Nonlinear Dynamics

A Primer

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Hardcover | 11 October 2001

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316 Pages

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A systematic and comprehensive introduction to the study of nonlinear dynamical systems, in both discrete and continuous time, for nonmathematical students and researchers working in applied fields. An understanding of linear systems and the classical theory of stability are essential although basic reviews of the relevant material are provided. Further chapters are devoted to the stability of invariant sets, bifurcation theory, chaotic dynamics and the transition to chaos. In the final two chapters the authors approach the subject from a measure-theoretical point of view and compare results to those given for the geometrical or topological approach of the first eight chapters. Includes about one hundred exercises. A Windows-compatible software programme called DMC, provided free of charge through a website dedicated to the book, allows readers to perform numerical and graphical analysis of dynamical systems. Also available on the website are computer exercises and solutions to selected book exercises. See www.cambridge.org/economics/resources

Industry Reviews

'... the book is well written and contains lots of useful information. It will serve as a good reference on dynamical systems both for students and researchers.' Zentralblatt fur Mathematik und ihre Grenzgebiete Mathematics Abstracts

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Non-Fiction Mathematics Calculus & Mathematical Analysis Differential Calculus & Equations Economics Economic History Economic Theory & Philosophy Applied Mathematics Chaos Theory Nonlinear Science Econometrics

Preface	p. xi
Statics and dynamics: some elementary concepts	p. 1
A static problem	p. 1
A discrete-time dynamic problem	p. 3
A continuous-time dynamic problem	p. 7
Flows and maps	p. 11
Exercises	p. 19
Review of linear systems	p. 22
Introduction	p. 22
General solutions of linear systems in continuous time	p. 25
Continuous systems in the plane	p. 34
General solutions of linear systems in discrete time	p. 43
Discrete systems in the plane	p. 47
An economic example	p. 52
Phase diagrams	p. 56
Exercises	p. 61
Stability of fixed points	p. 66
Some formal definitions of stability	p. 66
The linear approximation	p. 71
The second or direct method of Lyapunov	p. 76
General economic equilibrium	p. 85
Optimal economic growth	p. 92
Manifolds and tangent spaces	p. 98
Exercises	p. 99
Invariant and attracting sets, periodic and quasiperiodic orbits	p. 104
Invariant and limit sets	p. 105
Stability and attractiveness of general sets	p. 110
Attracting sets and attractors	p. 114
Periodic orbits and their stability	p. 118
Quasiperiodic orbits	p. 126
Conservative and dissipative systems	p. 128
Exercises	p. 130
Local bifurcations	p. 133
Introduction	p. 133
Centre manifold theory	p. 134
Local bifurcations for flows	p. 139
Local bifurcations for maps	p. 151
Bifurcations in two-dimensional maps	p. 160
Exercises	p. 161
Chaotic sets and chaotic attractors	p. 163
Basic definitions	p. 164
Symbolic dynamics and the shift map	p. 166
Logistic map with [mu] [gt] 2 + [square root]5	p. 171
Smale horseshoe	p. 173
Tent map and logistic map	p. 176
Doubling maps	p. 178
Chaotic attractors	p. 180
The Lorenz model	p. 182
Exercises	p. 189
Characteristic exponents, fractals, homoclinic orbits	p. 193
Lyapunov characteristic exponents	p. 193
Fractal dimension	p. 198
Horseshoes and homoclinic orbits	p. 204
Exercises	p. 211
Transition to chaos	p. 214
Period-doubling route to chaos	p. 214
Intermittency	p. 222
Crises	p. 226
Quasiperiodicity and chaos	p. 232
Exercises	p. 235
The ergodic approach	p. 237
Ergodic measures	p. 238
Lyapunov characteristic exponents revisited	p. 247
Natural, absolutely continuous, SBR measures	p. 249
Attractors as invariant measures	p. 252
Predictability, entropy	p. 253
Isomorphism	p. 260
Aperiodic and chaotic dynamics	p. 261
Mixing	p. 265
Shannon's entropy and Khinchin's axioms	p. 267
Exercises	p. 268
Deterministic systems and stochastic processes	p. 270
Bernoulli dynamics	p. 270
Markov shifts	p. 276
[alpha]-congruence	p. 277
Further reading	p. 282
Bibliography	p. 287
Subject index	p. 295
Table of Contents provided by Ingram. All Rights Reserved.

Nonlinear Dynamics

A Primer

At a Glance

Hardcover

Industry Reviews

Shipping

How to return your order

Defective items

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