 to n = 10000.

At n0 = 2000, the observer analyses the available data with the nearest-neighbortechnique. Suppose he attempts predictions only for one iteration (time step).At the end of n0 + 1, it will be obvious whether he was wrong or right. At least,he will know another actual (true) value of income, namely Yn0+1. Based on theknowledge of n0 + 1 true values of income, he will predict the next income value

42 The calculations were performed with the NLF program of Dynamical Software.

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230 Chapter 6

XPt

Xt

Actual vs. Predicted Values, Prediction: 1 PeriodFigure 6.12

Observed Predicted

Xt−T

Xt

Actual and Predicted Phase Spaces, Prediction: T = 10 PeriodsFigure 6.13

etc. The phase space in the right part of Figure 6.11 shows the evolution of thelagged pairs (Y p

t , Ypt−1) of predicted values.

Aside from a negligible fuzziness, the two objects in the phase spaces are aston-ishingly similar. Indeed, a statistical regression of the actual versus the predictedtime series uncovers a nearly one-to-one relation between the variables (cf. Figure6.12) with a tremendous r2 = 0.9984.

It has been stressed many times in previous sections that trajectories of nearbypoints stay together for some time even in systems with chaotic dynamics. Thus,the coincidence of actual and predicted values in Figures 6.11 and 6.12 is not really

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6.4. Predictability in the Face of Chaotic Dynamics 231

XPt

Xt

Actual vs. Predicted Values, Prediction: 10 PeriodsFigure 6.14

surprising. The more relevant question concerns the longer-run predictability ofchaotic systems. For this purpose the nearest-neighbor technique has been appliedto the above income time-series with longer prediction intervals. The phase spacesin Figure 6.13 show the evolution of the true values of income ten iterations aheadfrom the prediction period (left part) and the evolution of the associated predictedvalues Y p

n+10 predicted in n. While the structure of the phase space of the truesystem can still be recognized in the right part, the fuzziness has considerably grown.In fact, the regression in Figure 6.14 yields an r2 of only 0.44 which is obviouslynot suited to support the hypothesis of a strong correlation between the observedand predicted values. When even longer prediction intervals are assumed, thecorrelation between the actual and observed values becomes negligible.

Although research in predicting chaotic time series is still in its infancy,43 thefollowing conclusion can already be drawn: if a time series is chaotic it may bepossible to predict the short-run evolution with a sufficient accuracy. Economicsshould therefore concentrate on the detection of chaotic time series. The presenceof deterministic chaos encourages short-term predictions and should not lead todesperations in face of the complex behavior.44

The possibility of predicting a chaotic time series does not mean that standardeconometric procedures constitute worse forecasting techniques per se. In additionto the fact that linear or completely random systems can best be treated with thesetechniques, it may even be possible to approximate the short-term evolution of a

43 Cf. Casdagli (1989) and Sugihara/May (1990) for discussions.44 Compare, however, the results found by Frank/Stengos (1989b). Their application

of the nearest-neighbor technique described above to return rates of precious metalsuncover the need for more elaborated algorithms. Cf. also Prescott/Stengos (1991).

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232 Chapter 6

chaotic time series fairly well. However, when chaos prevails, the development offorecasting techniques which explicitly take the uncovered structure into accountis desirable.

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Chapter 7

Catastrophe Theory and Economic Dynamics

This final chapter deals with catastrophe theory and its role in economic dynam-ics. Catastrophe theory was very popular in the 1970s and was considered a

promising technique for the modeling of discontinuous jumps in the state variablesof a dynamical system. In applications of the theory such interesting empiricaltopics like the abrupt emergence of aggression in the behavior of various species,stock-market crashes, the capsizing of ships, etc. were studied. All these exam-ples describe phenomena which are characterized by an immediate, discontinuouschange in a variable.

After initial celebrations, it has been argued that catastrophe theory is not well-suited as an analytical tool and that it can at best serve as a heuristic tool in preparinga theory. Though several economic examples of applications of catastrophe theoryexist, it does not seem to be quite clear whether future work in dynamical eco-nomics will further elaborate on the theory. A short introduction will neverthelessbe presented in the following because there seems to exist a confusion regardingthe qualitative differences between chaos and catastrophes.

Though the label catastrophe theory suggests a discussion of disastrous events, itdeals with mathematically less spectacular behavior. Catastrophe theory constitutesan attempt to classify bifurcation phenomena in some families of structurally sta-ble functions. The choice of the term catastrophe theory will become apparentwhen it will be demonstrated that at singular points the state variables jump to newequilibrium values in an abrupt (catastrophic) fashion.1

1 Introductions to the theory can be found in, e.g., Saunders (1980), Arnold (1984),and Zeeman (1977), Chapters 1-2. See also Thom (1977).

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234 Chapter 7

7.1. Basic Ideas2

The subject of catastrophe theory is the classification of sudden jumps – or “catastro-phes” – in the behavior of dynamical systems. Consider a family of one-dimensionalfunctions which are parameterized by an n-dimensional vector α:

V = V (x,α), x ∈ R, α ∈ Rn (7.1.1)

Let V be an analytic function such that it can be written as a polynomial of theform3

V (x,α) = xn + α1xn−1 + . . .+ αnx

0, (7.1.2)

with someαi being possibly equal to zero. For a given n, the graph of the polynomial(7.1.2) has different geometric shapes when some parameters vanish. For example,consider the case n = 4 with αi = 0, i = 1, 2, 3, 4. The graph of x4 is quite differentfrom that of x4 +α1x

3. Depending on the number of vanishing αi’s, one or severalextrema of the function may occur.

Catastrophe theory concentrates on those forms of (7.1.2) which are structurallystable. A function like (7.1.2) with some αi being possibly equal to zero is said to bea structurally stable function if the number and the character of the function’s extremado not change when some of these αi change value.4 For example, the expressionh = x4 is not structurally stable because h = x4+α1x

3 has additional extrema. It canbe shown that for n = 4 the polynomial x4 + α2x

2 + α3x is structurally stable. Thisstructurally stable form of the polynomial (7.1.2) for a given n is called the universalunfolding of xn. The number of parameters which is necessary to “stabilize” xn fora given n is called the codimension of the unfolding, e.g., x4 has codimension two.

Catastrophe theory proves that for a codimension ≤ 4 exactly seven differentuniversal unfoldings exist, namely four unfoldings for the one-dimensional case(7.1.2) and three unfoldings in the two-dimensional case. This is the essential resultof Rene Thom’s famous classification theorem (cf. Thom (1977) for an introduc-tion), in which the universal unfoldings are labelled elementary catastrophes. Table7.1 lists these seven simplest universal unfoldings with codimension ≤ 4 togetherwith their pet names.

In order to demonstrate the relevance of the universal unfoldings for the behav-ior of dynamical systems consider the system

z = g(z), z ∈ Rn. (7.1.3)

2 Parts of the following two sections are essentially identical with material contained inSections 5.2.1 and 5.2.2 in Gabisch/Lorenz (1989).

3 Compare for the following Saunders (1980), pp. 17ff., and Poston/Stewart (1978),pp. 92ff.

4 Note that this definition of structural stability refers to a function and not to dynamicalsystems. Recall that a dynamical system is structurally stable if the solution curves aretopologically equivalent when a parameter is varied.

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7.1. Basic Ideas of Catastrophe Theory 235

Unfolding CoDim Pet Name

x3 + α1x 1 Foldx4 + α1x

2 + α2x 2 Cusp

x5 + α1x3 + α2x

2 + α3x 3 Swallowtail

x6 + α1x4 + α2x

3 + α3x2 + α4x 4 Butterfly

x3 − xy2 + α1(x2 + y2) + α2x+ α3y 3 Elliptic Umbilic

x3 + y3 + α1xy + α2x+ α3y 3 Hyperbolic Umbilic

y4 + x2y + α1x2 + α2y

2 + α3x+ α4y 4 Parabolic Umbilic

The Universal Unfoldings with Codimension ≤ 4Table 7.1

Assume that the variables can be divided into fast and slow variables. For example,let z1 be an extremely fast variable. In that case the other variables z2, · · · , zn can beinterpreted as “parameters” which change very slowly. The variable z1 immediatelyreacts to disequilibria and moves infinitely fast to an equilibrium value once it hasbeen displaced from an equilibrium value. Consequently,5

z1 = 0 = g1(z1, · · · , zn) ∀ t. (7.1.4)

The equation g1(z1, · · · , zn) = 0 describes an n− 1 – dimensional manifold in Rn.In the catastrophe-theoretic literature it is common to denote this manifold as anequilibrium surface. The idea that z1 = 0 ∀ t then implies that the motion of thesystem (7.1.3) is described by the n − 1 remaining differential equations for zi,i = 2, . . . , n, defined to take place on the z1 = 0 – surface.6

Assume the new symbols z1 = x, α = (z2, · · · , zn), and m = n − 1. Equation(7.1.4) can then be written

x = 0 = f(x,α), x ∈ R, α ∈ Rm.

5 Actually, the same arguments as those provided in the discussion of relaxation oscilla-tions in Section 2.5 can be applied to this case: it has to be assured that the variableindeed returns to the previous value.

6 The idea of distinguishing variables according to their different adjustment speeds isalso realized in the so-called adiabetic approximation, which has played a major role inthe synergetics literature (cf. Haken (1983b) for details). A variable like z1 is said to beslaved by slower variables because the motion of z1 on the manifold depends exclusivelyon the change in the slow variables. The adjustment equation for the slowest variableis called the master equation. Economic applications of this technique can be found inMedio (1984a) and Weidlich/Haag (1983).

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236 Chapter 7

Suppose that a function F (x,α) exists such that Fx ≡ f(x,α) = x. A dynamicalsystem which can be derived from such a function F (x,α) is called a gradient system.7

Catastrophe theory deals with dynamical systems x = f(x,α) for which F (x,α) isidentical with a member of the family of structurally stable universal unfoldingsV (x,α). In other words, it concentrates on those equilibrium surfaces which canbe interpreted as the first derivative of a universal unfolding, i.e., f(x,α) = 0 =Fx(x,α) ∼= Vx(x,α).

The properties of these specific equilibrium surfaces can be described by inspect-ing their singularity sets and their bifurcation sets. The singularity set S is definedas

S = {(x,α) ∈ R × Rm | Vxx = 0}, (7.1.5)

i.e., the set of all (x,α) for which the second derivative of the unfolding is equalto zero.8 Geometrically, the singularity set consists of all parameter combinationsfor which the equilibrium surface is tangent to the direction of the variable x. Theprojection of the singularity set on the parameter space is called the bifurcation setB:

B = {α ∈ Rm | Vxx = 0}.

The dynamic behavior in the fold catastrophe as the simplest catastrophe is essen-tially identical with the behavior in a continuous-time system undergoing a foldbifurcation (cf. Section 3.2.1): for α1 > 0, no equilibrium exists in the associateddynamical system. For α1 = 0, a bifurcation occurs at x = 0, such that for α1 < 0a stable and an unstable equilibrium branch exist.9 The rest of this section willtherefore deal with the second unfolding which has been coined cusp catastrophe.

The unfolding of the cusp catastrophe,

V (x) = x4 + α1x2 + α2x, (7.1.6)

has an equilibrium surface

M : 4x3 + 2α1x+ α2 = 0, (7.1.7)

and a singularity set

S : 12x2 + 2α1 = 0. (7.1.8)

7 Gradient systems are rare in economics because the so-called potentials from which theyare derived usually do not exist. The requirement of the existence of a potential canhowever be replaced by the weaker condition of the existence of a stable Lyapunovfunction (cf. Section 2.1).

8 In the multi-dimensional case, the determinant of the Hessian matrix, i.e., the matrixof second-order derivatives, must be equal to zero.

9 Cf. Gabisch/Lorenz (1989), pp. 205f., for a short discussion.

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7.1. Basic Ideas of Catastrophe Theory 237

The bifurcation set can be obtained by eliminating x from M and S, yielding

B : 8α31 + 27α2

2 = 0. (7.1.9)

Incidentally, (7.1.9) is exactly the formula for the discriminant of the equilibriumsurface equation (7.1.7). It follows that (7.1.7) has three real roots, which eitherall coincide if α1 = α2 = 0, or two of them coincide if α1 and α2 are distinct. Asthe unfolding (7.1.6) is an object in the four-dimensional space (V (x), x, α1, α2),a geometric presentation of the cusp catastrophe has to start with the equilibriumsurface (cf. Figure 7.1)

The Cusp CatastropheFigure 7.1

The term ‘cusp’ catastrophe is immediately obvious from the shape of the bifur-cation set. The state variable is always located on ‘top’ of the equilibrium surface.It becomes apparent that as soon as the parameters are changed in such a way thatthe state variable reaches the singularity set at B (cf. Figure 7.2) after having movedon the upper part of the surface, the variable x will jump down to the lower part ofM in Figure 7.1.

If the long-run movement of α1 and α2 is such that a motion on the lowerpart from C to D occurs, then there will be another jump back to the upper partat D, which again belongs to the singularity set. Considering these motions inthe parameter space only, it follows that catastrophes occur exactly every time thebifurcation set is crossed from the inside of the area delimited by this set (cf. Figure7.2).

The motion A-B-C-D-E has been drawn in Figure 7.1 under the implicit assump-tion that α1 changes only very slowly. When α1 is allowed to change with a higherspeed, other scenarios can occur in this model. Assume, for example, that the sys-tem is located at F. It is possible that a trajectory on the surface first moves toward

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238 Chapter 7

The Bifurcation Set of the Cusp CatastropheFigure 7.2

C and proceeds as described above. It might also be possible that the trajectory re-turns to the upper part of the surface via a route around the cusp point. In this case,no catastrophe occurs because the bifurcation set is never crossed. Which route willactually be followed depends, of course, on the specific forms of the equations in(7.1.3).

The other different elementary catastrophes are naturally more difficult to illus-trate and require the constancy of one or more parameters in order to be presentedgraphically. As most economic applications of catastrophe theory concentrate ongeometric aspects, it seem as if the higher catastrophes do not possess much rele-vance to economics. The interested reader is referred to Poston/Stewart (1978)for a detailed description of other elementary catastrophes.

It has been argued that catastrophe theory hardly deserves the label ‘theory’ atall. Indeed, catastrophe theory provides information on the possible types of behav-ior in a dynamical system, but can actually not answer the question of what preciselyhappens in a roughly specified system. The motion on the equilibrium surface de-pends on the dynamics of the slow variables, and without explicit knowledge ofthese slow dynamics it is impossible to say anything about the dynamics of the statevariables. However, once a dynamical system is precisely specified, it is unnecessaryto refer to catastrophe theory because the dynamic behavior can be studied moreeasily with the help of other tools. Catastrophe theory should therefore be viewedas a heuristic tool in studying problems for which little is known about the formaldynamics of the system.

Catastrophe theory has been applied to a variety of economic problems,10 in-cluding governmental behavior, stock-exchange crashes (cf. Zeeman (1977)), andsmooth dynamics in Malinvaud’s (1977) macro-model with rationing (cf. Blad(1981)). Birchenhall (1979) discussed a possible structural instability in the

10 A survey of the most relevant contributions can be found in Rosser (1991). Comparealso Balasko (1978) for a critical evaluation of catastrophe theory in economics.

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7.2. The Kaldor Model in the Light of Catastrophe Theory 239

Walrasian tatonnement in the light of catastrophe theory. A Cournot oligopolymodel was studied in Furth (1985). Intertemporal equilibria in an Arrow-Debreumodel were investigated in Weintraub (1980). Rosser (1983) re-considered there-switching debate in capital theory of the late 1960s with a form of the cusp catas-trophe. A critical discussion of the role of catastrophe in economics is containedin Balasko (1978). In the following, two economic examples of catastrophe mod-elling will be presented which seem to be particularly accessible.

7.2. The Kaldor Model in the Light of Catastrophe Theory

Varian’s (1979) extension of the Kaldor model constitutes one of the first economicapplications of catastrophe theory. Consider the three-dimensional system11

Y = α(I(Y,K) − S(Y,W )

),

K = I(Y,K) −D,

W = γ(W∗ −W ),

(7.2.1)

with W as ‘wealth’, W∗ as the long-run equilibrium value of wealth, γ > 0 asan adjustment coefficient, and D as autonomous and constant depreciation. Theinvestment function is of the well-known Kaldor type. Suppose that savings is nega-tively related to wealth in such a way that not only the income-independent part ofsavings but also the marginal propensity to save falls when wealth increases. Assumethat the usual Kaldor scenario with three partial equilibria in the goods market pre-vails when W = W∗. When wealth is displaced form its long-run equilibrium valuethe savings function changes its position and slope. For sufficiently strong displace-ments of wealth fromW∗ a single equilibrium exists in the goods market. It followsthat the fold in the Y = 0 – curve (cf. Figure 2.15) disappears for high and lowvalues of W .

A three-dimensional representation of the Y = 0 locus can be understood asthe combination of different Y = 0 layers belonging to different values of W . Forvalues of W in the neighborhood of W∗ the associated (Y,K) – planes display thetypical fold region; for high and low values of W the Y = 0 – curves in the (Y,K)planes possess a negative slope. Figure 7.3 shows the Y = 0 – surface for W < W∗.The underlying economic scenario implies that the second part of the surface forW > W∗ has a similar shape, implying that actually two cusp points exist with a foldregion in the middle.

Assume that wealth and the capital stock are relatively slowly changing variablesas compared with income. The model (7.2.1) then fulfills the requirements of catas-trophe theory, and the system always operates on top of the equilibrium surface.

Suppose that the long-run fixed point (Y ∗,K∗,W∗) is located on the uppersheet of the manifold (cf. point E in Figure 7.3). If a small disturbance of the

11 A longer presentation of the model can be found in Gabisch/Lorenz (1989), pp. 209ff.A similar model is described in George (1981).

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240 Chapter 7

A Three-Dimensional Kaldor-ModelFigure 7.3

equilibrium occurs, the system returns to the equilibrium monotonically accordingto the dynamics of K and W . However, once K is increased such that the pointB is crossed, a catastrophe occurs and income jumps down to the lower branch ofY = 0. A slow movement along Y = 0 is initiated until the bifurcation point C isreached where another catastrophe occurs and where Y jumps back to the upperbranch. Eventually, Y will approach the stable equilibrium E.

However, the shock may be extremely large, and wealth may decrease to a verylow value. Depending on the relative adjustment speeds of K and W , the motiontoward the fixed point may not be characterized by a catastrophe but by a smoothadjustment path around the cusp point. This latter path can be interpreted asreflecting a depression in contrast to the former recession because the adjustment tothe long-run equilibrium around the cusp point requires more time than the pathover the bifurcation set.

The catastrophe-theoretical extension of the Kaldor model made it possible tomodel a phenomenon which cannot occur in the original version. Note, however,that the above description of the possible behavior of the system has not mentionedwhich dynamical behavior actually prevails. Whether the long-run equilibrium ofa system is stable or unstable and whether a trajectory moves over the bifurcationset or around the cusp point, depends on the concrete specification of the modeland the values of the adjustment parameters. Catastrophe theory can only provideinformation on the necessary structure of the dynamical system in which a certainphenomenon should be modeled.

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7.3. A Catastrophe-Theoretical Approach to Stagflation 241

7.3. A Catastrophe-Theoretical Approach to Stagflation

Macroeconomic theory has had some problems (and still has) with a phenomenonthat was coined stagflation, i.e., the simultaneous presence of inflation and zerogrowth rates of the national product, coupled with nonzero and usually high unem-ployment rates. An attempt to model this phenomenon consists of modifying theoriginal Phillips curve by introducing additional influences like, e.g., the expectedinflation rate πe. A modified Phillips curve

π = f(u, πe), fu < 0, fπe > 0, (7.3.1)

with π as the actual inflation rate and u as the unemployment rate will thereforeshift in (u, π)-space for different values of πe.

The modified Phillips curve can explain the simultaneous presence of high in-flation and unemployment if inflationary expectations are high. As the stagflationphenomenon emerged in many western countries in the mid-1970s, an increasein inflationary expectations therefore should have been observed in this period ascompared with the late 1960s when inflation and unemployment exhibited the tra-ditional trade off. However, such an increase could not be established empirically,and the modified Phillips curve therefore does not constitute a satisfactory modelfor understanding stagflation.

An alternative way of modelling the stagflation phenomenon was proposed byWoodcock/Davis (1979) in the form of a catastrophe-theoretical approach. As-sume that actual inflation rates change according to

π = g(π, u, πe, . . .). (7.3.2)

Equation (7.3.2) may be thought of as a single law of motion among a set of dif-ferential equations describing the evolution of the other variables u, πe, etc. Inthe fashion of catastrophe theory, suppose that actual inflation rates adjust to their(partial) equilibrium values much faster than the remaining variables. If the adjust-ment speed is infinitely high, π = 0 ∀t and the remaining variables can be treatedas parameters. The equation g(π, u, πe) = 0 then describes the equilibrium surfaceof (7.3.2). Writing g(·) = 0 explicitly as π = f(u, πe) yields the same form as in(7.3.1), i.e., the modified Phillips curve.

This is a purely formal presentation, which only indicates the possible derivationof the Phillips curve from a dynamical system. However, the catastrophe-theoreticelements introduced above suggest that the equilibrium surface may have a compli-cated shape. In fact, Woodcock/Davis proposed a cusp-like equilibrium surfacewith a folding for high values of expected inflation rates (cf. Figure 7.4). Sup-pose that an economy is located at an initial point A on the upper sheet of thesurface. Whether or not the economy moves to different locations on the surfacedepends on the motion of the slow variables (parameters) u and πe. Assume thatthe unemployment rate can directly be influenced by fiscal policy. There are surelymultiple determinants of expected inflation, but it is possible that the governmentand monetary authorities can manipulate expectations to some degree.

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242 Chapter 7

The Woodcock/Davis Stagflation ModelFigure 7.4

If the government attempts to lower the high inflation rate at A by means ofexpenditure cuts, how fast the economy reaches a location on the lower sheet ofthe equilibrium surface depends on the change in the expected inflation rate. Ifexpectations do not change and if expenditure cuts are large, the economy movesalong A-C-D and experiences a catastrophe at C, such that the inflation rate dropsmore or less immediately. This situation may be considered unrealistic becausedrastic decreases in the inflation rate are typically unobservable.

Assume therefore that governmental and monetary authorities succeed in low-ering the expected inflation rate while the economy is still characterized by highactual rates. Governmental expenditure cuts will then imply a motion along A-E-Fif inflationary expectations decline immediately, or along A-B-E-F if expectationsbegin to decrease with a time lag.

The latter way around the cusp point requires more time to achieve the goal ofa low inflation rate than the fast way over the bifurcation set at F. For a considerabletime interval rising unemployment rates go hand in hand with a gradually decreas-ing inflation rate. Woodcock/Davis therefore claimed that motions around thecusp point are proper descriptions of the stagflation phenomenon.

The economic meaning of this scenario can be questioned. The change of theexpected inflation rate affects the results in a crucial way but the model does notexplain the determinants of expected inflation. Furthermore, as the slow motionis generated by governmental expenditure cuts, the government may abandon theanti-inflation program because results are not observable within a reasonable timeinterval.

While this model is therefore not completely convincing from a theoretical pointof view, it has turned out that this catastrophe-theoretic approach to the modifiedPhillips curve may fit observable data better than a traditional linear approach.Fischer/Jammernegg (1986) studied US data for the period 1966-1983 and found

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7.3. A Catastrophe-Theoretical Approach to Stagflation 243

7.5.a. Cusp Model 7.5.b. Linear ModelActual U.S. Inflation Rates 6:66 - 6:83 and Estimates

Source: Fischer/Jammernegg (1986), p.16 (redrawn from the original)Figure 7.5

that an appropriately modified, discrete version of the Woodcock/Davis approachto stagflation is superior to the standard investigation of the equation

πt = a+ bπet + cut−1. (7.3.3)

Figures 7.5.a-b show the actual inflation rates (solid lines) and the estimated infla-tion rates (dashed lines) for the cusp model (Figure 7.5.a) and the linear model(Figure 7.5.b). Obviously, the model inspired by catastrophe theory fits the actualdata much better, particularly with respect to peaks in the inflation rate.

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Chapter 8

Concluding Remarks

The foregoing presentation has, hopefully, made it evident that dynamical eco-nomics can be enriched by incorporating recent developments in the theory

of nonlinear dynamical systems. However, a few final remarks seem to be in order.The general tendency in all mathematical theorems and economic applications pre-sented in this short survey of nonlinear dynamics is that even the simplest dynamicalsystems may involve intuitively unexpected phenomena and highly complicated mo-tions of the state variables. While traditional investigations of an evolving economy(especially in business-cycle theory) have concentrated on regularity aspects, andwhile recent revivals of (new-) classical macroeconomics scroll the recognized irreg-ularities back to the noneconomic exogenous world, nonlinear dynamical systemsallow for an entirely new theoretical attitude toward an understanding of cycli-cal motion which must not necessarily be irregular or chaotic. By an appropriatechoice of nonlinearities it is almost always possible to model a particular dynamicalphenomenon which is believed to prevail in reality.

It can be argued that the subject of economic theorizing is not the search forcomplex dynamics in simple deterministic systems, but instead the abstraction fromunnecessary complications and the search for simple dynamics in complicated sys-tems. This is the same philosophy that justifies partial theorizing or highly aggre-gated macroeconomics. The procedure can imply useful results if an economy (atleast in tendency) follows these simplified rules. While abstraction and simplifi-cation dominate classroom economics for good reason, professional economistslike forecasters and advisers have to modify the basic models because reality obvi-ously cannot be grasped by, e.g., simple IS-LM models. The standard procedure inconstructing forecasting models consists in expanding the basic model by introduc-ing new variables, structural and behavioral equations, and stochastic exogenous

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Concluding Remarks 245

influences. Although most parts of large economic models are simply structuredingredients, the models in their entities are nevertheless highly complex systemswhose behavior might be unpredictable.

If nonlinear dynamical economics can teach a lesson to traditional theorizingin economic dynamics then it should run along the following lines: many basicstatements in dynamical economics are derived from the investigation of linear ornearly-linear dynamical systems. These statements have occasionally amounted toquasi-axioms in the sense that the results derived from linear models have paradig-matically been taken for granted in general cases which may involve nonlinearities.It has been attempted to demonstrate above that the introduction of numericallyslight nonlinearities may sometimes drastically change the dynamic behavior of astandard model. For example, a competitive economy may no longer be character-ized by the usual asymptotical stability of its equilibrium, but may instead exhibitperiodic orbits, quasiperiodic behavior, or even chaotic motion. Thus, the maincontribution of the recent developments in dynamical systems theory to economictheory may consist of a more sensitive attitude toward the role of nonlinearities ineconomics. A model which exhibits simple regular behavior in its linear version mayperform completely different once it is reformulated in order to include nonlinearaspects.

As the section on the empirical relevance of chaotic motion has demonstrated,it is not easy to establish the existence of deterministic chaos in an actual time se-ries, but there do exist examples of chaotic economic time series. Nevertheless, theresults still leave a suspicion about the involved statistics. An economy (as it is under-stood by the profession) is not an isolated system, acting without interference fromother abstracted subsystems of the society. Thus, influences from other subsystemscan never be avoided; they appear in a model in the form of noise, fluctuations,and exogenous shocks. Economics will therefore particularly gain from recent at-tempts to understand noisy chaos, i.e., deterministic complex motion disturbed bynoisy exogenous influences.1

Economic theory is always abstracting. It must necessarily abstract from thenumber of individual units in an economy, from qualitative differences betweengoods and services, from individual motivations to act in a certain way, etc., in orderto derive any results at all. Even if a theoretical economic model fits the world fairlywell in a numerical examination, this does not imply that the model is a perfectpicture of the real life. If a linear model with stochastic ingredients happens to fitchaotic data sufficiently well, it can be justified to use such a model in describingreality. Alternatively, a chaotic dynamic model can be useful even if the observedtime series are not chaotic. As complex phenomena like actual economic time seriescan be modeled more easily in nonlinear systems, these models seem to possess anadvantage over the traditional linear approach.

The dynamic phenomena presented in this book like local bifurcations to sev-eral fixed points or to closed orbits, the existence and uniqueness of limit cycles,

1 Compare Kapitaniak (1990) for an introduction to the mathematics of noisy chaos.An investigation of the influence of noise on the dynamics of the Goodwin model (cf.Section 2.4.2) can be found in Lines (1988).

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246 Concluding Remarks

or the chaos property allow one to model an empirical observation with relativeease and may contribute to a better understanding of reality. Nonlinear dynamicsmay be particularly useful in sub-disciplines of economic dynamics which still lackan explicit formal presentation. Recent formal work on evolutionary economicsand innovation2 constitute a major step in understanding the long-term behaviorof an economy. It may turn out that nonlinear dynamics are not especially rele-vant in traditional economic theory, but in fields still to be elaborated upon. Atthe least, these nonlinear phenomena may serve as an instrument in moving be-yond the restricted concentration on linear dynamical systems which was typical forthe mechanistic worldview outlined in Chapter 1. However, the fact that a modelof competitive processes or of macroeconomic business cycles exhibits strange dy-namics does in and of itself not imply that reality is indeed characterized by exactlythese dynamics. On the contrary, it is probable that actual economic time series aregoverned by an interaction of immanent nonlinear structures, stochastic noise, andexogenous shocks whose overall effect can never be estimated with precision. Thecontribution nonlinear dynamical economics has made to economic theory overthe last decade should be viewed as a provision of new and additional argumentswhy an estimation of the structure and the dynamical behavior of an economy canbe doomed to imperfection.

Nonlinear, deterministic systems like the ones considered in this book may con-tribute to a better understanding of economic motion but it should be stressedthat difference and differential equations do not represent the only formal toolsfor descriptions of economic motion. It has recently been emphasized that mixeddifference-differential equations may constitute more appropriate dynamical sys-tems for explaining a variety of phenomena.3 It can also be questioned whethereconomic motion is appropriately specified with these various functional descrip-tions. An alternative modeling procedure consists in establishing a set of discreterules which determine the evolution of an economy. So-called cellular automata rep-resent dynamical systems defined for discrete economic variables, the evolution ofwhich is determined by discrete decision rules, i.e., rules of the form: if the systemis in state Xi and if Xi fulfills a specific criterion, then change Xi to Xj .4 The in-teresting property of these automata in the present context can be seen in the factthat these systems are occasionally able to generate chaotic motion. The remarkson the possible relevance of chaotic motion made in the previous sections thereforedo not seem to be superfluous in the light of these recent developments.

In a somewhat speculative manner, it can be argued that chaos is an all-embrac-ing principle of life.5 When a stable stationary point is identified with dead matter(e.g., with inactive Schumpeterian innovators), then it is tempting to identify a vital

2 Cf. Arthur (1988, 1989) or Silverberg (1988).3 Economic applications can be found in, e.g., Jarsulic (1993) and Wen/Chen (1992).

Cf. Cushing (1977) for a survey of the involved mathematics.4 An overview of cellular automata systems with many examples form the natural sciences

can be found in Gutowitz (1991). An economic application is discussed in White(1992) in a model of urban evolution.

5 Cf. the general discussions in Gleick (1987) and Nicolis/Prigogine (1989).

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Concluding Remarks 247

organism with the opposite extreme, a chaotic state. In fact, conjectures exist that,for example, brain waves are chaotic.6 J.D. Farmer summarizes these ideas in theparable:7

Human beings have many of the properties of metastable chaotic solitary waves. (I say meta-stable because all of us eventually die and become fixed points.) Old age might be defined asthe onset of limit cycle behavior. May your chaos be always of high dimension.

It is surely too early to declare that chaos is the essential characteristic of economiclife. It can also not be excluded that economic reasoning will declare chaotic mo-tion as a theoretically interesting but empirically irrelevant phenomenon. However,the recent empirical research has uncovered the dominating presence of nonlin-earities in actual economic time series, implying that economic life is almost alwayscharacterized by complicated (though not necessarily chaotic) processes. It seemsas if the harmonic attitude toward life typical in the linear and mechanistic world-view can finally be rejected on the grounds of the current findings in many differentscientific disciplines. Once it has been accepted that the linear worldview is an ar-tificial and paradigmatically defected construction, complex dynamics will not beviewed as a destructive contribution to established truth anymore, but will be con-sidered as a promising concept in understanding real life phenomena.

These ideas and this book should therefore be concluded with a bonmot by Her-mann Haken, namely8

. . . (a) higher degree of order does not necessarily imply a higher content of meaning.

6 Cf. Glass/Mackey (1988) for a discussion of chaotic motion in biological systems.7 Farmer (1982b), p. 2448 Haken (1982), p. 2.

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Appendix

This appendix contains some material, the knowledge of which is useful (if not mandatory)for an evaluation of nonlinear dynamical systems but which is not directly related to thetopics mentioned in the main text. Besides, a discussion would have interrupted the lineof thought in an unnecessary way.

The first section recalls some fundamental properties of linear dynamical systems incontinuous and discrete time. Though this book deals with nonlinear dynamical systems,a thorough understanding of linear systems is nevertheless important because the localstability properties of a fixed point in nonlinear dynamical systems are studied with thehelp of linear approximation techniques. The section briefly recalls the standard methodsin solving linear one- and two-dimensional dynamical systems in continuous and discretetime. The Routh-Hurwitz criterion and the Schur criterion for determining the stabilityproperties of n-dimensional dynamical systems are included for the sake of completeness.The section also contains an outline of linear subspaces (eigenspaces) in n-dimensionalsystems. These spaces are relevant for the approximation of the behavior of a nonlinearsystem with the help of linear systems in the neighborhood of fixed points.

When a dynamical system is nonlinear its dimension constitutes a much more relevantaspect than in comparable linear systems. The difficulties involved in the calculation offixed points, bifurcation values of parameters, etc. increase with an increasing dimensionof a system. The investigation of the dynamic behavior of an n-dimensional system on itscenter manifold occasionally represents a method for a systematic reduction of the effectivedimension of the relevant, dynamic subsystem. The outline of center manifold theory in thesecond section attempts to illustrate the calculation of center manifolds for continuous-timesystem with and without a parameter dependence. The discrete-time case can be treatedmore or less analogously and is mentioned in passing.

The third section deals with different types of time lags in economic models. The avail-ability of a large number of mathematical results for ordinary differential equations seemsto have distracted the attention of economists from the modeling of delayed dynamicalsystems. In addition, as ordinary differential equations can implicitly represent a certaintime-lag structure, at least a rough knowledge of possible delay structures is desirable foran understanding of dynamical systems in economics. The section also includes a short dis-cussion of the advantages of the use of operators in investigating dynamical system. In par-

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A.1. Basic Properties of Linear Dynamical Systems 249

ticular, the introduction of operators can be very helpful in studying the relation betweendifferent types of lags or between continuous-time and discrete-time dynamical systems.

The appendix closes with a few remarks on the value of simulations in nonlinear dynam-ics. Numerical simulations are performed on digital computing devices with finite precisionand the question arises whether the detection of chaos is affected by this inaccurateness.Besides, in many cases it is mandatory to apply numerical approximation techniques inorder to obtain any results at all, implying that the immanent inaccurateness is further in-creased. While no way to overcome this phenomenon can be offered, the section containssome suggestions for properly interpreting the results of numerical simulations.

A.1. Basic Properties of Linear Dynamical Systems

The following section presents a very short survey of the phenomena observable in linear dy-namical systems. This section does not claim to be complete,1 but is intended as a reminderof the most important dynamic phenomena. It concentrates on those aspects of linear sys-tems which are especially interesting in comparison with analogous nonlinear systems. Assome important qualitative differences exist between continuous-time and discrete-time dy-namical systems, they will be presented separately.

A.1.1. Continuous-time Dynamical Systems

Consider an n – dimensional, linear, continuous-time, dynamical system with constant co-efficients

x1(t) = a11x1(t) + . . . + a1nxn(t) + c1,

x2(t) = a21x1(t) + . . . + a2nxn(t) + c2,

...xn(t) = an1x1(t) + . . . + annxn(t) + cn,

xi, αij ∈ R, (A.1.1)

with x(t) = dx(t)/dt, or, in vector notation,

x(t) = Ax(t) + c, x ∈ Rn, t ∈ R, (A.1.2)

with A as the n× n matrix

A =

a11 a12 . . . a1n

a21 a22 . . . a2n

......

. . ....

an1 an2 . . . ann

, (A.1.3)

1 Extensive treatments of linear dynamical systems with many economic examples can befound, e.g., in Allen (1963), Chapters 5 and 6, Brock/Malliaris (1989), Gandolfo(1983), or Takayama (1974). See also Hirsch/Smale (1974), Chapters 3 and 4. Asmost of the following subjects can be found in all of these standard references, detailedsources are rarely provided in this appendix.

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250 Appendix

and c as an n-dimensional column vector of constants.Consider first the simplest case with n = 1 and c = 0, i.e., the homogeneous, one-dimen-

sional differential equation

x(t) = ax(t). (A.1.4)

Equation (A.1.4) can be solved explicitly, namely2

x(t) = x(0)eat. (A.1.5)

Obviously, for, e.g., x(0) > 0, x(t) permanently increases (decreases) if a > 0 (< 0). Ifa = 0, x(t) stays at x(0) ∀ t.

If c �= 0, the solution to the one-dimensional, non-homogeneous differential equation

x(t) = ax(t) + c (A.1.6)

is

x(t) =(x(0) − x∗

)eat + x∗, (A.1.7)

where x∗ represents the fixed-point value of (A.1.6), i.e., the value of x which solves 0 =ax(t) + c. If (x(0)− x∗) �= 0, x(t) converges to (diverges from) the fixed-point value x∗ ifa < 0 (> 0). In both cases (A.1.4) and (A.1.6), the dynamic behavior of the equations ischaracterized by monotonically increasing or decreasing values of x.

Second, consider the case n = 2 and c = 0. The system of two linear differential equa-tions can easily be transformed into a second-order differential equation. Differentiatingthe first equation with respect to time and substituting for x2 and x2 leads to

x1(t) − (a11 + a22)x1(t) + (a11a22 − a12a21)x1(t) = 0. (A.1.8)

Obviously, the coefficients of x1 and x1 are the determinant and the negative value of thetrace of the coefficient matrix A = {aij}, i, j = 1, 2, respectively. The solution of (A.1.8) isgiven by3

x1(t) = m1eλ1t + m2e

λ2t, (A.1.9)

with mi as constants determined by the initial values x1(0) and x1(0), and λi as the eigen-values of A, i.e., the solutions of the equation |A − λI| = 0, where I is the 2 × 2-identitymatrix. Thus, the eigenvalues are the solutions of

λ2 − (a11 + a22)λ + (a11a22 − a21a12) = 0. (A.1.10)

Equation (A.1.10) is known as the characteristic equation. The coefficient of λ is the negativevalue of the trace of A while the absolute expression constitutes the determinant of A. The

2 Writing (A.1.4) as x/x = a and integrating over time yields lnx = at (recall the log-arithmic differentiation). Removing the natural log and considering the integrationconstant immediately leads to (A.1.5).

3 If the eigenvalues are identical, (A.1.9) must be replaced by x1(t) = (m1 + tm2)eλt.

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A.1. Basic Properties of Linear Dynamical Systems 251

A.1.a: Stable Node A.1.b. Unstable Node

A.1.c: Stable Focus A.1.d: Unstable Focus

A.1.e: Center A.1.f: Saddle Point

Types of Behavior in Continuous-Time Dynamical SystemsFigure A.1

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252 Appendix

roots of (A.1.10) are therefore given by

λ1,2 =tr A ±

√(tr A)2 − 4 det A

2. (A.1.11)

Substituting for λi, i = 1, 2, in (A.1.9) shows that these eigenvalues determine the dynamicalbehavior of the system. Basically, two cases can be distinguished:

i) Real roots: The eigenvalues λ1,2 in (A.1.11) are real when the discriminant, i.e., ∆ =

(tr A)2 − 4 det A, is positive or equal to zero. Consider first the case of a positive determi-nant, i.e., det A > 0, implying that both eigenvalues have the same sign. If the trace of Ais negative, both eigenvalues are negative, and the trajectory of the system monotonicallyapproaches a finite point (x∗1 , x∗2 ). The point (x∗1 , x∗2 ) is called a stable node. If the trace ispositive and both eigenvalues are positive, the trajectory monotonically diverges to +∞ or−∞, respectively. The system is said to possess an unstable node. Second, if the discriminantis positive but det A < 0, the eigenvalues are real and come in pairs of opposite sign. Inthat case the fixed point is said to be saddle point stable, i.e., the stable and unstable mani-folds which are asymptotes to all trajectories intersect at the fixed point. The unstable andstable manifold of the fixed point are also called the separatrixes of the saddle.

ii) Complex roots: The case of complex eigenvalues is the most interesting one from the pointof view of dynamical systems theory. If det A > 0 and ∆ < 0, the eigenvalues are complexconjugate, i.e., they can be written as λ1 = α + βi and λ2 = α − βi, with α = tr A/2,β =

√det A − (tr A)2/4, and i =

√−1. If the real parts Re λi are negative, dampenedoscillations occur such that a finite value will be approached in the limit. This value iscalled a stable focus of the system. If Re λi is positive, the amplitude of the oscillation willincrease over time. In this case the system is said to have an unstable focus. Finally, if Re λequals zero, the amplitude of the oscillation will be constant over time and the system issaid to exhibit center dynamics or to be neutrally stable.

This last case of center dynamics corresponds to the so-called harmonic oscillator which isespecially important in classical mechanics: if the trace of the coefficient matrix for n = 2is zero, (A.1.8) is formally identical with

x1(t) + ω2x1(t) = 0, (A.1.12)

with ω as the frequency of the oscillations. The solution of (A.1.12) is x(t) = a cos(ωt+t0),with a > 0 as a constant depending on the initial values of x1 and x2 at t0. In this case,every initial point

(x1(0), x2(0)

)is located in a closed orbit, the amplitude a of which is

determined by the distance between the initial point and the fixed point.The different possible types of behavior in two-dimensional continuous-time dynamical

systems are illustrated in Figure A.1. The stable focus and the stable node are also calledsinks, while their unstable correspondents are called sources.

The case n > 2 is naturally more difficult to analyze. Nevertheless, some results ex-ist though it may be difficult to establish the presence of the following necessary and/orsufficient conditions.

Consider the general system (A.1.1) with n > 2. If all eigenvalues λi of (A.1.1), i.e., theroots of the determinant |A − λI| = 0, are real and negative, the system converges mono-tonically toward a finite value of x. If there exists a pair of complex conjugate eigenvaluesλk, λk+1, the system oscillates with vanishing amplitude if the real parts of all λk ∈ C andthe real eigenvalues are negative. In both cases the system is called asymptotically stable.

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A.1. Basic Properties of Linear Dynamical Systems 253

If all eigenvalues are real and positive, the system diverges monotonically toward +∞or −∞, respectively. Saddle-point stability occurs if the eigenvalues are real and are of op-posite signs. If some eigenvalues are complex conjugate with positive real parts the systemoscillates with increasing amplitude. It depends, however, on the sign of the real eigen-values whether the oscillation is superimposed on an exploding or converging monotonetrajectory. Steady oscillations with constant amplitude occur if the real parts of the complexconjugate eigenvalues are zero.

When the dimension of the considered dynamical system is high, it will usually be dif-ficult to compute the eigenvalues. Without explicit calculations it will also be difficult todetermine whether the eigenvalues are real or complex. The Routh-Hurwitz criterion is aconvenient tool to establish the asymptotic stability of a fixed point, i.e., the negativity ofthe real eigenvalues or the real parts of the complex eigenvalues. However, the criteriondoes not provide an answer to the question whether the roots are indeed real or complex.4

Consider the characteristic equation written in the form

c0λn + c1λ

n−1 + c2λn−2 + . . . + cn−1λ + cn = 0. (A.1.13)

The n+ 1 coefficients ci, i = 0, . . . , cn are arranged in a Routh matrix in the following way:start with c1 as the first upper-left-hand entry and place the coefficients c3, c5, etc. in thesame row. All fictitious entries cn+k, k ≥ 1, are defined as being equal to zero. The secondrow starts with c0, c2, c4, etc. The resulting matrix for the general equation (A.1.13) hasthe form

R =

c1 c3 c5 c7 · · · 0

c0 c2 c4 c6 · · · 0

0 c1 c3 c5 · · · 0

0 c0 c2 c4 · · · 0

0 0 c1 c3 · · · 0...

......

.... . .

...

0 0 0 · · · cn−2 cn

. (A.1.14)

The minor matrices of R starting at the upper-left-hand corner are

R1 = c1, R2 =

(c1 c3

c0 c2

), R3 =

c1 c3 c5

c0 c2 c4

0 c1 c3

, (A.1.15)

etc. to Rn (which is identical with R).The Routh-Hurwitz criterion states that all real eigenvalues and all real parts of the com-

plex conjugate eigenvalues in (A.1.13) are negative if and only if the determinants of allthe matrices R1, R2, R3, . . .Rn are positive.

As an example, consider the case of a third-order differential equation with the charac-teristic equation

c0λ3 + c1λ

2 + c2λ + c3 = 0. (A.1.16)

4 If the presence of complex eigenvalues can be excluded, the negativity of the real eigen-values can be examined with the help of Descartes’ rule: the eigenvalues are negative whenall coefficients in the characteristic equation have the same sign.

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254 Appendix

The Routh matrix R is

R3 =

c1 c3 0

c0 c2 0

0 c1 c3

. (A.1.17)

The fixed point of the underlying third-order differential equation is asymptotically stableif the determinants of the minor matrices R1, R2, and R3 are positive, i.e., if

c1 > 0,c1c2 − c0c3 > 0,

c1c2c3 − c0(c3)2 > 0.

(A.1.18)

The conditions (A.1.18) and the appropriate conditions for higher-dimensional systems canbe combined and simplified. For example, it can be shown that the conditions (A.1.18) areequivalent with5

c1, c2, c3 > 0 and c1c2 − c0c3 > 0. (A.1.19)

There exist variants of the Routh-Hurwitz criterion and also different criteria for estab-lishing the negativity of the real eigenvalues or the real parts of the complex eigenvalues.Details can be found in Gandolfo (1983), pp. 250ff., and Hahn (1984), pp. 752f.

n = 1 n = 2 n ≥ 2

Monotone a < 0 det A > 0; λ1,2 ∈ R; λi ∈ R;Convergence λ1,2 < 0 λi < 0 ∀i.

Monotone a > 0 det A > 0; λ1,2 ∈ R; λi ∈ R;Divergence λ1,2 > 0 λi > 0 ∀i.

Saddle Point impossible det A < 0; λ1,2 ∈ R; λj , λk ∈ R ∀j, k;Stability λ1 > 0; λ2 < 0 λj > 0; λk < 0.

Converging impossible det A > 0; λ1,2 ∈ C; λj ∈ R; λk, λk+1 ∈ C;Oscillations Re λ1,2 < 0 λj < 0 ∧ Re λk < 0

Diverging impossible det A > 0; λ1,2 ∈ C; λj ∈ R; λk, λk+1 ∈ C;Oscillations Re λ1,2 > 0 λj > (<) 0; Re λk > 0

Steady impossible det A > 0; λ1,2 ∈ C; λj ∈ R; λk, λk+1 ∈ C;Oscillations Re λ1,2 = 0 λj = 0; Re λk = 0

Dynamic Behavior in Linear Continuous-Time SystemsTable A.1

5 Cf. Gandolfo (1983), p. 250, for details.

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A.1. Basic Properties of Linear Dynamical Systems 255

The above mentioned dynamical phenomena in continuous-time, linear, dynamical sys-tems are summarized in Table A.1.

A.1.2. Discrete-time Systems

The possible dynamic phenomena in linear, discrete-time dynamical systems are qualita-tively more or less equivalent to the continuous-time case with some important differencesespecially in one-dimensional systems. Consider an n-dimensional, linear, discrete-time dy-namical system with constant coefficients6

x1t+1 = a11x

1t + . . . + a1nx

nt + c1,

x2t+1 = a21x

1t + . . . + a2nx

nt + c2,

...

xnt+1 = an1x1t + . . . + annx

nt + cn,

(A.1.20)

or, in vector notation,

xt+1 = Axt + c, t ∈ Z, (A.1.21)

with A and c as defined in (A.1.3). Consider again first the simplest case n = 1 and c = 0.The solution to the homogeneous first order equation

xt+1 = axt (A.1.22)

is found by iterating (A.1.22), i.e., x1 = ax0 ⇒ x2 = ax1 = a2x0, etc., as:

xt = x0at, (A.1.23)

with x0 as the initial value. For example, if x0 > 0, xt increases (decreases) monotonicallyfor a > 1 (0 < a < 1). If a = 1, xt stays at the initial point for all t.

If c �= 0, the solution to the non-homogeneous equation

xt+1 = axt + c (A.1.24)

is

xt = (x0 − x∗)at + x∗, (A.1.25)

with x∗ as the fixed-point value of x, i.e., the value that solves (A.1.24) for xt+1 = xt.If (x0 − x∗) �= 0, xt converges to (diverges from) its fixed-point value monotonically if0 < a < 1 (a > 1).

If a < 0, a phenomenon arises in both cases (A.1.22) and (A.1.24) which is not possiblein the analogous continuous-time systems, namely that xt oscillates over time in a sawtoothpattern. For −1 < a < 0, the oscillations are dampened and xt approaches a finite value.

6 An introduction to linear difference equations can be found in Goldberg (1958).

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256 Appendix

If a < −1, the amplitude of the oscillations increases exponentially such that xt convergesalternatively to +∞ and −∞. Finally, if a = −1, xt oscillates with a constant amplitude.

Second, consider the case n = 2. As in the case of a continuous-time system, the sys-tem of two one-dimensional difference equations can be transformed into the second-orderdifference equation

x1t+2 − (a11 + a22)x1

t+1 + (a11a22 − a12a21)x1t = 0. (A.1.26)

The solution of (A.1.26) is given by7

x1t = m1λ1

t + m2λ2t, (A.1.27)

with mi and λi, i = 1, 2 having the same meaning as in (A.1.9). Depending on the signof the discriminant ∆ = (tr A)2 − 4 det A, the eigenvalues λ1,2 can be real or complexnumbers.

i) Real roots: The eigenvalues are real if the discriminant ∆ is positive or equal to zero.Depending on the values of det A and tr A, the eigenvalues λi can be positive or negative.An eigenvalue 0 < λi < 1 (λi > 1) implies a monotonic convergence (divergence) in oneof the two r.h.s. expressions in (A.1.27). An eigenvalue −1 < λi < 0 (λi < −1) implies aconverging (diverging) sawtooth oscillation in one of the r.h.s. expressions in (A.1.27). Asboth eigenvalues can have the same sign or can be of opposite sign, a variety of possibilitiesexists for the linear combination (A.1.27) of solutions. If the eigenvalues are distinct, thedominant root, i.e., the absolutely largest root, determines the qualitative behavior of thesystem for t → ∞. For example, if both eigenvalues are positive and smaller than 1 (largerthan 1), the system monotonically approaches a finite value (monotonically diverges). Ifthe eigenvalue λ1 is positive and larger than 1 and if λ2 is negative and larger than −1,the eigenvalue λ1 is the dominant root. The system is characterized by vanishing sawtoothoscillations around a divergent trend for t large.

The Gaussian PlaneFigure A.2

ii) Complex roots: When the discriminant is nega-tive, the roots can be complex conjugate num-bers. The system is characterized by converg-ing oscillations with vanishing amplitudes if themodulus of the complex eigenvalues is smallerthan 1, i.e., mod λi < 1. The modulus of acomplex number λ = α + βi is defined by theEuclidian distance between the origin and thepoint (α, β) in the Gaussian plane, i.e., mod =√

α2 + β2 (cf. Figure A.2). Simple geometryimplies that

λ1,2 = α± βi

= mod · (cos θ ± i sin θ),(A.1.28)

with θ as the angle between the distance line and the real axis. DeMoivre’s theorem impliesthat the solution (A.1.27) can be written in the form8

7 If both eigenvalues are identical, the solution (A.1.27) must be replaced by x1t = (m1 +

tm2)λt.8 DeMoivre’s theorem says that

(r(cos θ ± i sin θ)

)n= rn(cosnθ ± i sinnθ).

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A.1. Basic Properties of Linear Dynamical Systems 257

xt = m1λt1 + m2λ

t2,

= m1(mod(cos θ + i sin θ)

)t+ m2

(mod(cos θ − i sin θ)

)t,

= modt(m1(cos θt + i sin θt) + m2(cos θt− i sin θt)

),

= modt((m1 + m2) cos θt + (m1 −m2)i sin θt

),

= modt(n1 cos θt + n2i sin θt).

(A.1.29)

When the modulus is larger than 1, the amplitude of the system increases while xtconverges in an oscillating manner if the modulus is smaller than 1. Steady oscillationsoccur if the modulus equals 1.

Eigenvalues Inside and Outsideof the Unit Circle

Figure A.3

A fixed point of a 2-dimensional, discrete-time, dynamical system is thus obviously sta-ble when the modulus is smaller than one.A usual expression in this context is thatthe eigenvalues lie in the unit circle (inthe Gaussian plane). Figure A.3 shows thisplane with two different complex conjugateeigenvalues. The pair with the positive realpart lies outside the unit circle and rep-resents a scenario with an unstable fixedpoint. The second pair with a negative realpart resides inside the circle and thus hasa modulus smaller than one and depicts ascenario with a stable fixed point.

Finally, consider the n-dimensional case. If all eigenvalues are real, the behavior of thesystem is described by monotone convergence (divergence) if all eigenvalues are smaller(larger) than 1. If some eigenvalues are complex conjugate, the system oscillates. Accord-ing to the magnitude of the modulus of the complex conjugate eigenvalues and the mag-nitude of the real eigenvalues the oscillations are exploding or dampened, superimposedon a converging or diverging trend of the trajectory in dependence on the real eigenvalue.

In the n-dimensional case, it is usually difficult (if not impossible) to calculate the eigen-values, i.e., the roots of the characteristic equation when n > 3. It is also usually impossibleto determine whether some of the eigenvalues are complex. However, it is (in principle)possible to provide an answer to the question whether a fixed point is stable, i.e., whetherthe roots have modulus less than one.

Consider an n-dimensional, linear dynamical system with its characteristic equation

c0λn + c1λ

n−1 + c2λn−2 + . . . + cn = 0. (A.1.30)

The following matrices S1 and S2 are called Schur-matrices:

S1 =

c0 c1 c2 c3 · · · cn−2

c0 c1 c2 · · · cn−3

c0 c1 · · · cn−4

. . . · · · ...

0 c0 c1

c0

(A.1.31)

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258 Appendix

S2 =

cn

cn cn−1

0 cn cn−1 cn−2

. .. · · · · · · ...

cn cn−1 · · · c4 c3

cn cn−1 cn−2 · · · c3 c2

(A.1.32)

Consider the following minor matrices starting at the upper-left-hand corner of S1:

S11 = c0, S12 =

(c0 c1

0 c0

), S13 =

c0 c1 c2

0 c0 c1

0 0 c0

, (A.1.33)

etc. to S1n. The analogous minor matrices of S2 starting at the upper-right-hand corner ofS2 are

S21 = cn, S22 =

(0 cn

cn cn−1

), S23 =

0 0 cn

0 cn cn−1

cn cn−1 cn−2

, (A.1.34)

etc. to S2n.The Schur theorem establishes that all eigenvalues of a linear, n-dimensional, dynamical

system have a modulus < 1 if and only if the following properties are satisfied:

|S1i + S2i| > 0,|S1i − S2i| > 0,

for i = 1, . . . , n− 1,

and

c0 + c1 + c2 + . . . + cn > 0,c0 − c1 + c2 − c3 . . . cn > 0.

For example, in the case of a third-order differential equation with the characteristic equa-tion

c0λ3 + c1λ

2 + c2λ + c3 = 0, (A.1.35)

the stability conditions according to Schur’s theorem are:

c0 + c1 + c2 + c3 > 0,c0 − c1 + c2 − c3 > 0,|S11 + S21| = c0 + c3 > 0,|S11 − S21| = c0 − c3 > 0,

|S12 + S22| =∣∣∣ c0 c1 + c3c3 c0 + c2

∣∣∣ = c0(c0 + c2) − c3(c1 + c3) > 0,

|S12 − S22| =∣∣∣ c0 c1 − c3c3 c0 − c2

∣∣∣ = c0(c0 − c2) + c3(c1 − c3) > 0.

(A.1.36)

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A.1. Basic Properties of Linear Dynamical Systems 259

For n large, applications of this Schur criterion will obviously necessitate extensive computa-tional efforts. Furthermore, when an n-dimensional dynamical system of the form (A.1.20)is given, the computation of the coefficients ci in the characteristic equation will usually bedifficult and time-consuming.9

n = 1 n = 2 n ≥ 2

Monotone 0 < a < 1 λ1,2 ∈ R; λi ∈ R;Convergence λ1,2 < 1 |λi| < 1 ∀i

Monotone a > 1 λ1,2 ∈ R; λi ∈ R;Divergence λ1,2 > 1 |λi| > 1 ∀i

Converging −1 < a < 0 det A > 0; λ1,2 ∈ C; λj ∈ R; λk, λk+1 ∈ C;Oscillations (Sawtooth) mod λ1,2 < 1 λj < 1 ∧ modλk < 1

Diverging a < −1 det A > 0; λ1,2 ∈ C; λj ∈ R; λk, λk+1 ∈ C ;Oscillations (Sawtooth) mod λ1,2 > 1 modλk > 1

Steady a = −1 det A > 0; λ1,2 ∈ C; λj ∈ R; λk, λk+1 ∈ C;Oscillations (Sawtooth) mod λ1,2 = 1 λj = 1 ∧ modλk = 1

Dynamic Behavior in Linear Discrete-Time SystemsTable A.2

Some important dynamical phenomena in discrete-time, linear dynamical systems aresummarized in Table A.2. As was mentioned above, several other types of behavior are pos-sible in the real roots case when the eigenvalues have opposite signs and different absolutevalues.

A.1.3. Invariant Subspaces in Linear Dynamical Systems

The invariant subspaces briefly outlined in this subsection rely on the notion of an eigen-vector associated with an eigenvalue. It is necessary to recall the role of eigenvectors in thesolution of linear dynamical system at some length.10 The presentation concentrates onthe continuous-time case. The analogous subspaces emerging in discrete-time systems arebriefly mentioned at the end of the section.

9 A different version of the Schur criterion can be found in Gandolfo (1983), pp. 112f.That version requires the calculation of the determinants of up to 2n × 2n – matriceswhich implies an even more extensive computational expense.

10 Extensive treatments of the topics covered in this subsection can be found in Braun(1978), Gantmacher (1954), and Hirsch/Smale (1974). The following presentation isinspired by Rommelfanger (1977). Discussions of the topic concentrating on transfor-mation matrices can be found in Guckenheimer/Holmes (1983) and Wiggins (1990).

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260 Appendix

Eigenvectors and Solutions of Linear Dynamical Systems

Consider the linear, n-dimensional, homogeneous, dynamical system

x = Ax, x ∈ Rn, (A.1.37)

with A as an n× n - matrix of constant coefficients. A (fundamental) solution of (A.1.37)is found by attempting the same approach as in (A.1.5) or (A.1.9):

x(t) =

x1x2...xn

=

p1eλt

p2eλt

...pne

λt

= peλt, (A.1.38)

with p �= 0 as a vector of yet unspecified constants. Differentiating (A.1.38) with respect totime yields

x =

p1λeλt

p2λeλt

...pnλe

λt

= pλeλt. (A.1.40)

Substitution for x and x in (A.1.37) yields

λpeλt = Apeλt,λp = Ap,

(A.1.41)

or

(A − λI)p = 0. (A.1.42)

Equation (A.1.42) constitutes the definition of an eigenvector p associated with the eigenvalueλ: equation (A.1.42) possesses a non-trivial solution p �= 0 only if |A − λI| = 0, i.e., theconstituent equation for the determination of the eigenvalues of A.

The n×n - Matrix A possesses n eigenvalues, some of which might be identical or com-plex conjugate. Each of these eigenvalues possesses an associated eigenvector. However, theeigenvectors are determined only up to a multiplicative constant, i.e., if p = (p1, p2, . . . , pn)is an eigenvector associated with an eigenvalue λ, then p = (cp1, cp2, . . . , cpn) = c(p1, p2,. . . , pn) = cp is an eigenvector as well.

Assume that A possesses n (pair wise) different eigenvalues and that the associated eigen-vectors are linearily independent. Then (A.1.37) has n different (fundamental) solutions

xj(t) = cjpjeλjt, j = 1, . . . , n, (A.1.43)

where the indeterminacy of pj has been expressed by the introduction of the scalars cj .It has been mentioned in the previous subsections that the general solution of (A.1.37) isobtained by linear combinations of the n different fundamental solutions, i.e., if (A.1.43)are solutions of (A.1.37) then

x(t) =

n∑j=1

cjpjeλjt, j = 1, . . . , n, (A.1.44)

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A.1. Basic Properties of Linear Dynamical Systems 261

is also a solution of (A.1.37).

Invariant Subspaces

The eigenvectors described above are used for spanning invariant subspaces of Rn. Con-

sider the n-dimensional system (A.1.37). The eigenvalues λi, i = 1, . . . , n, can be dividedinto three classes:

Class S: Eigenvalues with negative real parts.Class U: Eigenvalues with positive real parts.Class C: Eigenvalues with zero real parts.

The eigenvectors belonging to the eigenvalues in these classes are denoted as pSh , h =

1, . . . , ns, pUk , k = 1, . . . , nu, and pC

� , � = 1, . . . , nc, respectively, with ns + nu + nc =n.11 The eigenvectors belonging to a particular class span subspaces in R

n (also known aseigenspaces):

Stable Subspace: ES = span{pS1 , . . . ,pS

ns}

Unstable Subspace: EU = span{pU1 , . . . ,pU

nu}

Center Subspace: EC = span{pC1 , . . . ,pC

nc}

(A.1.45)

The Rn can then be understood as the direct sum of the subspaces ES , EU , and EC :

Rn = ES ⊕ EU ⊕ EC . (A.1.46)

Of course, one or two of the subspaces may be empty in a given dynamical system. Thesubspaces represent invariant sets because a trajectory starting in one of the three sets willstay in that set forever.

A.4.a. A.4.b. A.4.c.Examples of Invariant Subspaces in R

2

Figure A.4

Figure A.4 depicts three examples of invariant subspaces in R2. The dynamical system

underlying Figure A.4.a is assumed to have one real positive and one real negative eigen-value. The eigenvectors are described by the upper two straight lines in the cross; the

11 If some eigenvectors are not linearily independent the eigenvectors have to be replacedby so-called generalized eigenvectors. Cf. Braun (1978) for details.

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262 Appendix

subspaces are formed by mirroring the lines because the span of the vectors includes mul-tiplication by negative scalars. The example in Figure A.4.a depicts the standard scenarioknown as a saddle point. The subspace EC is empty in this example. Figure A.4.b shows anexample with an empty subspace EU . The stable subspace is defined as in Figure A.4.a. Thecenter subspace is defined by the span of an eigenvector associated with a real eigenvaluewhich equals zero. Initial points located in this subspace do not move anymore. FigureA.4.c depicts a similar same case as in Figure A.4.b with the exception that ES = ∅ and anonempty unstable subspace.

A.5.a. A.5.b.Examples of Invariant Subspaces in R

3

Figure A.5

Two examples of subspaces in R3 are depicted in Figure A.5. The case of two negative

real eigenvalues and one positive real eigenvalue (and EC = ∅) is shown in Figure A.5.a.The unstable subspace is defined by the vertical line originating in the fixed point; thestable subspace is represented by the shaded plane. Figure A.5.b. contains an example ofa positive real eigenvalue and a pair of complex conjugate eigenvalues with negative realpart.

The eigenspace belonging to the complex eigenvalues is spanned by the real part, pR,and the imaginary part, pI, of the eigenvector. This can be explained by the followingconsideration: When an eigenvalue is complex the associated eigenvector and the solutionx(t) = cpeλt are complex as well. However, the complex solution x(t) defines two realsolutions: Differentiating the general form of the solution, i.e., x(t) = y(t) + iz(t), withrespect to time and substituting into (A.1.37) yields

x = Ax,y + iz = A(y + iz).

(A.1.47)

As the real and imaginary parts on both sides of the equation must be identical, it followsthat y = Ay and z = Az. Thus, y and z are (real) solutions of (A.1.37). The explicit formof the solutions is12

y(t) = eαt(

pR cosβt−(pI)2

sinβt),

z(t) = eαt(

pR sinβt +(pI)2

cosβt).

(A.1.48)

12 Cf. Braun (1978), Chapter 2.2.1, for details on the derivation of eiβt = cosβt+ i sinβt.

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A.1. Basic Properties of Linear Dynamical Systems 263

Thus, the solutions y(t) and z(t) are defined on a plane spanned by the real part pR andthe imaginary part pI of the complex eigenvector p associated with a complex eigenvalueλ = α+iβ. The conjugate eigenvalue and the associated eigenvector imply the same resultswith appropriate negative signs; the subspace is nevertheless unaltered because the negativeaxes are already included in the span of the previously derived vectors.

The case of discrete-time, linear, dynamical systems can be treated in a similar manner.Consider the n-dimensional system

xt+1 = Axt, x ∈ R, (A.1.49)

and A defined as above. Assume again that there exist n different eigenvalues of A. Thefundamental solution of (A.1.49) for an initial value x0 is

xt = Atx0. (A.1.50)

The relevance of eigenvectors cannot be seen as easily as in the continuous-time case. Con-sider the diagonal matrix L defined as

L =

λ1 0 . . . 00 λ2 . . . 0...

.... . .

...0 0 . . . λn

. (A.1.51)

The matrices A and L are similar if there exists a transformation matrix P with the propertydet P �= 0 and

L = P−1AP. (A.1.52)

Assume that such a matrix P exists. Then (A.1.52) can be written as

PL = AP. (A.1.53)

The diagonal form of L implies that (A.1.53) can also be written as

λipi = Api, i = 1, . . . , n, (A.1.54)

or

0 = (A − λiI)pi, i = 1, . . . , n, (A.1.55)

with pi as the ith column vector of the matrix P. Equation (A.1.55) is the constituentequation for the eigenvector pi associated with the eigenvalue λi. Thus, the eigenvectorspk, k = 1, . . . , n represent the kth columns of the transformation matrix P. With At =PLtP−1, the fundamental solution (A.1.50) can be written as

xt = PLtP−1x0. (A.1.56)

As L is a diagonal matrix, the matrix Lt can be calculated by simply exponentiating theentries λi.

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264 Appendix

The invariant subspaces can be determined by categorizing the eigenvalues in a wayanalogous to the case of continuous-time systems:

Class S: Eigenvalues with modulus less than 1.

Class U: Eigenvalues with modulus greater than 1.

Class C: Eigenvalues with modulus equal to 1.

The definition of the invariant subspaces is identical with the definition provided above forthe case of continuous-time systems (cf. (A.1.45)).

A.2. Center Manifolds and the Reduction of (Effective) Dimensions

In diverse sections of the main text various dynamical systems with a particular dimensionhave been studied. When an n-dimensional system was investigated, a mathematical resultdefined for such an n-dimensional system was applied to it. For example, most types of bifur-cations in Sections 3.2 and 3.2 of the main text were discussed in a one- or two-dimensionalcontext with the appropriate theorems formulated for the one- or two-dimensional case,respectively. While many theorems mentioned in this chapter can be extended to the n-dimensional case, severe analytical problems are usually involved when systems with morethan two or three state variables are considered. Besides, some theorems (like the Hopf-bifurcation theorem for discrete maps) are restricted to the case of a particular dimension.It would thus be desirable if methods existed that allow for a reduction of the effective dimen-sion of a given n-dimensional dynamical system, i.e., to investigate a system with a dimensionm < n which nevertheless provides sufficient information on the dynamics of the originaln-dimensional system.

A particular method for the reduction of the dimension of a given system has been de-scribed in Chapter 2 in the context of relaxation oscillations. This method, not uncommonin economics, usually relies on variations in assumed adjustment coefficients and thus onfundamental changes of the nature of the dynamical system under investigation. Anothermethod consists in applying the center manifold theorem to be described below. Basically, ap-plications of this theorem require that the dynamical system under investigation is availablein a precise algebraic form; dynamical systems in general forms like x = f(x) with f de-scribed qualitatively cannot be investigated with the help of the center manifold theorem.While such general formulations dominate economic dynamic models, the method shouldnevertheless be outlined in the following for two reasons: First, in a few examples precisealgebraic forms are indeed available in economic dynamic systems; second, in all economicexamples (including the graphical examples) it has implicitly been assumed that such aprecise formulation is (in principle) possible. It should, however, be kept in mind thata center manifold can only be calculated for given algebraic forms of a system and thatgeneralizations of the results are inappropriate.

The concepts of stable, unstable, and center manifolds have already been mentionedin Section 2.1 in the context of the equivalence between the local behavior of a nonlineardynamical system and the associated linearized system. It has also been mentioned thatthe local behavior of a nonlinear dynamical system cannot be described with the help ofits associated linear system when a fixed point is not hyperbolic, i.e., when one or severaleigenvalues are equal to zero or have zero real parts (or have a modulus equal to one in thediscrete-time case).13 However, all bifurcation types described in the previous sections deal

13 Recall that nonlinear dynamical systems would not be really interesting if their localbehavior could entirely be described by the behavior of the associated linear system.

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A.2. Center Manifolds and the Reduction of Dimensions 265

with situations in which eigenvalues take on these values. It follows that the characteristiclocal behavior of nonlinear dynamical systems is related to this case of zero real roots orpurely imaginary roots (or roots with modulus 1 in the discrete-time case). As this chapterconcentrates on local bifurcations with the described properties of the eigenvalues, take thescenario with zero real roots etc. for granted and assume that an n-dimensional, nonlinear,dynamical system of the general form

x = f(x), x ∈ Rn, (A.2.1)

can be written as

y = Ay + g(y, z),z = Bz + h(y, z),

y ∈ Rc, z ∈ R

s, c + s = n. (A.2.2)

The matrix A is a c × c - matrix with real eigenvalues equal to zero (or purely imaginaryeigenvalues); the matrix B is an s×s - matrix with negative real eigenvalues (or negative realparts of complex eigenvalues).14 In the formulation (A.2.2) it has implicitly been assumedthat the unstable manifold (and the unstable eigenspace) is empty. The center manifoldtheorem guarantees15 that a center manifold for (A.2.2) exists but that it may not necessarilybe unique.

Stable and Center Manifoldsof a 2D System

Figure A.6

As a motivation for the concentration on cen-ter manifolds consider Figures A.6 and A.7. InFigure A.6 it has been assumed that the dynam-ical system is two-dimensional and that it pos-sesses a negative real eigenvalue and a zero realeigenvalue. The negative real eigenvalue im-plies that the motion of the system is dominatedby a convergence toward the center manifold.If the dynamical system starts at initial pointswhich are located further away from the cen-ter manifold the presence of the stable manifoldguarantees that trajectories will eventually con-verge toward the center manifold. Figure A.7 de-picts the case of a three-dimensional system witha one-dimensional stable manifold and a two-dimensional center manifold derived from com-plex conjugate eigenvalues with zero real parts.16

Once again, the motion in a distance of the cen-ter manifold is dominated by the stable manifold

and trajectories converge toward the center manifold. Thus, it can be suspected that in thecases of Figures A.6 and A.7 the motion of the system is eventually dominated by the centermanifold.

Suppose the center manifold can be described by an equation system of the form

z = k(y). (A.2.3)

14 Compare Section 2.1 for this procedure.15 Cf. Guckenheimer/Holmes (1983), p. 127.16 Compare the discussion in the appendix A.1.3 for a description of the appropriate

eigenspaces.

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266 Appendix

A One-Dimensional Stable Manifold and a Two-Dimensional Center ManifoldFigure A.7

There exist theorems17 saying that the local dynamic behavior of (A.2.2) is equivalent tothe behavior of

y = Ay + g(y, k(y)

). (A.2.4)

If one succeeds in deriving the proper expression for z = k(y) it is thus possible to studythe local behavior near the fixed point with the help of a c < n – dimensional dynamicalsystem.

The problem remains to calculate the specific form of (A.2.3) for a given dynamicalsystem (A.2.1).18 Differentiating (A.2.3) with respect to time yields

z = Jk y, (A.2.5)

with Jk as the Jacobian matrix of first-order derivatives of the vector-valued function k. Theabove-mentioned theorems establish that the dynamics of the original system (A.2.2) even-tually takes place on the center manifold; thus the original dynamics can be described bythe system

y = Ay + g(y, k(y)

),

z = Bk(y) + h(y, k(y)

),

y ∈ Rc, z ∈ R

s. (A.2.6)

Substituting (A.2.6) into (A.2.5) yields

Bk(y) + h(y, k(y)

)= Jk

(Ay + g

(y, k(y)

)), (A.2.7)

17 Cf. Carr (1981). More detailed information can be obtained from Arrowsmith/Place(1990), pp. 93ff., Guckenheimer/Holmes (1983), pp. 127ff., and Wiggins (1990), pp.195ff.

18 The following procedure is described in detail in Wiggins (1990), pp. 195ff.

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A.2. Center Manifolds and the Reduction of Dimensions 267

or

Jk

(Ay + g

(y, k(y)

))− Bk(y) − h

(y, k(y)

)= 0. (A.2.8)

A procedure for calculating (A.2.3) with any desired degree of precision is described inGuckenheimer/Holmes (1983), pp. 131f.: Assume that the manifold (A.2.3) can be ap-proximated by polymomials of the form

z = k(y) = a2y2 + a3y3 + a4y4 + . . . + ajyj + O(yj+1). (A.2.9)

The expression O(yj+1) includes all terms with powers greater than or equal to j + 1. Inmany cases it suffices to consider only powers of 2 and 3 in approximations of (A.2.3).

Substituting (A.2.9) in (A.2.8) yields polynomials with powers of 2 and greater. The un-known coefficients ai, i = 2, 3, . . ., are found by equating the coefficients of all expressionsof the same power. Substitution of the resulting equation z = k(y) with the proper coeffi-cients into (A.2.4) yields the desired dynamical system defined on the center manifold.

The following simple example of a two-dimensional system illustrates the procedureoutlined above for the general n-dimensional case.19 Consider the system

x = x2y − x5,

y = x2 − y,x, y ∈ R. (A.2.10)

The system has a fixed point at the origin. The Jacobian, evaluated at the origin, is

J |(x,y)=(0,0) =( 0 0

0 −1

), (A.2.11)

implying that the eigenvalues are λ1 = 0 and λ2 = −1. Thus, the fixed point (0, 0) isnon-hyperbolic.

Writing (A.2.10) in the form (A.2.2) yields

x = 0 + x2y − x5 = 0 + g(x, y),

y = −y + x2 = −y + h(x, y).(A.2.12)

The center manifold of (A.2.12) is a one-dimensional curve tangent to the linear centereigenspace y = 0, i.e., the x-axis, at the origin. The particular form of (A.2.8) for thisexample is

Jk ·(

0 · x + g(x, k(x)

))− (−1)k(x) − h

(x, k(x)

)= 0,

Jk ·(x2k(x) − x5)+ k(x) − x2 = 0.

(A.2.13)

When the manifold is approximated by

k(x) = a2x2 + a3x

3 + O(x4), (A.2.14)

19 The example is described in detail in Wiggins (1990), pp. 196f.

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268 Appendix

(A.2.13) turns into(2a2x + 3a3x

2 + . . .)(

x2(−x5 + a2x2 + a3x

3 + . . .)

− x2 + a2x2 + a3x

3 + . . . = 0. (A.2.15)

Equation (A.2.15) can be rearranged as an equation of the form

b2x2 + b3x

3 + b4x4 + . . . = 0. (A.2.16)

The equation is fulfilled when all coefficients bi, i = 2, 3, . . . , equal zero. For (A.2.15) thisimplies that a2 − 1 = 0 and a3 = 0. Thus, the center manifold is approximated by

y = k(x) = 1 · x2 + 0 · x3 + O(x4) = x2 + O(x4). (A.2.17)

The motion on the center manifold is described by20

x = x2k(x) − x5 = x2(x2 + O(x4))− x5,

= x4 + O(x5).(A.2.18)

Equation (A.2.18) is a simple one-dimensional equation. The eigenvalue at the origin stillequals zero but the stability property of the origin can now be determined by a simpleargument. For all x(0) �= 0 in a neighborhood of the origin x is positive. Thus, the originis unstable (half-stable) in the sense that for x(0) < 0 the trajectory converges towardthe origin and that for x(0) > 0 the trajectory diverges from the origin. As the dynamicbehavior of the original system (A.2.10) can locally be represented by the reduced system(A.2.18), the origin of (A.2.10) is unstable as well.

Center manifolds for discrete maps of the form

xt+1 = f(xt), x ∈ Rn, (A.2.19)

can be derived in a way very similar to the procedure outlined above. The system (A.2.19)can be written as

yt+1 = Ayt + g(yt, zt),zt+1 = Bzt + h(yt, zt),

(A.2.20)

where A and B have the same properties as in (A.2.2). The procedure for the determinationof the reduced system on the center manifold is identical with the case outlined above withthe exception that the equation analogous to (A.2.8) or (A.2.13) has to be calculated asfollows. Substitution of z = k(y) in (A.2.20) yields

yt+1 = Ayt + g(yt, k(yt)

),

zt+1 = Bk(yt) + h(yt, k(yt)

)= k(yt+1).

(A.2.21)

20 The reader will notice that in this particular example the differential equation (A.2.18)can also be determined by a simple substitution: consider the case y = 0 in (A.2.10)and substitute the resulting y = x2 in the first equation of (A.2.10). However, in mostother examples the procedure described above has to be applied in order to derive thelaws of motion on the manifold.

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A.2. Center Manifolds and the Reduction of Dimensions 269

Substituting yt+1 from the first equation of (A.2.21) into the r.h.s. of the second equationand rearranging terms yields

k(

Ayt + g(yt, k(yt)

))− Bk(yt) − h

(yt, k(yt)

)= 0. (A.2.22)

The rest of the procedure is identical with the procedure outlined for continuous-timesystems.

Many dynamical systems include one or more parameters. Indeed, the entire Chapter3 deals with situations in which a parameter is changed and the topological nature of thesolution curves changes when the parameter takes on a particular value. The questionarises whether center manifolds can be derived for this case as well. This can in fact bedone when the parameters are treated in a particular way. Let µ be a p-dimensional vectorof parameters. A parameter-dependent dynamical system of the general form

x = f(x,µ), x ∈ Rn, µ ∈ R

p, (A.2.23)

can be rewritten as

y = Ay + g(y, z,µ),z = Bz + h(y, z,µ),

(y, z) ∈ Rc × R

s, c + s = n, µ ∈ Rp. (A.2.24)

The center manifold of (A.2.24) can be determined when the parameters µ are interpretedas dynamic variables with µ = 0. The system

y = Ay + g(y, z,µ),z = Bz + h(y, z,µ),µ = 0,

(y, z) ∈ Rc × R

s, c + s = n, µ ∈ Rp. (A.2.25)

will be treated as an c+ s+p – dimensional system which has c+p zero real eigenvalues oreigenvalues with zero real part. The center manifold of (A.2.25) is represented as a graphof the variables with associated zero real eigenvalues, i.e., over y and µ. The analogousexpression of (A.2.3) in this case is

z = k(y,µ). (A.2.26)

Performing the same procedure as above in (A.2.5) - (A.2.8) yields

Jk,y ·(

Ay + g(y, k(y,µ),µ

))− Bk(y,µ) − h

(y, k(y,µ),µ

)= 0, (A.2.27)

with Jk,y as the matrix of partial derivatives of k(y,µ) with respect to k and µ. In the calcu-lation of (A.2.27) it has been made use of the fact that µ = 0. The rest of the procedure isidentical with the one outlined above. Center manifolds for parameter-dependent discrete-time maps can be derived analogously.

The consideration of parameters in (A.2.23) implies that the dimension of the centermanifold is increased by the dimension of the vector of parameters, p. All solution curvesin a small neighborhood of the fixed point are contained in this manifold. Thus, when asingle parameter p takes on a bifurcation value, the bifurcating solution curve for a slightlychanging parameter is contained in the center manifold as well.

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270 Appendix

A.3. A Brief Introduction to the Theory of Lags and Operators

Economic dynamics deals with models that contain at least one equation of the generalform

yt = f(xt−1, xt−2, . . . , xt−m), (A.3.1)

with x ∈ Rn as a vector of variables. The vector can contain y, i.e., the variable under

consideration. The examples mentioned in the main text include investment functions,price expectation hypotheses, etc. Most models in economic dynamics deal with very simplelag structures in the form of, e.g.,

yt = f(yt−1, xt−1), (A.3.2)

with y, x ∈ R or

yt = f(xt−T ), (A.3.3)

with y, x ∈ R, T ≥ 1. When the value of the (dependent) variable yt depends on the valueof x delayed by a constant number of periods the system is said to posses a constant lag.

In addition to these simple constant lags, several different lag structures are occasionallyassumed in models of economic dynamics.21 The majority of economic examples can befound in models with learning behavior. In order to simplify the exposition, the depen-dence of a variable yt on its past values will be omitted in the sequel.

Discrete Time

In discrete-time models, the distributed lag belongs to the most common lag structures. Alag is called a distributed lag when the value of a variable Yt is a weighted average of the npast values of another variable X:

Yt = λ1Xt−1 + λ2Xt−2 + . . . + λnXt−n, λ ∈ (0, 1),n∑

i=1

λi = 1. (A.3.4)

The geometric lag represents a special form of distributed lags:

Yt = (1 − λ)(Xt−1 + λXt−2 + λ2Xt−3 + . . . + λnXt−n−1), λ ∈ (0, 1). (A.3.5)

The sum of the coefficients constitutes a geometric sum and converges toward 1 for n → ∞,i.e.,

limn→∞ (1 − λ)

n∑i=0

λi = 1. (A.3.6)

21 Extensive discussions of different types of time lags can be found in Allen (1963), pp.23ff., and Koyck (1954).

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A.3. An Introduction to the Theory of Lags 271

Types of Lags Formal Description

Constant Lag Yt = Xt−T , T ∈ N

Distributed Lags • general

Yt = λ1Xt−1 + λ2Xt−2 + . . . + λnXt−n;

λ ∈ (0, 1),∑n

i=1 λi = 1

• geometric

Yt = (1 − λ)(Xt−1 + λXt−2+

+λ2Xt−3 + . . . + λnXt−n−1),

λ ∈ (0, 1)

Types of Time Lags in Discrete-Time Dynamical SystemsTable A.3

The presence of geometrically distributed lags has a consequence which is used in somemodels discussed in the main text.22 Consider the equation

Yt = (1 − λ)(Xt−1 + λXt−2 + λ2Xt−3 + . . . + λnXt−n−1). (A.3.7)

Substituting t− 1 for t and multiplying the equation with λ yields

λYt−1 = (1 − λ)(λXt−2 + λ2Xt−3 + λ3Xt−4 + . . . + λn+1Xt−n−2). (A.3.8)

Subtraction of (A.3.7) from (A.3.8) yields

Yt − λYt−1 = (1 − λ)(Xt−1 − λn+1Xt−n−2). (A.3.9)

For n large, the expression λn+1Xt−n−2 converges toward 0 because λ is smaller than 1.Re-arranging terms yields

Yt − Yt−1 = (1 − λ)(Xt−1 − Yt−1). (A.3.10)

The geometric lag (A.3.7) results in a linear, first-order, difference equation in the variableY .23

Continuous Time

When a model is formulated in continuous time, basically the same lag structures as inthe discrete-time case can be assumed. Table A.4 contains the forms of the constant and

22 Compare Koyck (1954), pp. 22, for the following.23 A well-known economic example of equation (A.3.10) is provided by the hypothesis of

adaptive price expectations, i.e., ∆pet = α(pt−1 − pet−1), α > 0.

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272 Appendix

continuously distributed lags in continuous-time models. The time indices t, T , and τ arenon-negative real numbers.

Types of Lags Formal Description

Constant Lag Y (t) = X(t− T ), T ∈ R

Continuously Distributed Lags • General Case

Y (t) =∫∞

0 f(τ)X(t− τ) dτ ;∫∞0 f(τ) dτ = 1

• Exponential Lag

Y (t) = λ∫∞

0 e−λτX(t− τ) dτ ;∫∞0 λe−λτ dτ = 1

Types of Time Lags in Continuous-Time Dynamical SystemsTable A.4

The presence of an exponentially distributed lag allows a similar transformation like theone described above for discrete-time systems.24 Consider the exponential lag

Y (t) = λ

∫ ∞

0e−λτX(t− τ) dτ. (A.3.11)

Replacing t− τ with x yields

Y (t) = λ

∫ t

−∞e−λ(t−x)X(x) dx = λe−λt

∫ t

−∞eλxX(x) dx. (A.3.12)

Differentiation of (A.3.12) with respect to t yields

λY eλt + Y eλt = λd

dt

(∫ t

−∞eλxX(x) dx

)= λeλtX(t). (A.3.13)

It immediately follows that

Y = −λ(Y −X), (A.3.14)

i.e., an ordinary first-order differential equation in Y . In other cases, similar differentialequations can be derived from a continuous lag with the help of the Laplace transformation.25

24 Cf. Allen (1963), pp. 26f.25 Cf. Allen (1963), pp. 155ff.

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A.3. An Introduction to the Theory of Lags 273

The Use of Operators

Occasionally, differential equations like (A.3.14) are written as

DY = −λ(Y −X), (A.3.15)

with D = d/dt as the differential operator. In the main text, a difference like xt − xt−1 issometimes abbreviated as ∆xt with ∆ as the difference operator. Another operator is the shiftoperator E which shifts a variable yt to the next period:

Eyt ≡ yt+1. (A.3.16)

The advantage of the use of operators consists in the fact that they can (with some restric-tions) be treated as variables that follow the standard rules of algebra. In particular, whenO denotes one of the three operators D, ∆, or E, the rules

commutative : O2 + O = O + O2

associative : OO2 = O2O

distributive : O(O + 1) = O2 + O

(A.3.17)

apply for the operators. The rules do not apply for combinations of the operators withother variables.26 In the following, a few examples will demonstrate the usefulness of theintroduction of operators in investigating dynamical systems.

a) Operators in Discrete-timeThe shift operator E is particularly well-suited to demonstrate the advantages of the use ofoperators. With the rules of algebra mentioned above it is possible to transform (A.3.16)into

yt = E−1yt+1, (A.3.18)

with E−1 as the shift of the variable yt+1 into the previous period. Obviously, the statementimplied by (A.3.16) is preserved by this algebraic operation. A dynamic relation like

yt+2 = ayt (A.3.19)

can thus be written as

yt+2 = Eyt+1 = E2yt = ayt, (A.3.20)

or

yt = aE−2yt ≡ ayt−2. (A.3.21)

It is simple to derive the relation between the shift operator E and the common differenceoperator ∆:

∆yt ≡ yt − yt−1,

= Eyt−1 − yt−1 = (E − 1)yt−1 = E−1(E − 1)yt =E − 1E

yt.(A.3.22)

26 Cf. Allen (1963), pp. 725ff., for a discussion of the allowed operations.

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274 Appendix

Thus, the relation between the two operators is

∆ =E − 1E

or E =1

1 −∆. (A.3.23)

In order to examine the correctness of this result consider the difference equation

∆yt = yt − yt−1 = ayt−1. (A.3.24)

Substituting for ∆ yields

E − 1E

yt = ayt−1,

(E − 1)yt = aEyt−1,

Eyt − yt = aEyt−1,

yt+1 − yt = ayt.

(A.3.25)

Replacing t by t− 1 immediately yields (A.3.24).The following example uncovers that operators can be particularly useful in models with

specific lag structures. Consider the geometric lag (A.3.7) and let Yt = pet and Xt = pt.The equation can then be interpreted as a price expectation hypothesis with pt as actualprices in t and pet as the prices expected to prevail in period t. With the help of the shiftoperator E, the hypothesis can be written as

pet = (1 − λ)(pt−1 + λpt−2 + λ2pt−3 + . . . + λnpt−n−1),

= (1 − λ)(E−1pt + λE−2pt + λ2E−3pt + . . . + λnE−n−1pt

),

= (1 − λ)E−1(λ0E0 + λ1E−1 + λ2E−2 + . . . + λnE−n

)pt,

(A.3.26)

with λ ∈ (0, 1).The expression in parentheses is a geometric sum, implying that (A.3.26) can be written

as

pet = (1 − λ)E−1(λnE−n − 1λE−1 − 1

)pt. (A.3.27)

For n → ∞, (A.3.27) converges toward

pet = −(1 − λ)E−1( 1λE−1 − 1

)pt. (A.3.28)

because of λ < 1. Equation (A.3.28) can be transformed by using the above mentionedalgebraic rules for the operator:

(λE−1 − 1)pet = −(1 − λ)E−1pt,

λE−1pet − pet = −(1 − λ)E−1pt.(A.3.29)

Expanding the operator yields

λpet−1 − pet = −(1 − λ)pt−1. (A.3.30)

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A.3. An Introduction to the Theory of Lags 275

Re-arranging terms leads to

pet − pet−1 = (1 − λ)(pt−1 − pet−1

), (A.3.31)

i.e., the standard presentation form of adaptive expectations with λ < 1. This is not verysurprising after the calculations in (A.3.7) – (A.3.10) but the example shows that a straightapplication of operators in a given system allows for the derivation of interesting results.

b) Operators in Continuous TimeIn continuous-time systems, basically the same results as in the previous section can be de-rived. For example, it is permitted to perform the following operation with the differentialequation (A.3.15):

(D + λ)Y = λX =⇒ Y =λ

D + λX. (A.3.32)

This representation of the underlying exponential lag structure is useful because it allowsfor a definition of more complicated exponential lags. The the so-called multiple exponentiallag is defined as

Y (t) =

(λn

D + λn

)n

X(t), (A.3.33)

with n as a natural number defining the degree of the exponential lag. For n = 1, theresulting lag (A.3.32) can be called a simple exponential lag.

The exponential term in (A.3.33) converges toward eD for n → ∞. The expressionacts like a shift operator E−1 described above in the context of discrete-time systems anddecreases the time argument t by 1:27

Y (t) = e−DX(t) = X(t− 1). (A.3.34)

Though t in (A.3.34) is a real variable, it is possible to measure time only in equidistant in-tervals. When the time interval between two measurements is ∆t = 1, (A.3.34) is equivalentwith a standard, discrete-time, one-dimensional dynamical system.

Sparrow (1980) has demonstrated that the differential equation system

x1 = n(f(xn) − x1

),

x2 = n(x1 − x2),...

xn = n(xn−1 − xn),

(A.3.35)

can be written as

xn =

(n

D + n

)n

f(xn), (A.3.36)

27 Cf. Yosida (1984), pp. 74ff.

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276 Appendix

i.e., the system implicitly inhibits a multiple exponential lag structure. For n → ∞ theexponential term converges toward e−D. Equation (A.3.36) turns into

xn = f(xt−1), (A.3.37)

for n → ∞, i.e., an ordinary one-dimensional difference equation. An intensive discussionof this relation between continuous-time and discrete-time dynamical systems is containedin Invernizzi/Medio (1991) and Medio (1991a).

A.4. Numerical Simulations and Chaotic Dynamics in Theoretical Economics

The detection of chaotic dynamics and other phenomena in nonlinear dynamical systemscould not have been possible without the availability of fast electronic computing devices.However, the wide interest in the behavior of these nonlinear systems was initiated not onlyby the advances made in the hardware equipment but also by the availability of completesoftware libraries and easy-to-use packages, relieving the end-user from endless struggleswith implementations of sophisticated numerical algorithms. Though it is always wise tolook behind the scenes when a particular algorithm is chosen from a software package itis probably true that an end-user cares less about the working of a program than someonewho implements an algorithm for himself. Algorithms for the computation of a particularmathematical task differ in speed, accuracy, controllability, etc., and in many cases a simplealgorithm approximates the mathematically correct result with sufficient precision withina reasonable amount of time. When the numerical results are unsatisfactory, at least somequalitative properties like the convergence of a root-finding algorithm to any solution maybe observed.

Nonlinear dynamical systems can be different. Many examples of dynamical systemsintroduced in the main text lack an entirely analytical treatment such that a numerical sim-ulation of the system appears to be helpful in understanding its dynamic behavior. Usually,the simulator is not interested in the particular value of an endogenous variable at a cer-tain point in time but in the global behavior of the system and the geometric shape of anattractor (if it exists), i.e., the simulator focuses on the qualitative behavior of a dynamicalsystem. The observed behavior in a simulation might, however, depend on the underlyingparticular algorithm in a crucial way. This is especially relevant in the numerical integrationof nonlinear, continuous-time dynamical system.

The potential relevance of the assumption of continuous-time dynamical systems in eco-nomics has been stressed occasionally in the course of the book. As the analytical solutionof a differential equation (or systems of which) can be found only in exceptional cases,numerical integration techniques are mandatory in the inspection of a system’s behavior.However, a time continuum cannot be constructed on a computing device with a finiteprecision. Thus, the true solution of a dynamical system can only be approximated by cal-culating the values of the endogenous variables in finite (possibly variable) time steps.28

No numerical algorithm is able to calculate the true value of a variable at all points in (fic-titious) time but on average the differences between calculated and true values may cancelout. On the contrary it may be possible that the calculated solution has nothing in com-mon with the true solution. This can also be the case for discrete-time dynamical systemswhere no specific algorithm for calculating the solution is necessary but where the imma-nent inaccurateness of the computing device can have identical consequences. Regarding

28 Compare, e.g., Parker/Chua (1989), Chapter 4, for a discussion of local and globalerrors in different integration algorithms.

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A.4 Numerical Simulations and Chaotic Dynamics 277

chaotic dynamical systems, the problem initiated by numerical simulations is obvious: whena system’s trajectory depends sensitively on initial conditions, the exponential divergence oftwo true trajectories is amplified by the repeated incorrectness of the numerical algorithm.Thus, it might be possible that the simulated trajectory does not reflect the influences ofthe system’s nonlinearities but represents a fictitious evolution as a consequence of the re-strictions in digital, finite-precision computing devices.

If someone who is interested in the dynamic behavior of a given nonlinear system never-theless decides to simulate the system numerically, the choice of the algorithm deserves par-ticular attention. For example, assume that the following three-dimensional, continuous-time dynamical system should be simulated:29

y1 = 80.5(−0.1 (y1 − 2.8)3 + 0.4 (y1 − 2.8) + 1.0 − y2

),

y2 = 0.5(y1 − y2

ln (1.0 + arctan y3)

),

y3 = 0.1 (1 + y3)2 (y1 − 2.0 − arctan y3),

(A.4.1)

and that the search is for chaotic dynamics. The number of computer runs with varyingcoefficients necessary to encounter chaotic behavior suggests a fast algorithm with a not toosmall time step. The object in Figure A.8 represents a projection of the calculated points(y1, y2, y3) of (A.4.1) in phase space as calculated by the Runge/Kutta algorithm onto they1-y2 plane. This is a fixed step-size algorithm without error correction and it is certainly themost popular integration algorithm.30 The object consists of 15000 calculated points witha time step of 0.05 time units.31 The complexity of the object together with its observablestructure indicates the potential presence of chaos. Indeed, the largest Lyapunov exponent,calculated from the generated time series with the Wolf et al. (1985) method32 is stronglypositive. One has to conclude that simulating (A.4.1) with the Runge-Kutta method yieldschaotic trajectories.

As was pointed out above, different algorithms can imply different qualitative results.Figure A.9 shows the object generated by simulating equation (A.4.1) with the same co-efficients as in the previous simulation but with the Adams/Gear method instead.33 TheAdams/Gear method is a variable-step-size algorithm with error correction34 which gen-erally leads to more accurate results but which is typically slower than the Runge/Kuttamethod. The observable object is not quite a single, closed orbit but the wide attractor ofFigure A.8 has shrunk to negligible noise. The evidence of chaos derived from the consid-eration of Figure A.8 has disappeared by using a different algorithm.

The lesson from this simple example is obvious and can be summarized in the followingprinciple:

29 This example has not been constructed on an ad hoc basis for the purpose of this sec-tion. Similar dynamical systems emerge in the context of simultaneous price-quantityadjustment processes. Cf. Lorenz (1992a) for a discussion of a specific process of thiskind.

30 Cf. Hairer/Nørsett/Wanner (1987), pp. 130ff., for an intensive discussion of variousRunge/Kutta and alternative methods.

31 This time step does not appear to be too large when it is taken into account that thesystem can be interpreted as a discrete-time system when the time step is 1.0.

32 Compare Section 6.2.4. for the definition of Lyapunov exponents.33 Cf. Hairer/Nørsett/Wanner (1987), pp. 347ff.34 The absolute and relative error allowances were chosen to be 0.1 · 10−5.

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278 Appendix

Y2

Y1

The Simulation of (A.4.1); 4th-order Runge-Kutta MethodFigure A.8

Y2

Y1

The Simulation of (A.4.1); Variable Step-Size Adams/Gear MethodFigure A.9

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A.4 Numerical Simulations and Chaotic Dynamics 279

Principle: Never trust a single numerical simulation of a nonlinear continu-ous-time dynamical system.

These remarks may appear to be in contrast with the numerical robustness of the oftenquoted attractors of the prototype systems like the Lorenz or Rossler systems. The fact thattheir global behavior seems to be robust with respect to different integration techniques canindeed be considered as evidence for the presence of chaos in the true solution. However,the researcher who investigates the dynamic behavior of a genuinely new system can only beadvised to check the behavior of that system with as many as possible or available algorithms,keeping in mind that even the application of a larger number of different algorithms toa seemingly chaotic system does not constitute a definite numerical proof of chaos in thetrue system.

Figure A.10

The differences in the dynamic behavior ofthe differential equation system in the examplementioned above are due to the use of differ-ent integration techniques which in themselvesalways constitute discrete approximations of acontinuous-time process. On a first glimpse itmight seem that genuine discrete-time systemsbehave quite robust when they are simulated ondigital computers. However, consider the so-called doubling map (A.4.2), i.e., the map that as-signs only the fractional part of 2xt to the valuext+1 in the next period.35 Figure A.10 shows thegraph of the map with its two pieces consistingof straight lines with a slope of 2. The map is de-fined on the interval (0, 1] and maps the intervalonto itself. The trajectory outlined in the figuredepicts a period-three cycle. Thus the prerequi-sites of the Li/Yorke theorem are fulfilled and

chaotic motion prevails in the sense of that theorem. The Lyapunov exponent of

xt+1 = 2xmod 1 (A.4.2)

is calculated as λL = log2(2N )/N = log2 2 = 1, i.e., there exists a sensitive dependence oninitial conditions. There is no immediate reason to expect anything else than a confirmationof this analytical result in a numerical simulation.

Nevertheless, numerical simulations of (A.4.2) typically yield oscillatory (and occasion-ally complex) behavior within the first iterations and an eventual jump to the fixed point(x∗ = 0.0). If one concentrates only on the simulations, one has to conclude that thedynamic behavior of (A.4.2) is far from being chaotic.

The reason for this divergence of the numerical simulations and an analytic consider-ation can be found in the immanent features of digital computers. Basically, two storageproperties can be made responsible for the results.

1. The orbit in Figure A.10 constitutes a period-three cycle. A numerical example of suchan orbit is given by the sequence 4/7, 1/7, 2/7, 4/7, . . . All three components of thiscycle represent rational numbers with infinitely many digits following the period. How-ever, the standard floating-point arithmetic on digital computers considers only a finite

35 Cf. Devaney (1992) for a more detailed description of this map and its dynamic behav-ior.

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280 Appendix

Period True CalculatedValues Values

1 4/7 0.57142862 1/7 0.14285723 2/7 0.28571444 4/7 0.57142885 1/7 0.1428576...

......

18 2/7 0.289062519 4/7 0.578125020 1/7 0.156250021 2/7 0.312500022 4/7 0.625000023 1/7 0.250000024 2/7 0.500000025 4/7 0.0000000

Period True CalculatedValues Values

1 0.6 0.60000002 0.2 0.20000003 0.4 0.40000014 0.8 0.80000025 0.6 0.6000004...

......

17 0.6 0.601562518 0.2 0.203125019 0.4 0.406250020 0.8 0.812500021 0.6 0.625000022 0.2 0.250000023 0.4 0.500000024 0.8 0.0000000

Table A.5: Divergence of True and Simulated Values in the Doubling Map

number of digits and truncates the possibly infinite series of digits of a rational number.The left part of Table A.5 lists the true values {xt} of the period-three cycle and thevalues calculated with the help of a REAL*4 arithmetic. It follows that an orbit con-sisting of such numbers cannot be calculated on a digital computer with the help ofthe standard floating-point arithmetic. The same is true for those numbers whose exactrepresentation requires m digits following the period but which can be represented by acomputer only with n < m digits. Only the use of exact rational arithmetics can preventthe calculation of incorrect results.

2. It might be suspected that this phenomenon is restricted to the case n < m as definedabove. However, this is not true. Consider the period-four cycle 0.6, 0.2, 0.4, 0.8, 0.6, . . ..The right part of Table A.5 lists the true values and the values calculated by iteratingthe map numerically. It can be observed that the third value deviates already from itstrue representation. It depends to some degree on the used programming languagewhether deviations in the output’s last digit reflect a change in the internal represen-tation of the number or just its output formatting. Usually, however, the deviation isa consequence of the internal binary representation of a number. Iterating a numberoften implies that the contribution of the last bit (the so-called least significant bit) tothe represented values decreases. In many cases the implied internal change in the rep-resented number is insignificant; in the present case of the doubling map an error issystematically produced from one iteration step to the other.

The seeming “convergence” of both calculated time series in the two parts of Table A.5 isa consequence of the particular map and these two properties of digital computers, i.e.,truncating and increasing insignificance of the last bit. The map possesses the two fixedpoints x∗ = 0 and x∗ = 1. The pre-image of x∗ = 0 in the interval (0, 1] is x = 0.5: oncex = 0.5 is reached, the trajectory jumps to x∗ = 0 and stays there forever. The pre-imageof x = 0.5 consists of two values, namely x = 0.25 and x = 0.75. This branching treecan be followed for a while and it will turn out that all numbers in the interval with ‘25’as the last two digits in the representation of the number constitute the basin of x = 0.0.Thus, when the two storage procedures described above incidentally generate an internal

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A.4 Numerical Simulations and Chaotic Dynamics 281

representation of a number with ‘25’ as the last digits, the iteration of the map will lead toa quick convergence toward the fixed point x∗ = 0.0.

In other examples the errors generated by digital computers may not have those drasticconsequences as in the example described above. However, it is worthwhile (if not manda-tory) to check the internal representations of numbers in digital computers (controlled bya specific programming language with its own characteristics) before any conclusions onpossibly chaotic dynamic behavior in numerically calculated time series are drawn. For ex-ample, simple facts (like rounding in integer division) can easily be overlooked. A carefulexamination of the limitations of digital computers can be helpful in avoiding severe mis-perceptions of the computations. Without such an investigation it cannot be excluded thatthe observed chaotic or non-chaotic trajectories are not the results of the inherent dynamicsof the considered systems but a consequence of disregarded computer architecture.

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Woodford, M. (1987): “Equilibrium Models of Endogenous Fluctuations.” Mimeo. Uni-versity of Chicago.

Woodford, M. (1989): “Imperfect Financial Intermediation and Complex Dynamics.” In:Barnett, W.A./Geweke, J./Shell, K. (eds.): Economic Complexity: Chaos, Sunspots,Bubbles, and Nonlinearity, pp. 309-334. Cambridge: Cambridge University Press.

Woodford, M. (1990): “Learning to Believe in Sunspots.” Econometrica 58, pp. 277-307.

Woodcock, A.E.R./Davis, M. (1979): Catastrophe Theory. London: Penguin.

Yan–Qian, Y. (1986): Theory of Limit Cycles. Translations of Mathematical Monographs 66.Providence: The American Mathematical Society.

Yosida, K. (1984): Operational Calculus. A Theory of Hyperfunctions. Berlin–Heidelberg–New York: Springer–Verlag.

Zeeman, E.C. (1977): Catastrophe Theory. Selected Papers 1972-77. Reading, MA: Addison–Wesley.

Zhang, W-B. (1988): “Hopf bifurcations in multisector models of optimal economicgrowth.” Economics Letters 26, pp. 329-334.

Zhang, W-B. (1990a): Economic Dynamics. Berlin–Heidelberg–New York: Springer–Verlag.

Zhang, W-B. (1990b): Synergetic Economics – Dynamics, Nonlinearity, Instability, Non–Equilib-rium, Fluctuations and Chaos. Berlin–Heidelberg–New York: Springer–Verlag.

Zhang, Z./Wen, K.-H./Chen, P. (1992): “Complex Spectral Analysis of Economic Dynamicsand Correlation Resonances in Market Behavior.” Mimeo. University of Texas, Austin.

Zhifen, Z. (1986): “Proof of the Uniqueness Theorem of Limit Cycles of GeneralizedLienard Equations.” Applicable Analysis 23, pp. 63-76.

Page 323: Lorenz, Chaos

Name Index

Abraham, R.H., 82, 121, 182, 283Abramowitz, M., 284Alexander, J.C., 96, 283Allen, R.G.D., 249, 273, 283Amano, A., 48, 283Ames, W.F., 298Ancot, J.P., 289Anderson, G.S., 121, 283Anderson, P.W., 284, 286, 290, 291Andronov, A.A., 36, 42, 63, 73, 82, 283Aoki, M., 285Arneodo, A., 193, 194, 198, 283, 288Arnold, V.I., 63, 66, 82ff., 233, 283f.Arrow, K.J., 21, 47, 286, 286, 284, 284,

284, 290, 291, 295, 306Arrowsmith, D.K., 40f., 63, 66, 266, 284Arthur, W.B., 246, 284Azariadis, C., 148, 284

Baek, E.G., 287Bajo-Rubio, O., 225, 284Bala, V., 148, 284Balasko, Y., 238f., 284Barnett, W.A., 205, 226, 284, 286f.,

289, 292, 294, 304, 308Batten, D., 293, 302Baumol, W.J., 21, 121, 124, 228, 284f.,Beckmann, M.J., 48, 50, 285Begg, D.K.H., 148, 285Bellman, R., 145, 285Benassy, J.P., 47, 285Benedicks, M., 150, 285Benettin, G., 218, 285Benhabib, J., 98, 101, 108f., 121, 124,

143, 146f., 285

Page 324: Lorenz, Chaos

310 Name Index

Berge, P., 121, 133, 172, 179, 195, 202,203, 205, 206, 212, 285

Berndt, E.R., 284 , 287Bernoulli, D., 14Beyn, W.-J., 194, 285Birchenhall, C.R., 238, 285Blackwell, D., 145, 285Blad, M.C., 238, 285Blatt, J.M., 24, 28f., 285f.Blaug, M., 11, 286Bohm, V., 157, 286Boldrin, M., 20, 105, 143, 148, 286Boyce, W.E., 40ff., 52, 286Braun, M., 259, 261f., 286Brezinski, C., 298Brock, W.A., 23f., 53, 121, 145, 147,

205f., 218, 223f., 249, 286f.Brody, A., 25, 72, 73, 287

Cairns, S.S., 305Candela, G., 148, 287Carleson, L., 150, 285Carr, J., 266, 287Cartwright, M.L., 18, 182, 186, 287Casdagli, M., 231, 287Casti, J., 287, 293, 302Chaikin, C.E., 36, 42, 63, 73, 82, 283Chang, W.W., 43f., 287Chen, P., 121, 205, 226, 246, 284, 288,

307f.Chiarella, C., 72f, 76, 107, 148, 288Choi, S., 226, 284Chow, S.-N., 36, 288Chua, L.O., 120, 276, 301Clark, C.W., 62, 288

Page 325: Lorenz, Chaos

Name Index 311

Coddington, E.A., 40, 288Coleman, D., 306Collet, P., 121f., 127, 130, 136ff., 288Coullet, P., 193f., 283f., 288Crutchfield, J.P., 8, 288Cugno, F., 71, 118, 288Cushing, J.M., 246, 289

Dale, C., 202, 289Dana, R.A., 102, 153, 289Davies, D.G., 48, 289Davis, M., 241, 308Dawkins, R., 12, 289Day, R.H., 130, 136f., 139, 146ff., 159f.,

285, 289Debreu, G., 9, 21, 41, 49, 289Dechert, W.D., 121, 147, 218, 224, 287,

289, 293Delli Gatti, D., 156, 289Dendrinos, D.S., 149, 290Deneckere, R.J., 145, 286, 290Dernburg, T.F., 106, 290Dernburg, J.D., 106, 290Desai, M., 69, 290Descartes, R., 7Devaney, R.L., 121f., 124, 150, 279, 290Diamond, P., 150, 290Diamond, P.A., 101, 290Dierker, E., 36, 49, 290DiPrima, R.L., 40ff., 52, 286Dockner, E.J., 101, 290Dopfer, K., 148, 290Dosi, G., 304

Eckalbar, J.C., 121, 290Eckmann, J.-P., 29, 121f., 127, 130,

136ff., 205, 215, 217, 219, 225,288, 290

Edgeworth, F.Y., 11, 290Eliasson, G., 289Euler, L. 14

Falconer, K., 291Farkas, M., 25, 72f., 287Farmer, J.D., 215, 228, 247, 288, 291Farmer, R.E.A., 118, 291Fatou, P., 18, 120, 291Feichtinger, G., 101, 148, 288, 290f.,

296, 298f., 302Feigenbaum, M., 127, 291Feinstein, C.H., 293Fernandez-Rodriguez, F., 284

Page 326: Lorenz, Chaos

312 Name Index

Filippov, A., 53Fischer, E.O., 242, 291Fischer, P., 283, 295, 297Fisher, I., 10, 291Flaschel, P., 49, 69f., 291ff.Foley, D.K., 110, 292Frank, M.Z., 205, 224, 226, 231, 292Franke, R., 73, 101, 148, 292Frisch, R., 23, 292Fudenberg, D., 101, 290Furstenberg, G.M., 286Furth, D., 239, 292

Gabisch, G., 24, 43, 47, 69, 148, 234,236, 239, 292

Gaertner, W., 148, 156, 292Galeotti, M., 53, 58, 286, 292, 298f.,

307Galgani, L., 218, 285Galileo, G. 7Gallegati, M., 156, 289Gandolfo, G., 27, 64, 69, 106, 195, 249,

254, 259, 293Gantmacher, F.R., 107, 259, 293Gardini, L., 148, 156, 287, 289, 293Garrido, L., 173, 293Gencay, R., 218, 224, 289, 292f.George, D., 239, 293Georgescu-Roegen, N., 9, 13, 293Geronazzo, L., 286, 298f., 307Geweke, J., 284, 286, 289, 292, 294,

304, 308Glass, L., 247, 293Gleick, J., 246, 293Glendinning, P., 194, 293Glombowski, J., 69, 293Goldberg, S., 255, 293Goodwin, R.M., 19, 49, 67, 69, 182f.,

186, 288, 293, 300, 307Gori, F., 53, 58, 286, 292, 298, 299, 307Grandmont, J.-M., 122, 147f., 161, 293,

294Granger, C.W.J., 202, 224, 294Grassberger, P., 158, 211, 218f., 221,

294, 297Grebogi, C., 158, 170, 211, 294, 299f.Guckenheimer, J., 29f., 33f., 39f., 54,

73, 76, 81, 84, 87, 96, 99f., 107,115f., 121, 133, 135, 138, 161, 182,184, 193, 205, 215, 259, 265ff., 294

Guesnerie, R., 143, 148, 284, 294Guillemin, V., 38, 294

Page 327: Lorenz, Chaos

Name Index 313

Gutowski, H., 246, 294

Haag, G., 235, 307Haavelmo, T., 141, 294Hahn, F.H., 31, 47, 254, 284, 295Hahn, F.R., 148, 295Hahn, W., 34, 295Hairer, E., 277, 295Haken, H., 176, 219, 235, 247, 291,

295, 295, 303, 306Hale, J.K., 36, 288Hammer, G., 292Harcourt, G.C., 19, 295Hassard, B.D., 96, 295Hatanaka, M., 202, 294Haxholdt, C., 186, 295Hayek, F.A., 300Hegel, G.W.F., 8Heiner, R.A., 147, 295Helleman, R., 297Herrmann, R., 103, 153, 295Herzel, H., 218, 297Hicks, J.R., 24, 156, 295Hilbert, D., 51, 295Hildenbrand, W., 49, 287, 295Hinich, M.J., 226, 284Hirsch, M.W., 34, 40f., 52, 64, 84, 249,

259, 295Hodgson, G.M., 11, 295Holden, A.V., 294, 297, 306, 308Holmes, P.J., 29f., 33f., 39f., 54, 73, 76,

81, 84, 87, 96, 99f., 107,115f., 121,133, 161, 182, 184, 193, 205, 215,259, 265ff., 294, 302

Hommes, C., 148, 156f., 295, 296, 301Hopf, E., 95, 296Hsieh, D., 121, 287, 296Hunt, E.K., 293Hurwitz, L., 47, 284

Ichimura, S., 55, 296Intriligator, M., 295f., 300, 304, 306Invernizzi, S., 276, 296Iooss, G., 115, 296f.,Ipaktchi, A., 294

Jacobson, M.V., 136, 296Jammernegg, W., 242, 291Jarsulic, M., 157, 246, 296Jensen, R.V., 148, 296Jevons, W.S., 2, 9f., 296Johansson, B., 293, 302

Page 328: Lorenz, Chaos

314 Name Index

Jojima, K., 12, 296Julia, G., 18, 120, 296Jungeilges, J., 156, 292

Kaldor, N., 59, 153, 296Kalecki, M., 23, 43, 222, 297Kamphurst, S.O., 290Kant, I. 6Kantz, H., 158, 297Kapitaniak, T., 245, 297Karlin, S., 301Karlquist, A., 287Kazarinoff, N.D., 96, 295Kelsey, D., 121, 147, 297Kelso, J.A.S., 291Kepler, J. 7Kim, K.-H., 148, 289Kirman A.P., 49, 295, 297Kocak, H. 120, 297Koch, B.-P., 297Kostelich, E.J., 221, 306Koyck, L.M., 297Kruger, M., 69, 288, 291, 292, 293, 300,

307Krasner, S., 304Kurihara, K.K., 296Kurths, J., 218, 297

Lagrange, J. de, 14Lanford, O.E., 161, 297Langford, W.F., 177, 297Laplace, P.S., 7f., 14, 77Larsen, E.R., 157, 182, 186, 227, 295,

297, 301, 305Lasota, A., 136, 297Lassalle, J.P., 34, 297Lauwerier, H.A., 122, 297LeBaron, B., 121, 224, 226, 287, 304Lefschetz, S., 34, 297Lehnert, D., 157, 304Leibniz, G.W., 6Leven, R.W., 297Levi, M., 182, 184, 297Levinson, N., 18, 40, 52, 182, 288Li, T.Y., 18, 135, 298Lichtenberg, A.J., 298Liebermann, M.A., 298Lines, M., 245, 298Littlewood, J.E., 18, 182, 287, 298Lienard, A., 298Lorenz, E.N., 18, 120, 298

Page 329: Lorenz, Chaos

Name Index 315

Lorenz, H.-W., 24, 43, 47, 49, 55, 58,69, 161, 180, 182, 187, 200, 208,234, 236, 239, 277, 292, 298f.

Lotka, A.Y., 62, 299Lucas, R.E., 22, 299Lux, T., 73, 101, 292, 299Lyapunov, A.M., 299

Mackey, M.C., 247, 293Majumdar, M., 148, 284Malgrange, P., 102, 153, 289, 294Malinvaud, E., 238, 299Malliaris, A.G., 53, 249, 287Malthus, T.R., 21, 113, 299Marotto, F.R., 150, 155, 299Marschak, J., 48, 299Marsden, J.E., 82, 96, 99, 283, 299Marshall, A., 2, 11, 21, 299Martinengo, G., 27, 293Marx, K. 8Marzollo, A., 285Mas-Colell, A., 48f., 299May, R.M., 122, 231, 299, 305Mayer-Kress, G., 306, 308McCracken, M., 96, 99, 299McDonald, S.W., 158, 299f.McKenzie, L.W., 145, 300Medawar, P.B., 11, 300Medio, A., 108, 121, 136, 202, 205,

236, 276, 296, 300Mees, A.I., 161, 300Meese, R., 205, 300Melese, F., 136, 300Menger, C., 11, 300Merton, R.C., 303Metzler, L.A., 195, 300Mill, J.St., 7, 10f., 300Milnor, J.W., 36, 300Mira, C., 158, 300Mirowski, P., 9, 300Misiurewicz, M., 135, 298Miyao, T., 98, 101, 285Montrucchio, L., 71, 118, 143, 286,

288, 301Morecroft, J.D.W., 186, 297Morishima, M., 49, 301Mosekilde, E., 157, 182, 186, 227, 295,

297, 301, 305f.

Nell, E.J., 300v. Neumann, J., 122, 306Newhouse, S., 179, 301

Page 330: Lorenz, Chaos

316 Name Index

Newman, P., 47, 301Newton, I., 6, 14Nicolis, G., 246, 301Nijkamp, P., 149, 290, 299, 301, 303Nishimura, K., 108f., 143, 285Nitecki, Z., 161, 301Novak, A., 101, 291Novshek, W., 49, 301Nusse, H.E 135, 137, 148, 156ff., 296,

301Nørsett, S.P., 277, 295

Oster, G., 294Ott, E., 121, 150, 158, 170, 220, 291,

294, 294, 299ff.

Packard, N.H., 288Padoan, P.C., 27, 293Pallaschke, D., 292Papell, D.H., 225, 301Pareto, V., 2, 9, 21Parker, T.S., 120, 276, 301Peixoto, M.M., 84, 294, 301, 305Pelikan, S., 145, 170, 290, 294Peters, E.E., 205, 302Phillips, A.W., 55, 302Pianigiani, G., 130, 135ff., 148, 159,

289, 298, 302Pines, D., 284, 286, 290f.,Place, C.M., 40f., 63, 66, 266, 284Ploeg, F., van der 69, 148, 302Pohjola, M.J., 69, 148, 302Poincare, H., 17, 120, 302Pollack, A., 38, 294Pomeau, Y., 195, 285Pompe, B., 297Popper, K.R., 15, 302Poston, T., 234, 238, 302Potter, S.M., 225, 302Prescott, D.M., 231, 302Preston, C., 122, 137, 302Prigogine, I., 246, 301Procaccia, I., 211, 219, 221, 294Prskawetz, A., 148, 302Puu, T., 148, 180, 186, 302

Ramsey, J.B., 226, 302Rand, D., 148, 302, 306Rand, R.H., 302Rasmussen, S., 182, 297Reggiani, A., 149, 299, 301, 303Reichlin, P., 107, 118, 302

Page 331: Lorenz, Chaos

Name Index 317

Reuter, G.E.H., 182, 186, 287Ricardo, D., 6f., 21Ricci, G., 19, 301, 307Rommelfanger, H., 259, 303Rose, A., 205, 300Rosser, J.B., 149, 238f., 303Rossler, O.E., 303Rothman, P., 226, 302Ruelle, D., 18, 29, 115, 121, 136, 170,

205, 215, 217, 219, 290, 301, 303Ryder, H.E., 48, 50, 285

Saari, D.G., 49, 303Salmon, M., 304Samuelson, P.A., 23, 47, 71, 121, 148,

183, 303Sargent, T.J., 22, 299Sarkovskii, A.N., 303Saunders, P.T., 233f., 303Sayers, C.L., 205, 225f., 287, 301ff.,Scheinkman, J.A., 145, 205, 224, 226,

287, 290, 304Schinasi, G.J., 55, 304Schmidt, K., 225, 304Schofield, R., 306Schuster, H.G., 121, 304Schwodiauer, G., 287Schwarz, J.G., 293Scott, K.A., 182, 283Sebba, G., 12, 304Semmler, W., 49, 101, 292, 299, 304f.Shafer, W., 49, 147, 289, 304Shaw, C.D., 121, 283Shaw, R.S., 288Shell, K., 284, 286, 289, 292, 294, 304,

308Shil’nikov, L.P., 193, 304Sidorowich, J.J., 228, 291Silverberg, G., 157, 246, 304Simonovits, A., 25, 156f., 296, 304f.,Simo, C., 173, 293Sinai, J.G., 305Singer, D., 122, 137, 305Slutzky, E., 23, 222, 305Smale, S., 34, 40f., 52, 64, 84f., 158,

161, 165, 182, 249, 259, 295, 305Smith, A., 6f., 21Smith, O.K., 52, 298Smith, W.R., 283, 295, 297Smyth, D.J., 43f., 287Sonnenschein, H., 49, 287, 299, 301,

304f.,

Page 332: Lorenz, Chaos

318 Name Index

Sargent, T.J., 22, 299Sorger, G., 101, 291, 305Sosvilla-Rivero, S., 284Sotomayor, J., 95, 305Sparrow, C., 194, 275, 293, 305Stahlecker, P., 225, 304Stengos, T., 205, 224, 226, 231, 292,

302Sterman, J.D., 157, 182, 186, 227, 297,

301, 305f.Stewart, I., 234, 238, 302Stewart, H.B., 161, 306Stoker, J.J., 305Stora, R., 297Strelcyn, J.M., 218, 285Stutzer, M., 141, 305Sugihara, G., 231, 305Suppes, P., 301Swift, J.B., 308Swinney, H.L., 306, 308Szegœ, G.P., 288

Takayama, A., 31, 48, 249, 306Takens, F., 18, 115, 179, 205f., 301,

303, 306Thoben, H., 12, 306Thom, R., 233f., 306Thompson, J.M.T., 161, 306Thomsen, J.S., 157, 182, 186, 297, 301,

306Tomita, K., 182, 306Torre, V., 180, 306Transue, W., 136, 300Tresser, C., 193f., 283, 288, 306v. Tunzelmann, G.N., 70, 306Tvede, M., 186, 295

Ueda, Y., 120, 306Ulam, S.M., 122, 306Urban, R., 148, 296Uzawa, H., 47, 306

Varian, H.R., 36, 38, 40, 283, 290, 306Vastano, J.A., 221, 306, 308Velupillai, K., 19, 69, 77, 301, 303.,

306f.Vercelli, A., 82, 84, 288, 300, 307Verhulst, P.-F., 114, 307Vidal, C., 195, 285Voltaire, 6Volterra, V., 62, 307

Page 333: Lorenz, Chaos

Name Index 319

Walker, D.A., 47, 307Walras, L., 2, 9, 21, 47, 159, 307Walter, J.-L., 148, 289Wan, Y-H., 96, 295Wanner, G., 277, 295Weber, W.E 226, 284Weghorst, W., 148, 292Weidlich, W., 235, 307Weintraub, E.R., 239, 307Wen, K.-H., 246, 307f.West, B.J., 13f., 113f., 307Westlund, A.H., 224, 307White, H., 284, 287White, R.W., 149, 246, 307Whitley, D., 87, 110f., 122, 307Wiggins, S., 28,f., 38f., 121, 259, 266f.,

307f.Wirl, F., 101, 291Wolf, A., 215f., 218, 220f., 308Wolfstetter, E., 69f., 308Woodcock, A.E.R., 241, 308Woodford, M., 20, 143, 147f., 286, 294,

308

Yan-Qian, Y., 51, 53f., 308Yorke, J.A., 18, 96, 120, 135f., 158, 170,

283, 291, 294, 297ff.Yosida, K., 275, 308Young, L., 306

Zeeman, E.C., 233, 238, 308Zhang, W-B., 101, 108, 308Zhifen, Z., 52, 58, 308

Page 334: Lorenz, Chaos

Subject Index

Accelerator, 24Adams/Gear algorithm, 277adiabatic approximation, 235,adjustment coefficients, 73α-limit set, 39amplitude, 256approximation, linear, 22, 26area preserving system, 65arithmomorphic system, 13attracting set, 29attractor, 28, 65

cyclical 35strange, 150, 170

autocorrelation function, 203auctioneer, 47averaging, 54

Backward iteration, 140basin boundary, 29basin of attraction, 29, 156, 199BDS test, 224Bellman’s equation, 145Bendixson criterion, 45Bernoulli differential equation, 142bifurcation, 680ff.,

bifurcations, continuous time, 81bifurcations, discrete time, 110bifurcation diagram, 81bifurcation point, 81bifurcation set, 236bifurcation value, 82flip, 111, 123fold, 87, 110f.,Hopf, see Hopf bifurcationpitchfork, 91, 110f.

Page 335: Lorenz, Chaos

Subject Index 321

subcritical, 93, 97, 113supercritical, 93, 98, 113transcritical, 89, 110f.

biology 11branch of fixed points, 81broad band noise, 203

Cantor set, 158, 164middle-third set, 210

capital stock, desired, actual 55catastrophe theory, 232ff.catastrophes, elementary, 233

fold, 234,cusp, 234f.,swallowtail, 234butterfly, 234elliptic umbilic, 234hyperbolic umbilic, 234parabolic umbilic, 234

ceiling 24cellular automata, 246center dynamics, 252center manifold, 34, 100, 107center manifold theorem, 264chaos,

in discrete-time models, 121ff.in continuous-time models, 167ff.empirical results, 223ff.topological, 135

chaotic regime, 129characteristic equation, see equationcharacteristic roots, see eigenvaluesclass struggle, 67classical mechanics, 5ff., 252closed orbit, 35, 252

Page 336: Lorenz, Chaos

322 Subject Index

cobweb model, 148codimension, 234complex roots, see roots, complexconservative system, 61, 71, 77consumption frontier, 71, 108contraction, on an attractor, 162correlation,

dimension, 208, 211f.integral, 212

corridor stability, 98coupling of oscillators, 174ff.,critical value, 122cross-dual adjustment, 48

Degree theory, 36DeMoivre’s theorem, 256depression, 240Descartes’ rule, 253determinant, 250deterministic theory, 15f.deterministic worldview, 13ff.diffeomorphism, 150, 162difference equations, see equation, dif-

ferencedifferential operator, 117, 142, 273dimension,

of an attractor, 181ff.,correlation, see correlation dimen-

sion,embedding, see embedding dimen-

sion,Euclidian, see Euclidian dimension,fractal, see fractal dimension,Hausdorff, see Hausdorff dimension,

discount factor, 144discount rate, 108discriminant, 237dissipative system, 62, 71, 216divergence, 65, 214Divisa index, 226doubling map, 279

Endomorphism, 123eigenspaces, 32eigenvalues, 56, 97, 178, 260ff.

complex, 103, 252, 256dominant, 256real, 252, 256

eigenvector 260eigenvectors, generalized, 261elementary catastrophes, 234

Page 337: Lorenz, Chaos

Subject Index 323

embedding, 205embedding dimension, 205,endomorphism, 123enlightenment, 6entropy, metric, 218

Kolmogorov, see Kolmogorov en-tropy,

equilibrium surface, 235ergodic behavior, 131equation,

Bernoulli, 142characteristic eq., 250difference eq., 110ff., 122ff., 255ff.differential eq., 27ff., 87ff., 168ff.,

234f., 249ff.Lienard eq., 52, 56, 189logistic eq., 122, 146, 170, 204Rayleigh eq., 186van der Pol eq., 52, 76, 182

equivalence, 83Euclidian dimension, 208exchange of stability, 90expanding fixed point, 150extrapolative expectations 196

Feigenbaum number, 127financial intermediation 147financial markets, 148finite differences, 117, 142first integral, 63first return map, 172fixed point,

stable, 31expanding fixed point, 150of order k, 123

floor, 24flow, 28, 61focus, 252folding, on an attractor, 213forced oscillation, see oscillator, forced,Fourier transform, 202fractal dimension, 210ff.,frequency, 175, 202. 252friction, 61

Gaussian plane, 96, 256general equilibrium analysis, 25, 147generations, overlapping, 147f.Goodwin model,

nonlinear accelerator, 155ff.,predator-prey, 67, 118, 148

gradient systems, 236

Page 338: Lorenz, Chaos

324 Subject Index

Grassberger/Procaccia plot, 213growth model, 91, 138

Hamiltonian, 108harmonic oscillator, 252Hartman-Grobman theorem, 33Hausdorff dimension, 209, 220heaviside function, 212Henon map, 150Hicks model, 24histogram, 130homeomorphism, 83homoclinic,

orbits, 39, 152, 194points, 165

Hopf bifurcation, 178f.discrete time, 115f.continuous time, 95f.

horseshoe map, 158, 161, 182hyperbolic fixed point, 82

Implicit function theorem, 96Inada conditions, 91index theory, 36ff.intermittency, 133invariant measure, 136invariant set, 28, 164, 192, 198invariant subspaces, 32inverse Fourier transform 202invertibility, 104, 130, 169f.,investment behavior, 43irreversible processes, 77,IS-LM model, 105, 147isolation technique, 13iterate, 113

Jacobian matrix, 33f.joint probability, 219

Kaldor model 43, 93, 101, 117, 239Keynesian demand policy, 187f.Kolmogorov entropy, 218,

Labor market 88laboratory experiment, 227lag, 72, 269

constant, 271distributed, 270exponential, 271multiple exponential, 174

lag operator, 272

Page 339: Lorenz, Chaos

Subject Index 325

Landau scenario, 179Lausanne school, 9Lebesgue measure, 135f.Levi/Poincare map, 184Levinson/Smith theorem, 189Li/Yorke theorem 134, 141Liapunov, see LyapunovLie derivative, 65, 198Lienard equation, see equation,Lienard transformation, 55limit cycle, 35, 65, 217

uniqueness, 51ff.limit set, 39linear regression 24Lipschitz condition 53Lorenz attractor, 168Lotka/Volterra equations, 62, 69Lyapunov,

dimension, 220exponent, 198, 208, 213function, 34numbers, 214spectrum, 214stability, 31

Manifold, 33, 65, 235master equation, 235maximum principle, 108measure theory, 136measure, absolutely continuous invari-

ant, 136mechanistic worldview, 12mercantilistic policy, 6Metzler model, 195m-history, 206mixing behavior, 132modulus, 117, 256multiplicative ergodic theorem, 215multiplier-accelerator model, 24, 55,

148

Nearest neighbor, 228neutrally stable, 62New Classical Macroeconomics, 23Newhouse/Ruelle/Takens theorem,

179node, 252noisy chaos, 245nonlinear accelerator, 183, 186non-wandering set, 30normal form, 100

Page 340: Lorenz, Chaos

326 Subject Index

Offer price, 48ω-stability, 39optimal control, 25, 143orbit, 27order equation, 204oscillator,

coupled, 174forced, 182

overlapping generations 118, 147

Peixoto’s theorem, 85perfect foresight, 147period doubling, 113, 127Pesin’s identity, 220Phillips curve, 68, 71, 241Phillips Model 54Poincare index, 36Poincare-Bendixson theorem, 39, 45,

61, 105Poincare map, 80, 110, 171, 185Poincare section, 170Poincare-Hopf theorem, 38policy function, 143, 145population dynamics, 113positive invariant set, 28potential, 236power spectrum, 203predator-prey system, 61, 66predictability, 7, 228prediction, 78principal minors, 106psychology, 11

Quantum mechanics, 15, 17quasi-linear, 19quasi-periodic motion, 176

Random process, 100, 202random process, 28, 119, 228rational expectations, 22, 222Rayleigh eq., see equation,reaction speed, 55recession, 240reductionism, 11relativity theory, 17repelling set, 29residual diagnostic, 223reversibility, 61f.roots,

characteristic, 256complex, 103, 252, 256

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dominant, 256real, 252, 256

Rossler attractor, 169, 195routes to turbulence, 180f.Routh matrix, 253Routh-Hurwitz criterion, 106, 253Ruelle-Takens scenario, 180f.Runge/Kutta algorithm, 277

Saddle loop, 39saddle node, 88saddle point, 108, 252Sarkovskii theorem, 134saturation, 114, 139sawtooth oscillation, 256Schur matrices, 257Schwarzian derivative, 113, 137, 141,

143scientific progress, 20sensitive dependence, 132separatrix, 40, 252set,

bifurcation set, 236connected set, 42limit, 39singularity set, 236uncountable, 135

shift operator, 273Shilnikov theorem, 193shuffle diagnostics, 224sink, 252slaving principle, 235Smale-Birkhoff homoclinic theorem,

165snap-back repeller, 150solution curve, 27source, 252spectral analysis, 202stability,

asymptotic, 31 252global, 34neutral, 252structural, 70, 82structural, of a function 234

stagflation, 241stochastics, 222strange attractor, 120, 150, 169f., 180,

208stretching, 162, 213structural stability, 82f.subcritical, see bifurcationS-unimodal map, 137

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supercritical, see bifurcationsuperposition, principle of, 13f.synergetics, 204

Taylor expansion, 33, 44tatonnement, 47, 148, 159, 239tent map, 132, 136time constant, 74topological chaos, 135topologically transitive, 29torus, 175, 217trace, 62, 250trajectory, 27transient, 30, 78, 130, 157transient chaos, 165, 186transversal homoclinic orbit, 165transversality, 88trapping region, 29, 186turbulence, 179

Unfolding, universal, 234unimodal map, 143uniqueness, of Limit Cycles 51unit circle, 116unit root processes, 224universal constant, 107unsharpness relation, 15urban decline, 149

Value function, 145van-der-Pol oscillator, 52, 76, 182vector field, 27Verhulst dynamics, 122volume preserving system, 65

Wandering set, 30weakly forced oscillator 183weltanschauung, 10windows, 129