ISBN: 3540402268
TITLE: Attractors, Bifurcations, & Chaos
AUTHOR: Puu
TOC:

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: Poincar-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 Poincar Sections and Return Maps 79
2.15 A Short History of Chaos 90
3 Differential Equations: Partial 95
3.1 Vibrations and Waves 95
3.2 Time and Space 96
3.3 Travelling Waves in 1D: 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-Poincar 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
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 Cournot Model 262
7.3 Stackelberg Equilibria 265
7.4 The Iterative Process 266
7.5 Stability of the Cournot 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: Coutinuous 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
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
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
10.21 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
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
13 Development: Multiple Attractors 503
13.1 Population Dynamics 504
13.2 Diffusion 509
13.3 Stability 514
13.4 The Dynamics of Capital and Labour 519
References 529
List of Figures 535
Index 543
END
