# Lorenz, Chaos

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Chaos theoryTRANSCRIPT

Nonlinear Dynamical Economicsand

Chaotic Motion

Second Edition

by

Hans-Walter Lorenz

Volkswirtschaftliches SeminarGeorg-August-Universitat

Platz der Gottinger Sieben 3W-3400 Gottingen, Germany

VTo My Parents

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)

Prefaces

Preface to the Second Edition

Usually, the rst 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 Poppers 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 difcult if not impossible to keep track of all the advances made in the last

X Prefaces

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 elds 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 aws. Particular thanks

Prefaces XI

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. Thenal manuscript was typeset in PCTEX.

Gottingen, March 1989 Hans-Walter Lorenz

Contents

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

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

1.1. Some Reections 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 Modied 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. Goodwins Predator-Prey Model of the Class Struggle . . . . . . . . . . . .67

XIV Contents

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 . . . . . . . . . . . . . . . . .

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