Attractors, Bifurcations, & Chaos: Nonlinear Phenomena in Economics

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Springer Science & Business Media, Jul 10, 2003 - Mathematics - 549 pages
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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.
 

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Contents

1 Introduction
1
12 Linear Versus Nonlinear Modelling
2
13 Modelling Nonlinearity
4
15 Perturbation Analysis
6
16 Numerical Experiment
7
17 Structural Stability
8
19 Chaos and Fractals
9
110 Layout of the Book and Reading Strategies
10
82 The Original Model
308
83 Nonlinear Investment Functions and Limit Cycles
309
Existence
312
Asymptotic Approximation
315
Transients and Stability
320
87 The TwoRegion Model
325
88 The Persistence of Cycles
326
89 Perturbation Analysis of the Coupled Model
328

Ordinary
13
22 Linear Systems
20
23 Structural Stability
28
24 Limit Cycles
32
25 The Hopf Bifurcation
37
26 The SaddleNode Bifurcation
39
PoincareLindstedt
41
TwoTiming
47
Lyapunovs Method versus Linearisation
53
210 Forced Oscillators Transients and Resonance
56
van der Pol
60
Duffing
69
213 Chaos
76
214 Poincare Sections and Return Maps
79
215 A Short History of Chaos
90
Partial
95
32 Time and Space
96
dAlamberts Solution
97
34 Initial Conditions
99
35 Boundary Conditions
101
Variable Separation
103
37 The General Solution and Fouriers Theorem
106
38 Friction in the Wave Equation
109
39 Nonlinear Waves
111
Gradient and Divergence
114
311 Line Integrals and Gausss Integral Theorem
118
Eigenfunctions
124
313 The Square
127
314 The Circular Disk
132
315 The Sphere
136
316 Nonlinearity Revisited
141
317 Tessellations and the EulerPoincare Index
143
318 Nonlinear Waves on the Square
145
319 Perturbation Methods for Nonlinear Waves
150
4 Iterated Maps or Difference Equations
161
42 The Logistic Map
162
43 The Lyapunov Exponent
171
44 Symbolic Dynamics
174
45 Sharkovskys Theorem and the Schwarzian Derivative
178
46 The Henon Model
180
47 Lyapunov Exponents in 2D
184
48 Fractals and Fractal Dimension
187
49 The Mandelbrot Set
192
410 Can Chaos be Seen?
196
411 The Method of Critical Lines
199
412 Bifurcations and Periodicity
209
5 Bifurcation and Catastrophe
217
51 History of Catastrophe Theory
218
52 Morse Functions and Universal Unfoldings in 1 D
219
53 Morse Functions and Universal Unfoldings in 2 D
223
Fold
228
Cusp
229
Swallowtail and Butterfly
232
Umblics
235
6 Monopoly
239
62 The Model
241
63 Adaptive Search
244
64 Numerical Results
246
65 Fixed Points and Cycles
248
66 Chaos
252
67 The Method of Critical Lines
254
68 Discussion
259
7 Duopoly and Oligopoly
261
72 The Cournot Model
262
73 Stackelberg Equilibria
265
74 The Iterative Process
266
75 Stability of the Cournot Point
269
76 Periodic Points and Chaos
271
77 Adaptive Expectations
275
78 The Neimark Bifurcation
276
79 Critical Lines and Absorbing Area
283
710 Adjustments Including Stackelberg Points
285
711 Oligopoly with Three Firms
287
712 Stackelberg Action Reconsidered
295
713 Back to Duopoly
296
714 True Triopoly
303
Continuous Time
307
810 The Unstable Zero Equilibrium
331
811 Other Fixed Points
333
812 Properties of Fixed Points
337
813 The Arbitrary Phase Angle
338
814 Stability of the Coupled Oscillators
340
815 The Forced Oscillator
342
817 The Small Open Economy
344
819 Catastrophe
346
820 Period Doubling and Chaos
347
821 Relaxation Cycles
351
The Autonomous Case
354
The Forced Case
355
Continuous Space
357
92 Interregional Trade
358
93 The Linear Model
360
94 Coordinate Separation
362
95 The Square Region
364
96 The Circular Region
366
97 The Spherical Region
367
98 The Nonlinear Spatial Model
370
99 Dispersive Waves
372
910 Standing Waves
374
911 Perturbation Analysis
376
Discrete Time
381
102 Investments
382
103 Consumption
384
104 The Cubic Iterative Map
385
105 Fixed Points Cycles and Chaos
386
106 Formal Analysis of Chaotic Dynamics
393
108 The Three Requisites of Chaos
394
109 Symbolic Dynamics
395
1010 Brownian Random Walk
396
1011 Digression on Order and Disorder
400
1012 The General Model
401
1013 Relaxation Cycles
402
1014 Lyapunov Exponents and Fractal Dimensions
405
1015 Numerical Studies of the General Case
408
1016 The Neimark Bifurcation
411
1017 Critical Lines and Absorbing Areas
418
The Model
426
Fixed Points
429
Invariant Spaces
430
1021 Processes in Three Dimensions
437
11 Dynamics of Interregional Trade
443
112 The Basic Model
444
113 Structural Stability
449
114 The Square Flow Grid
451
115 Triangular Hexagonal Grids
454
116 Changes of Structure
457
117 Dynamisation of Beckmanns Model
463
118 Stability
464
119 Uniqueness
467
Increasing Complexity
471
121 The Development Tree
473
122 Continuous Evolution
475
123 Diversification
476
124 Lancasters Property Space
478
126 Bifurcations
479
127 Consumers
481
128 Producers
484
129 Catastrophe
486
1210 Simple Branching in 1 D
487
1211 Branching and Emergence of New Implements in 1 D
489
1212 Catastrophe Cascade in 1 D
492
1213 Catastrophe Cascade in 2 D
494
1214 Fast and Slow Processes
497
1215 Alternative Futures
499
Multiple Attractors
503
131 Population Dynamics
504
132 Diffusion
509
133 Stability
514
134 The Dynamics of Capital and Labour
519
References
529
List of Figures
535
Index
543
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