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Nonlinear Dynamics : A Primer - Alfredo Medio

Nonlinear Dynamics

A Primer

Hardcover Published: 19th November 2001
ISBN: 9780521551861
Number Of Pages: 316

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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

'... 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

Prefacep. xi
Statics and dynamics: some elementary conceptsp. 1
A static problemp. 1
A discrete-time dynamic problemp. 3
A continuous-time dynamic problemp. 7
Flows and mapsp. 11
Exercisesp. 19
Review of linear systemsp. 22
Introductionp. 22
General solutions of linear systems in continuous timep. 25
Continuous systems in the planep. 34
General solutions of linear systems in discrete timep. 43
Discrete systems in the planep. 47
An economic examplep. 52
Phase diagramsp. 56
Exercisesp. 61
Stability of fixed pointsp. 66
Some formal definitions of stabilityp. 66
The linear approximationp. 71
The second or direct method of Lyapunovp. 76
General economic equilibriump. 85
Optimal economic growthp. 92
Manifolds and tangent spacesp. 98
Exercisesp. 99
Invariant and attracting sets, periodic and quasiperiodic orbitsp. 104
Invariant and limit setsp. 105
Stability and attractiveness of general setsp. 110
Attracting sets and attractorsp. 114
Periodic orbits and their stabilityp. 118
Quasiperiodic orbitsp. 126
Conservative and dissipative systemsp. 128
Exercisesp. 130
Local bifurcationsp. 133
Introductionp. 133
Centre manifold theoryp. 134
Local bifurcations for flowsp. 139
Local bifurcations for mapsp. 151
Bifurcations in two-dimensional mapsp. 160
Exercisesp. 161
Chaotic sets and chaotic attractorsp. 163
Basic definitionsp. 164
Symbolic dynamics and the shift mapp. 166
Logistic map with [mu] [gt] 2 + [square root]5p. 171
Smale horseshoep. 173
Tent map and logistic mapp. 176
Doubling mapsp. 178
Chaotic attractorsp. 180
The Lorenz modelp. 182
Exercisesp. 189
Characteristic exponents, fractals, homoclinic orbitsp. 193
Lyapunov characteristic exponentsp. 193
Fractal dimensionp. 198
Horseshoes and homoclinic orbitsp. 204
Exercisesp. 211
Transition to chaosp. 214
Period-doubling route to chaosp. 214
Intermittencyp. 222
Crisesp. 226
Quasiperiodicity and chaosp. 232
Exercisesp. 235
The ergodic approachp. 237
Ergodic measuresp. 238
Lyapunov characteristic exponents revisitedp. 247
Natural, absolutely continuous, SBR measuresp. 249
Attractors as invariant measuresp. 252
Predictability, entropyp. 253
Isomorphismp. 260
Aperiodic and chaotic dynamicsp. 261
Mixingp. 265
Shannon's entropy and Khinchin's axiomsp. 267
Exercisesp. 268
Deterministic systems and stochastic processesp. 270
Bernoulli dynamicsp. 270
Markov shiftsp. 276
[alpha]-congruencep. 277
Further readingp. 282
Bibliographyp. 287
Subject indexp. 295
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9780521551861
ISBN-10: 0521551862
Audience: Tertiary; University or College
Format: Hardcover
Language: English
Number Of Pages: 316
Published: 19th November 2001
Publisher: Cambridge University Press
Country of Publication: GB
Dimensions (cm): 24.7 x 17.4  x 1.9
Weight (kg): 0.7