Chaotic dynamics : theory and applications to economics

This book is a tool for the theoretical and numerical investigation of nonlinear dynamical systems modelled by means of ordinary differential and difference equations. The work is divided into two parts: a book, comprising a theoretical overview of the subject matter and a number of applications; and an integrated software program. Although the emphasis is laid on dynamical systems arising from economic motivation, and the applications are derived from these systems, both the text and the program will also be of use to researchers in other fields of study. The book first discusses the fundamental concepts and methods of chaos theory, and then applies these theoretical results and the facilities provided by the companion software program to models suggested by economic problems.

Preface	p. xiii
Acknowledgements	p. xvii
Part I Theory
1 General introduction: chaos and economics	p. 3
1.1 Introduction	p. 3
1.2 An intuitive definition of chaos	p. 4
1.3 The discovery of chaos	p. 6
1.4 Chaotic dynamics and business cycles	p. 8
1.5 Equilibrium and disequilibrium dynamics	p. 11
1.6 The impact of chaos theory on economics	p. 15
1.7 Chaos theory and statistical inference	p. 18
1.8 A few caveats	p. 22
1.9 The content of the book	p. 24
2 Basic mathematical concepts	p. 27
2.1 Vector fields, flows and maps	p. 27
2.2 Conservative and dissipative systems	p. 32
2.3 Invariant [small alpha]- and [small omega]-limit, attracting sets. Attractors	p. 37
2.4 Basic types of attractors	p. 44
2.5 Stability	p. 47
2.5.1 Linearization	p. 48
2.5.2. Lyapunov direct method	p. 54
2.6 Dynamically equivalent systems. Conjugacy	p. 55
2.7 Bifurcations	p. 59
2.7.1 Hopf bifurcation for flows	p. 63
2.7.2 Hopf bifurcation for maps (or Neimark bifurcation)	p. 67
2.8 Ergodic theory of chaos	p. 70
2.8.1 Probability space. Invariant measure	p. 70
2.8.2 Ergodicity	p. 72
2.8.3 Absolute continuity and observability	p. 73
2.8.4 Some examples of invariant measures	p. 75
2.8.5 Entropy and Lyapunov characteristic exponents	p. 77
3 A user's guide	p. 81
4 Surfaces of sections and Poincare maps	p. 87
4.1 Introduction	p. 87
4.2 'Canonical' Poincare maps	p. 88
4.3 Numerical Poincare maps	p. 94
5 Spectral analysis	p. 101
5.1 The Fourier transform	p. 101
5.2 Power spectral density	p. 103
5.3 Applications of spectral analysis	p. 104
5.4 The estimate of PSD. Aliasing	p. 110
5.5 Estimating PSD with finite samples	p. 111
6 Lyapunov characteristic exponents	p. 115
6.1 The theory of LCEs	p. 115
6.2 Lyapunov characteristic exponents and chaos	p. 121
6.3 Numerical computation of LCEs	p. 123
6.4 Computation of LCEs from experimental data	p. 124
7 Dimensions	p. 127
7.1 Fractal dimension	p. 127
7.2 Numerical computations	p. 133
7.3 Other measures of fractal dimension	p. 135
8 Symbolic dynamics	p. 139
8.1 The space of symbol sequences. The shift map	p. 139
8.2 The Smale horseshoe and the geometry of chaos	p. 143
9 Transition to chaos. Theoretical predictive criteria	p. 149
9.1 Introduction	p. 149
9.2 Homoclinic orbits and horseshoes	p. 150
9.3 Codimension one routes to chaos: one-dimensional maps	p. 153
9.4 Period-doubling	p. 158
9.5 Intermittency	p. 165
9.6 Saddle connection, or 'blue sky catastrophe'	p. 169
9.7 Quasi-periodic route to chaos	p. 173
10 Analysis of experimental signals - some theoretical problems	p. 179
10.1 The theoretical problem	p. 179
10.2 The reconstruction of the attractor - some difficulties	p. 181
10.3 Singular spectrum analysis	p. 183
10.4 Filtering away noise	p. 186
Part II Applications to economics
11 Discrete and continuous chaos	p. 193
11.1 Introduction	p. 193
11.2 Probability distributions of lags	p. 195
11.3 A feedback representation of lags	p. 197
11.4 Symptomatology of chaos	p. 201
11.5 A reconstructed 1D map	p. 210
11.6 Conclusions	p. 213
12 Cycles and chaos in overlapping-generations models with production	p. 215
12.1 The basic model	p. 215
12.2 Constant relative risk aversion: forward dynamics	p. 218
12.3 Constant absolute risk aversion: backward dynamics	p. 221
13 Chaos in a continuous-time model of inventory business cycles	p. 241
13.1 The model	p. 241
13.2 Lag structure and complex behaviour	p. 244
13.3 Parameter analysis and numerical simulations	p. 247
13.4 A few complications and conclusions	p. 253
14 Analysis of experimental signals - applications	p. 259
14.1 Detecting deterministic chaos	p. 259
14.2 Applications of SSA. Noiseless data	p. 264
14.3 Reconstruction of attractors with noisy data	p. 268
14.4 Application of the SSA analysis to financial data	p. 273
14.4.1 Differentiated series	p. 273
14.4.2 Detrended series	p. 278
14.5 Conclusions	p. 279
Part III Software
15 DMC Manual	p. 285
15.1 Installation	p. 285
15.1.1 MS-DOS	p. 285
15.1.2 Macintosh	p. 287
15.1.3 What to do if DMC does not work	p. 288
15.2 Graphical user interface	p. 288
15.3 The editor	p. 293
15.4 Functions	p. 295
15.4.1 MODEL	p. 295
15.4.2 EVAL	p. 298
15.4.3 PLOT	p. 303
15.4.4 STAT	p. 307
15.4.5 FILES	p. 313
15.4.6 UTIL	p. 315
15.4.7 OPTS	p. 317
15.4.8 QUIT	p. 318
15.4.9 Internal menu commands	p. 318
15.5 DMC internal compiler	p. 321
References	p. 329
Index	p. 339

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