Table Of ContentAttractors, Bifurcations, & Chaos
Springer-Verlag Berlin Heidelberg GmbH
Tonu Puu
Attractors,
Bifurcations,
& Chaos
Nonlinear Phenomena
in Economics
Second Edition
with 209 Figures
i
Springer
Professor Dr. Tonu Puu
Umeă University
Centre for Regional Science
90187 Umeă, Sweden
ISBN 978-3-642-07296-3 ISBN 978-3-540-24699-2 (eBook)
DOI 10.1007/978-3-540-24699-2
Cataloging-in-Publication Data applied for
A catalog record for this book is available from the Library of Congress.
Bibliographic information published by Die Deutsche Bibliothek
Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie;
detailed bibliographic data available in the internet at http.!ldnb.ddb.de
This work is subject to copyright. AII rights are reserved, whether the whole or part of the material
is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,
broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication
of this publication or parts thereof is permitted only under the provisions of the German Copyright
Law of September 9,1965, in its current version, and permission for use must always be obtained
from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law.
http://www.springer.de
O Springer-Verlag Berlin Heidelberg 2003
Originally published by Springer-Verlag Berlin Heidelberg New Ymk in 2003
Softcover reprint of tbe bardcover 2nd edition 2003
The use ot general descriptive names, registered names, trademarks, etc. in this publication does
not imply, even in the absence of a specific statement, tbat such names are exempt from the
relevant protective laws and regulations and therefore free for general use.
Cover design: Erich Kirchner, Heidelberg
Preface to the Second Edition
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, and in a Russian translation as "Nelineynaia
Economicheskaia Dinamica".
The first three editions were focused on applications. The last was differ
ent, as it also included some chapters with mathematical background mate
rial - ordinary differential equations and iterated maps - so as to make the
book self-contained and suitable as a textbook for economics students of
dynamical systems. To the same pedagogical purpose, the number of illus
trations were expanded.
The book published in 2000, with the title "A ttractors, Bifurcations, and
Chaos -Nonlinear Phenomena in Economics", was so much changed, that
the author felt it reasonable to give it a new title. There were two new math
ematics chapters -on partial differential equations, and on bifurcations and
catastrophe theory -thus making the mathematical background material fairly
complete.
The author is happy that this new book did rather well, but he preferred to
rewrite it, rather than having just a new print run. Material, stemming from
the first versions, was more than ten years old, while nonlinear dynamics has
been a fast developing field, so some analyses looked rather old-fashioned
and pedestrian. The necessary revision turned out to be rather substantial.
The new edition is somewhat extended, but the total change is more exten
sive than this indicates, as a substantial amount of text was removed.
The author is indebted to several people, without whom the new edition
would not have been possible, in particular Professor Laura Gardini, Univer
sity of Urbino, and Dr. Irina Sushko, National Academy of Sciences of
Ukraine, Kiev.
VI Preface
There have been innumerable working sessions in Professor Gardini's
delightful country house, overlooking the hills ofUrbino, which once inspired
Raffaello. Though these sessions were devoted to specific joint projects, the
author managed to pick up some mathematics from Professor Gardini and
Dr. Sushko, which was indispensable for the present work. It goes without
saying that they are not responsible for any misunderstandings that the author,
as a non-mathematic an, might have gathered.
Deep gratitude is also due to an old friend, Professor Martin J. Beckmann,
Brown University, whose never ending generous support and appreciation of
the author's work, even after it took another route than the earlier joint work
on phenomena in the continuous space, has been an important motive force
for the author's continued work.
The spatial models, of which there are still reminiscences in several chap
ters of this book, originated in Professor Beckmann's seminal work from the
early 1950's, and provided the very basis for a fruitful collaboration over
several decades. This work also was the origin of the present author's inter
est in nonlinearity and dynamical systems in general.
Urbino in March 2003
Tonu Puu
Contents
1 Introduction 1
1.1 Dynamics Versus Equilibrium Analysis 1
1.2 Linear Versus Nonlinear Modelling 2
1.3 Modelling Nonlinearity 4
1.4 Some Philosophy of Modelling 4
1.5 Perturbation Analysis 6
1.6 Numerical Experiment 7
1.7 Structural Stability 8
1.8 The Critical Line Method 8
1.9 Chaos and Fractals 9
1.10 Layout of the Book and Reading Strategies 10
2 Differential Equations: Ordinary 13
2.1 The Phase Portrait 13
2.2 Linear Systems 20
2.3 Structural Stability 28
2.4 Limit Cycles 32
2.5 The Hopf Bifurcation 37
2.6 The Saddle-Node Bifurcation 39
2.7 Perturbation Methods: Poincare-Lindstedt 41
2.8 Perturbation Methods: Two-Timing 47
2.9 Stability: Lyapunov's Direct Method versus Linearization 53
2.10 Forced Oscillators, Transients and Resonance 56
2.11 Forced Oscillators: van der Pol 60
2.12 Forced Oscillators: Duffing 69
2.13 Chaos 76
2.14 Poincare Sections and Return Maps 79
2.15 A Short History of Chaos 90
vm Contents
3 Differential Equations: Partial 95
3.1 Vibrations and Waves 95
3.2 Time and Space 96
3.3 Travelling Waves in ID: d'Alambert's Solution 97
3.4 Initial Conditions 99
3.5 Boundary Conditions 101
3.6 Standing Waves: Variable Separation 103
3.7 The General Solution and Fourier's Theorem 106
3.8 Friction in the Wave Equation 109
3.9 Nonlinear Waves 111
3.10 Vector Fields in 2D: Gradient and Divergence 114
3.11 Line Integrals and Gauss's Integral Theorem 118
3.12 Wave Equation in Two Dimensions: Eigenfunctions 124
3.13 The Square 127
3.14 The Circular Disk 132
3.15 The Sphere 136
3.16 Nonlinearity Revisited 141
3.17 Tessellations and the Euler-Poincare Index 143
3.18 Nonlinear Waves on the Square 145
3.19 Perturbation Methods for Nonlinear Waves 150
4 Iterated Maps or Difference Equations 161
4.1 Introduction 161
4.2 The Logistic Map 162
4.3 The Lyapunov Exponent 171
4.4 Symbolic Dynamics 174
4.5 Sharkovsky's Theorem and the Schwarzian Derivative 178
4.6 The Henon Model 180
4.7 Lyapunov Exponents in 2D 184
4.8 Fractals and Fractal Dimension 187
4.9 The Mandelbrot Set 192
4.10 Can Chaos be Seen? 196
4.11 The Method of Critical Lines 199
4.12 Bifurcations and Periodicity 209
5 Bifurcation and Catastrophe 217
5.1 History of Catastrophe Theory 218
5.2 Morse Functions and Universal Unfoldings in 1 D 219
5.3 Morse Functions and Universal Unfoldings in 2 D 223
5.4 The Elementary Catastrophes: Fold 228
Contents IX
5.5 The Elementary Catastrophes: Cusp 229
5.6 The Elementary Catastrophes: Swallowtail and Butterfly 232
5.7 The Elementary Catastrophes: Umblics 235
6 Monopoly 239
6.1 Introduction 239
6.2 The Model 241
6.3 Adaptive Search 244
6.4 Numerical Results 246
6.5 Fixed Points and Cycles 248
6.6 Chaos 252
6.7 The Method of Critical Lines 254
6.8 Discussion 259
7 Duopoly and Oligopoly 261
7.1 Introduction 261
7.2 The Coumot Model 262
7.3 Stackelberg Equilibria 265
7.4 The Iterative Process 266
7.5 Stability of the Coumot Point 269
7.6 Periodic Points and Chaos 271
7.7 Adaptive Expectations 275
7.8 The Neimark Bifurcation 276
7.9 Critical Lines and Absorbing Area 283
7.10 Adjustments Including Stackelberg Points 285
7.11 Oligopoly with Three Firms 287
7.12 Stackelberg Action Reconsidered 295
7.13 Back to "Duopoly" 296
7.14 True Triopoly 303
8 Business Cycles: Continuous Time 307
8.1 The Multiplier-Accelerator Model 307
8.2 The Original Model 308
8.3 Nonlinear Investment Functions and Limit Cycles 309
8.4 Limit Cycles: Existence 312
8.5 Limit Cycles: Asymptotic Approximation 315
8.6 Limit Cycles: Transients and Stability 320
8.7 The Two-Region Model 325
8.8 The Persistence of Cycles 326
8.9 Perturbation Analysis of the Coupled Model 328
X Contents
8.10 The Unstable Zero Equilibrium 331
8.11 Other Fixed Points 333
8.12 Properties of Fixed Points 337
8.13 The Arbitrary Phase Angle 338
8.14 Stability of the Coupled Oscillators 340
8.15 The Forced Oscillator 342
8.16 The World Market 342
8.17 The Small Open Economy 344
8.18 Stability of the Forced Oscillator 344
8.19 Catastrophe 346
8.20 Period Doubling and Chaos 347
8.21 Relaxation Cycles 351
8.22 Relaxation: The Autonomous Case 354
8.23 Relaxation: The Forced Case 355
9 Business Cycles: Continuous Space 357
9.1 Introduction 357
9.2 Interregional Trade 358
9.3 The Linear Model 360
9.4 Coordinate Separation 362
9.5 The Square Region 364
9.6 The Circular Region 366
9.7 The Spherical Region 367
9.8 The Nonlinear Spatial Model 370
9.9 Dispersive Waves 372
9.10 Standing Waves 374
9.11 Perturbation Analysis 376
10 Business Cycles: Discrete Time 381
10.1 Introduction 381
10.2 Investments 382
10.3 Consumption 384
10.4 The Cubic Iterative Map 385
10.5 Fixed Points, Cycles, and Chaos 386
10.6 Formal Analysis of Chaotic Dynamics 393
10.7 Coordinate Transformation 393
10.8 The Three Requisites of Chaos 394
10.9 Symbolic Dynamics 395
Contents XI
10.10 Brownian Random Walk 396
10.11 Digression on Order and Disorder 400
10.12 The General Model 401
10.13 Relaxation Cycles 402
10.14 Lyapunov Exponents and Fractal Dimensions 405
10.15 Numerical Studies of the General Case 408
10.16 The Neimark Bifurcation 411
10.17 Critical Lines and Absorbing Areas 418
10.18 Two Regions: The Model 426
10.19 Two Regions: Fixed Points 429
10.20 Two Regions: Invariant Spaces 430
1021 Processes in Three Dimensions 437
11 Dynamics of Interregional Trade 443
11.1 Interregional Trade Models 443
11.2 The Basic Model 444
11.3 Structural Stability 449
11.4 The Square Flow Grid 451
11.5 Triangular/Hexagonal Grids 454
11.6 Changes of Structure 457
11.7 Dynamisation of Beckmann's Model 463
11.8 Stability 464
11.9 Uniqueness 467
12 Development: Increasing Complexity 471
12.1 The Development Tree 473
12.2 Continuous Evolution 475
12.3 Diversification 476
12.4 Lancaster's Property Space 478
12.5 Branching Points 478
12.6 Bifurcations 479
12.7 Consumers 481
12.8 Producers 484
12.9 Catastrophe 486
12.10 Simple Branching in 1 D 487
12.11 Branching and Emergence of New Implements in 1 D 489
12.12 Catastrophe Cascade in 1 D 492
12.13 Catastrophe Cascade in 2 D 494
1214 Fast and Slow Processes 497
12.15 Alternative Futures 499
