Attractors, Bifurcations, & Chaos

Nonlinear Phenomena in Economics

By: TÃ¶nu Puu

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Hardcover | 10 July 2003 | Edition Number 2

At a Glance

Hardcover
566 Pages

Edition Type
Revised

Dimensions(cm)
23.5 x 16.51 x 3.81

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Attractors, Bifurcations, & Chaos - now in its second edition - begins with an introduction to mathematical methods in modern nonlinear dynamics and deals with differential equations. Phenomena such as bifurcations and deterministic chaos are given considerable emphasis, both in the methodological part, and in the second part, containing various applications in economics and in regional science. Coexistence of attractors and the multiplicity of development paths in nonlinear systems are central topics. The applications focus on issues such as business cycles, oligopoly, interregional trade dynamics, and economic development theory.

Industry Reviews

From the reviews of the second edition:

"Its unifying theme is the analysis of economic dynamics when we drop the assumptions of linearity and use the results of recently developing nonlinear dynamic science. ... the book is carefully organized and written in a friendly style. It is interesting and smooth reading. ... It is useful to graduate students, researchers and academics ... . The book is written in a readable fashion, all details are explained clearly. Therefore, the book is a must for anyone interested in nonlinear economic dynamics." (Akio Matsumoto, Journal of Economic Behavior and Organization, Vol. 58, 2005)

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Non-Fiction Mathematics Applied Mathematics Chaos Theory Optimisation Game Theory Society & Social Sciences Economics Economic Theory & Philosophy

Introduction	p. 1
Dynamics Versus Equilibrium Analysis	p. 1
Linear Versus Nonlinear Modelling	p. 2
Modelling Nonlinearity	p. 4
Some Philosophy of Modelling	p. 4
Perturbation Analysis	p. 6
Numerical Experiment	p. 7
Structural Stability	p. 8
The Critical Line Method	p. 8
Chaos and Fractals	p. 9
Layout of the Book and Reading Strategies	p. 10
Differential Equations: Ordinary	p. 13
The Phase Portrait	p. 13
Linear Systems	p. 20
Structural Stability	p. 28
Limit Cycles	p. 32
The Hopf Bifurcation	p. 37
The Saddle-Node Bifurcation	p. 39
Perturbation Methods: Poincaré-Lindstedt	p. 41
Perturbation Methods: Two-Timing	p. 47
Stability: Lyapunov's Direct Method versus Linearization	p. 53
Forced Oscillators, Transients and Resonance	p. 56
Forced Oscillators: van der Pol	p. 60
Forced Oscillators: Duffing	p. 69
Chaos	p. 76
Poincaré Sections and Return Maps	p. 79
A Short History of Chaos	p. 90
Differential Equations: Partial	p. 95
Vibrations and Waves	p. 95
Time and Space	p. 96
Travelling Waves in ID: d'Alambert's Solution	p. 97
Initial Conditions	p. 99
Boundary Conditions	p. 101
Standing Waves: Variable Separation	p. 103
The General Solution and Fourier's Theorem	p. 106
Friction in the Wave Equation	p. 109
Nonlinear Waves	p. 111
Vector Fields in 2D: Gradient and Divergence	p. 114
Line Integrals and Gauss's Integral Theorem	p. 118
Wave Equation in Two Dimensions: Eigenfunctions	p. 124
The Square	p. 127
The Circular Disk	p. 132
The Sphere	p. 136
Nonlinearity Revisited	p. 141
Tessellations and the Euler-Poincaré Index	p. 143
Nonlinear Waves on the Square	p. 145
Perturbation Methods for Nonlinear Waves	p. 150
Iterated Maps or Difference Equations	p. 161
Introduction	p. 161
The Logistic Map	p. 162
The Lyapunov Exponent	p. 171
Symbolic Dynamics	p. 174
Sharkovsky's Theorem and the Schwarzian Derivative	p. 178
The Hénon Model	p. 180
Lyapunov Exponents in 2D	p. 184
Fractals and Fractal Dimension	p. 187
The Mandelbrot Set	p. 192
Can Chaos be Seen?	p. 196
The Method of Critical Lines	p. 199
Bifurcations and Periodicity	p. 209
Bifurcation and Catastrophe	p. 217
History of Catastrophe Theory	p. 218
Morse Functions and Universal Unfoldings in 1 D	p. 219
Morse Functions and Universal Unfoldings in 2 D	p. 223
The Elementary Catastrophes: Fold	p. 228
The Elementary Catastrophes: Cusp	p. 229
The Elementary Catastrophes: Swallowtail and Butterfly	p. 232
The Elementary Catastrophes: Umblics	p. 235
Monopoly	p. 239
Introduction	p. 239
The Model	p. 241
Adaptive Search	p. 244
Numerical Results	p. 246
Fixed Points and Cycles	p. 248
Chaos	p. 252
The Method of Critical Lines	p. 254
Discussion	p. 259
Duopoly and Oligopoly	p. 261
Introduction	p. 261
The Cournot Model	p. 262
Stackelberg Equilibria	p. 265
The Iterative Process	p. 266
Stability of the Cournot Point	p. 269
Periodic Points and Chaos	p. 271
Adaptive Expectations	p. 275
The Neimark Bifurcation	p. 276
Critical Lines and Absorbing Area	p. 283
Adjustments Including Stackelberg Points	p. 285
Oligopoly with Three Firms	p. 287
Stackelberg Action Reconsidered	p. 295
Back to ""Duopoly""	p. 296
True Triopoly	p. 303
Business Cycles: Continuous Time	p. 307
The Multiplier-Accelerator Model	p. 307
The Original Model	p. 308
Nonlinear Investment Functions and Limit Cycles	p. 309
Limit Cycles: Existence	p. 312
Limit Cycles: Asymptotic Approximation	p. 315
Limit Cycles: Transients and Stability	p. 320
The Two-Region Model	p. 325
The Persistence of Cycles	p. 326
Perturbation Analysis of the Coupled Model	p. 328
The Unstable Zero Equilibrium	p. 331
Other Fixed Points	p. 333
Properties of Fixed Points	p. 337
The Arbitrary Phase Angle	p. 338
Stability of the Coupled Oscillators	p. 340
The Forced Oscillator	p. 342
The World Market	p. 342
The Small Open Economy	p. 344
Stability of the Forced Oscillator	p. 344
Catastrophe	p. 346
Period Doubling and Chaos	p. 347
Relaxation Cycles	p. 351
Relaxation: The Autonomous Case	p. 354
Relaxation: The Forced Case	p. 355
Business Cycles: Continuous Space	p. 357
Introduction	p. 357
Interregional Trade	p. 358
The Linear Model	p. 360
Coordinate Separation	p. 362
The Square Region	p. 364
The Circular Region	p. 366
The Spherical Region	p. 367
The Nonlinear Spatial Model	p. 370
Dispersive Waves	p. 372
Standing Waves	p. 374
Perturbation Analysis	p. 376
Business Cycles: Discrete Time	p. 381
Introduction	p. 381
Investments	p. 382
Consumption	p. 384
The Cubic Iterative Map	p. 385
Fixed Points, Cycles, and Chaos	p. 386
Formal Analysis of Chaotic Dynamics	p. 393
Coordinate Transformation	p. 393
The Three Requisites of Chaos	p. 394
Symbolic Dynamics	p. 395
Brownian Random Walk	p. 396
Digression on Order and Disorder	p. 400
The General Model	p. 401
Relaxation Cycles	p. 402
Lyapunov Exponents and Fractal Dimensions	p. 405
Numerical Studies of the General Case	p. 408
The Neimark Bifurcation	p. 411
Critical Lines and Absorbing Areas	p. 418
Two Regions: The Model	p. 426
Two Regions: Fixed Points	p. 429
Two Regions: Invariant Spaces	p. 430
Processes in Three Dimensions	p. 437
Dynamics of Interregional Trade	p. 443
Interregional Trade Models	p. 443
The Basic Model	p. 444
Structural Stability	p. 449
The Square Flow Grid	p. 451
Triangular/Hexagonal Grids	p. 454
Changes of Structure	p. 457
Dynamisation of Beckmann's Model	p. 463
Stability	p. 464
Uniqueness	p. 467
Development: Increasing Complexity	p. 471
The Development Tree	p. 473
Continuous Evolution	p. 475
Diversification	p. 476
Lancaster's Property Space	p. 478
Branching Points	p. 478
Bifurcations	p. 479
Consumers	p. 481
Producers	p. 484
Catastrophe	p. 486
Simple Branching in 1 D	p. 487
Branching and Emergence of New Implements in 1 D	p. 489
Catastrophe Cascade in 1 D	p. 492
Catastrophe Cascade in 2 D	p. 494
Fast and Slow Processes	p. 497
Alternative Futures	p. 499
Development: Multiple Attractors	p. 503
Population Dynamics	p. 504
Diffusion	p. 509
Stability	p. 514
The Dynamics of Capital and Labour	p. 519
References	p. 529
List of Figures	p. 535
Index	p. 543
Table of Contents provided by Publisher. All Rights Reserved.

Attractors, Bifurcations, & Chaos

Nonlinear Phenomena in Economics

At a Glance

Hardcover

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