Attractors, Bifurcations, and Chaos: Nonlinear Phenomena in Economics

512 Pages·2000·15.698 MB·English

by  Prof. Tönu Puu (auth.)

Attradors, Bifurcations,
and Chaos
Nonlinear Phenomena in Economics
Springer-Verlag Berlin Heidelberg GmbH
Tonu Puu
Attractors, Bifurcations,
and Chaos
Nonlinear Phenomena in Economics
With 186 Figures
and 2 Tables
, Springer
Prof. T6nu Puu
Umea University
Department of Economics
SE-90187 Umea
Sweden
Library of Congress Cataloging-in-Publication Data
Die Deutsche Bibliothek - CIP-Einheitsaufnahme
Attractors, bifurcations, and chaos: nonlinear phenomena in economics; with 2 tables I 'idnu
Puu. - Berlin; Heidelberg; New York; Barcelona; Hong Kong; London; Milan; Paris; Singapore;
Tokyo: Springer, 2000
ISBN 978-3-662-04096-6 ISBN 978-3-662-04094-2 (eBook)
DOI 10.1007/978-3-662-04094-2
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© Springer-Verlag Berlin Heidelberg 2000
Originally published by Springer-Verlag Berlin Heidelberg New York in 2000.
Softcover reprint of the hardcover 1st edition 2000
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Preface
The present book relies on various editions of my earlier book "Nonlinear
Economic Dynamics", first published in 1989 in the Springer series "Lecture
Notes in Economics and Mathematical Systems", and republished in three
more, successively revised and expanded editions, as a Springer monograph,
in 1991, 1993, and 1997. The frrst three editions were focused on applications.
The last was considerably different, by including some chapters with
mathematical background material (ordinary differential equations and iterated
maps), so as to make the book more self-contained and suitable as a textbook
for economics students of dynamical systems. To the same pedagogical
purpose, the number of illustrations were expanded. The author also prepared
some of the software used in producing the illustrations for use by the readers
on the PC, by making the programs interactive and providing them with a
user interface. Simulations are essential, when dealing with nonlinear systems,
where closed form solutions do not exist. Even theoretical science then
becomes experimental. (The software prepared for that book can still be
acquired directly from the author at the address tonu. puu@econ. umu. se.)
The present book has been so much changed, that I felt it reasonable to
give it a new title. There are two new mathematics chapters (on partial dif
ferential equations and on catastrophe theory), making the mathematical back
ground material fairly complete. There is also an account of the recently
emergent method of critical lines and absorbing areas for non-invertible maps
added to the chapter on maps.
As for the application part, the inclusion of partial differential equations
made it possible to discuss topics of pattern formation in the space economy,
which is another interest of the author, stemming from collaborative work
over many years with Professor Martin 1. Beckmann of Brown University
and the TV Miinchen, last given account of in my other Springer monograph
"Mathematical Location and Land Use Theory" also dating from 1997.
VI Preface
The critical line method is put to use in most application models, for drawing
attractors and for better understanding their bifurcations. A short account of
collaborative work in progress with the Urbino group, aiming at an extension
of the critical line method to critical surfaces in three dimensions, is also
included in the application to oligopoly. In writing these parts the author is
very much indebted for the expert advice from Professor Laura Gardini of
the University of Urbino.
The mathematical background chapters should provide enough of up to
date methodological tools for any economics student of nonlinear dynamics,
no matter what the particular field of aimed application is. A unique feature in
this book, among literature for economics students, is a fairly complete ac
count of the perturbation methods.
As for the applications presented in this volume, most of the material on
business cycles stems from various editions of my previous dynamics book,
though much new information has been added using the critical line method.
The same holds true for the discussion of oligopoly.
Completely new in this book is the attempt to understand economic devel
opment from the viewpoint of increasing diversity, in contrast to the normal
focus on quantitative growth. Coexistence of different attractors provides for
divergence in development, due to minor differences in the dynamical proc
ess itself, for alternative histories and futures if we so wish. The question is
whether this result of nonlinearity should not be at least as intriguing for eco
nomics as unpredictability due to deterministic chaos. Among other things the
cherished relation between steady states and optimality becomes problematic
once they are no longer unique.
Linearity for dynamical processes goes together with convexity of the back
ground structure. Expressed more explicitly, unique (optimal) steady states
and predictability, so comfortable for the belief in a well ordered reality, be
long together, and they have to leave the stage together, once we admit that
things are not so simple.
The author gratefully acknowledges the financial support over many years
of his research by the Swedish Research Council for the Humanities and
the Social Sciences, a support which definitely has been a necessary condi
tion for the earlier stages of this work. The sponsorship has now shifted to the
Swedish Transport & Communications Research Board, to which the au
thor is equally indebted.
Umea in October 1999
TonuPuu
Contents
1 Introduction 1
1.1 Dynamics Versus Equilibrium Analysis 1
1.2 Linear Versus Nonlinear Modelling 2
1.3 Perturbation Methods 4
1.4 Structural Stability 5
1.5 Chaos and Fractals 6
1.6 The Choice of Topics Included 7
2 Differential Equations: Ordinary 9
2.1 The Phase Portrait 9
2.2 Linear Systems 16
2.3 Structural Stability 24
2.4 Limit Cycles 28
2.5 The Hopf Bifurcation 33
2.6 The Saddle-Node Bifurcation 35
2.7 Perturbation Methods: Poincare-Lindstedt 37
2.8 Perturbation Methods: Two-Timing 43
2.9 Stability: Lyapunov's Direct Method versus Linearization 49
2.10 Forced Oscillators, Transients and Resonance 52
2.11 Forced Oscillators: van der Pol 56
2.12 Forced Oscillators: Duffing 65
2.13 Chaos 72
2.14 A Short History of Chaos 75
3 Differential Equations: Partial 81
3.1 Vibrations and Waves 81
3.2 Time and Space 82
3.3 Travelling Waves in 1D: d'Alambert's Solution 83
3.4 Initial Conditions 85
VIII Contents
3.5 Boundary Conditions 87
3.6 Standing Waves: Variable Separation 89
3.7 The General Solution and Fourier's Theorem 92
3.8 Friction in the Wave Equation 95
3.9 Nonlinear Waves 97
3.10 Vector Fields in 2D: Garadient and Divergence 100
3.l1 Line Integrals and Gauss's Integral Theorem 104
3.12 Wave Equation in Two Dimensions: Eigenfunctions 110
3.13 The Square 113
3.l4 The Circular Disk 118
3.l5 The Sphere 122
3.l6 Nonlinearity Revisited 127
3.17 Tessellations and the Euler-Poincare Index 129
3.18 Nonlinear Waves on the Square 131
3.19 Perturbation Methods for Nonlinear Waves 136
4 Iterated Maps or Difference Equations 147
4.l Introduction 147
4.2 The Logistic Map 148
4.3 The Lyapunov Exponent 157
4.4 Symbolic Dynamics 160
4.5 Sharkovsky's Theorem and the Schwarzian Derivative 164
4.6 The Henon Model 166
4.7 Lyapunov Exponents in 2D 170
4.8 Fractals and Fractal Dimension 173
4.9 The Mandelbrot Set 178
4.10 Can Chaos be Seen? 182
4.11 The Method of Critical Lines 185
5 Bifurcation and Catastrophe 195
5.l History of Catastrophe Theory 196
5.2 Morse Functions and Universal Unfoldings in 1 D 197
5.3 Morse Functions and Universal Unfoldings in 2 D 201
5.4 The Elementary Catastrophes: Fold 206
5.5 The Elementary Catastrophes: Cusp 207
5.6 The Elementary Catastrophes: Swallowtail and Butterfly 210
5.7 The Elementary Catastrophes: Umblics 213
6 Monopoly 217
6.1 Introduction 217
Contents IX
6.2 The Model 219
6.3 Adaptive Search 222
6.4 Numerical Results 224
6.5 Fixed Points and Cycles 225
6.6 Chaos 229
6.7 The Method of Critical Lines 231
6.8 Discussion 236
7 Duopoly and Oligopoly 239
7.1 Introduction 239
7.2 The Cournot Model 240
7.3 Stackelberg Equilibria 243
7.4 The Iterative Process 244
7.5 Stability of the Cournot Point 245
7.6 Periodic Points and Chaos 246
7.7 Adaptive Expectations 248
7.8 Adjustments Including Stackelberg Points 253
7.9 Oligopoly with Three Firms 255
7.10 Stackelberg Action Reconsidered 263
7.11 The Iteration with Three Oligopolists 264
7.12 Back to "Duopoly" 265
7.13 True Triopoly 270
7.14 Changing the Order of Adjustment 275
8 Business Cycles: Continuous Time 277
8.1 The Multiplier-Accelerator Model 277
8.2 The Original Model 278
8.3 Nonlinear Investment Functions and Limit Cycles 279
8.4 Limit Cycles: Existence 282
8.5 Limit Cycles: Asymptotic Approximation 285
8.6 Limit Cycles: Transients and Stability 290
8.7 The Two-Region Model 295
8.8 The Persistence of Cycles 296
8.9 Perturbation Analysis of the Coupled Model 298
8.10 The Unstable Zero Equilibrium 301
8.11 Other Fixed Points 303
8.12 Properties of Fixed Points 307
8.13 The Arbitrary Phase Angle 308
8.14 Stability of the Coupled Oscillators 310
8.15 The Forced Oscillator 312
X Contents
8.16 The World Market 312
8.17 The Small Open Economy 314
8.18 Stability of the Forced Oscillator 314
8.19 Catastrophe 316
8.20 Period Doubling and Chaos 317
8.21 Relaxation Cycles 320
8.22 Relaxation: The Autonomous Case 321
8.23 Relaxation: The Forced Case 323
8.24 Three Identical Regions 325
8.25 On the Existence of Periodic Solutions 327
8.26 Stability of Three Oscillators 332
8.27 Simulations 333
9 Business Cycles: Continuous Space 337
9.1 Introduction 337
9.2 Interregional Trade 338
9.3 The Linear Model 340
9.4 Coordinate Separation 342
9.5 The Square Region 344
9.6 The Circular Region 346
9.7 The Spherical Region 347
9.8 The Nonlinear Spatial Model 350
9.9 Dispersive Waves 352
9.10 Standing Waves 354
9.11 Perturbation Analysis 356
10 Business Cycles: Discrete Time 361
10.1 Introduction 361
10.2 Investments 362
10.3 Consumption 364
10.4 The Cubic Iterative Map 365
10.5 Fixed Points, Cycles, and Chaos 366
10.6 Formal Analysis of Chaotic Dynamics 373
10.7 Coordinate Transformation 373
10.8 The Three Requisites of Chaos 374
10.9 Symbolic Dynamics 375
10.10 Brownian Random Walk 376
10.11 Digression on Order and Disorder 380
10.12 The General Model 381