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Dynamical Systems and Chaos

By: Floris Takens, Henk Broer

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Hardcover | 28 October 2010

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Hardcover
332 Pages

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24.4 x 16.4 x 2.4

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Over the last four decades there has been extensive development in the theory of dynamical systems. This book aims at a wide audience where the first four chapters have been used for an undergraduate course in Dynamical Systems. Material from the last two chapters and from the appendices has been used quite a lot for master and PhD courses. All chapters are concluded by an exercise section. The book is also directed towards researchers, where one of the challenges is to help applied researchers acquire background for a better understanding of the data that computer simulation or experiment may provide them with the development of the theory.

Industry Reviews

From the reviews:

"Today numerous books dealing with either dynamical systems and/or chaos but this one stands out in many ways. Its scope, depth and breath give it a feeling of a must read. ... The exercises per chapter run from simple and straightforward to extended research questions forming time-consuming open challenges for the interested reader. All in all a great text written in a very good style for which the authors deserve to be complemented." (Henk Nijmeijer, Nieuw Archief voor Wiskunde, Vol. 14 (4), December, 2013)

"This work ... takes a different tack compared to others. ... geared for upper-level undergraduate and graduate students interested in this area of mathematics ... . Anyone interested in the field of dynamical systems might like to have this book because of its organization and the ease of finding the often-used examples. Summing Up: Recommended. Upper-division undergraduates through professionals." (M. D. Sanford, Choice, Vol. 48 (9), May, 2011)

"This book gives a clear and accessible exposition of some of the central concepts addressed by the classical theory of dynamical systems. ... The book is very good at bringing out the essence of each concept without unnecessary technical clutter. ... this is an excellent book, conveying deep understanding of the subject. It should definitely find its way onto the shelves of the experts, but I also very much recommend it as an excellent supplementary text to a differential equations or dynamical systems course." (Gregor KovaÄiÄ, SIAM Review, Vol. 53 (4), 2011)

"This is a skillfully written guide to the fundamentals of the theory of dynamical systems and chaos aimed at a wide audience. ... the book appeals to a wide audience. It can be successfully used by students, researchers and readers with a good background in differential equations, analysis and topology looking for a concise but informative modern reference on dynamical systems and chaos." (Svitlana P. Rogovchenko, Zentralblatt MATH, Vol. 1218, 2011)

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Non-Fiction Mathematics Applied Mathematics Chaos Theory Calculus & Mathematical Analysis Differential Calculus & Equations Nonlinear Science Reference, Information & Interdisciplinary Subjects Research & Information Information theory Cybernetics & Systems Theory

Examples and definitions of dynamical phenomena	p. 1
The pendulum as a dynamical system	p. 2
The free pendulum	p. 2
The free undamped pendulum	p. 3
The free damped pendulum	p. 7
The forced pendulum	p. 9
Summary and outlook	p. 13
General definition of dynamical systems	p. 14
Differential equations	p. 15
Constructions of dynamical systems	p. 19
Restriction	p. 19
Discretisation	p. 20
Suspension and poincaré map	p. 21
Further examples of dynamical systems	p. 25
A Hopf bifurcation in the Van der Pol equation	p. 25
The Van der Pol equation	p. 25
Hopf bifurcation	p. 26
The Hénon map: Saddle points and separatrices	p. 27
The logistic system: Bifurcation diagrams	p. 31
The Newton algorithm	p. 37
R U {∞} as a circle: Stereographic projection	p. 39
Applicability of the Newton algorithm	p. 40
Nonconvergent Newton algorithm	p. 40
Newton algorithm in higher dimensions	p. 42
Dynamical systems defined by partial differential equations	p. 44
The 1-dimensional wave equation	p. 44
Solution of the 1-dimensional wave equation	p. 45
The 1-dimensional heat equation	p. 46
The Lorenz attractor	p. 48
The Lorenz system; the Lorenz attractor	p. 48
Sensitive dependence on initial state	p. 50
The Rössler attractor; Poincaré map	p. 50
The Rössler system	p. 51
The attractor of the Poincaré map	p. 52
The doubling map and chaos	p. 53
The doubling map on the interval	p. 53
The doubling map on the circle	p. 54
The doubling map in symbolic dynamics	p. 54
Analysis of the doubling map in symbolic form	p. 56
General shifts	p. 59
Exercises	p. 61
Qualitative properties and predictability of evolutions	p. 67
Stationary and periodic evolutions	p. 67
Predictability of periodic and stationary motions	p. 68
Asymptotically and eventually periodic evolutions	p. 70
Multi- and quasi-periodic evolutions	p. 71
The n-dimensional torus	p. 73
Translations on a torus	p. 75
Translation systems on the 1 -dimensional torus	p. 75
Translation systems on the 2-dimensional torus with time set R	p. 77
Translation systems on the n-dimensional torus with time set R	p. 78
Translation systems on the n-dimensional torus with time set Z or Z+	p. 80
General definition of multi- and quasi-periodic evolutions	p. 81
Multi- and quasi-periodic subsystems	p. 82
Example: The driven Van der Pol equation	p. 83
The prediction principle l'histoire se répète	p. 85
The general principle	p. 86
Application to quasi-periodic evolutions	p. 87
Historical remarks	p. 89
Chaotic evolutions	p. 91
Badly predictable (chaotic) evolutions of the doubling map	p. 91
Definition of dispersion exponent and chaos	p. 94
Properties of the dispersion exponent	p. 97
'Transition' from quasi-periodic to stochastic	p. 98
'Transition' from periodic to chaotic	p. 99
'Transition' from chaotic to stochastic	p. 99
Chaotic evolutions in the examples of Chapter 1	p. 99
Chaotic evolutions of the Thom map	p. 101
Exercises	p. 105
Persistence of dynamical properties	p. 109
Variation of initial state	p. 109
Variation of parameters	p. 112
Persistence of stationary and periodic evolutions	p. 115
Persistence of stationary evolutions	p. 115
Persistence of periodic evolutions	p. 118
Persistence for the doubling map	p. 119
Perturbations of the doubling map: Persistent chaoticity	p. 119
Structural stability	p. 121
The doubling map modelling a (fair) coin	p. 125
Exercises	p. 128
Global structure of dynamical systems	p. 133
Definitions	p. 133
Examples of attractors	p. 136
The doubling map and hyperbolic attractors	p. 137
The doubling map on the plane	p. 137
The doubling map in 3-space: The solenoid	p. 139
Digression on hyperbolicity	p. 143
The solenoid as a hyperbolic attractor	p. 145
Properties of hyperbolic attractors	p. 146
Nonhyperbolic attractors	p. 149
Hénon-like attractors	p. 149
The Lorenz attractor	p. 150
Chaotic systems	p. 155
Basin boundaries and the horseshoe map	p. 156
Gradient systems	p. 156
The horseshoe map	p. 157
Symbolic dynamics	p. 161
Structural stability	p. 162
Horseshoelike sets in basin boundaries	p. 164
Exercises	p. 166
On KAM theory	p. 173
Introduction, setting of the problem	p. 173
KAM theory of circle maps	p. 175
Preliminaries	p. 175
Formal considerations and small divisors	p. 178
Resonance tongues	p. 181
KAM theory of area-preserving maps	p. 183
KAM theory of holomorphic maps	p. 185
Complex linearisation	p. 186
Cremer's example in Herman's version	p. 187
The linear small divisor problem	p. 188
Motivation	p. 188
Setting of the problem and formal solution	p. 190
Convergence	p. 193
Exercises	p. 197
Reconstruction and time series analysis	p. 205
Introduction	p. 205
An experimental example: The dripping faucet	p. 206
The reconstruction theorem	p. 207
Generalisations	p. 209
Continuous time	p. 209
Multidimensional measurements	p. 209
Endomorphisms	p. 210
Compactness	p. 210
Historical note	p. 211
Reconstruction and detecting determinism	p. 211
Box-counting dimension and its numerical estimation	p. 213
Numerical estimation of the box-counting dimension	p. 215
Box-counting dimension as an indication for 'thin' subsets	p. 216
Estimation of topological entropy	p. 216
Stationarity and reconstruction measures	p. 217
Probability measures defined by relative frequencies	p. 218
Definition of stationarity and reconstruction measures	p. 218
Examples of nonexistence of reconstruction measures	p. 219
Correlation dimensions and entropies	p. 220
Definitions	p. 220
Miscellaneous remarks	p. 222
Compatibility of the definitions of dimension and entropy with reconstruction	p. 222
Generalised correlation integrals, dimensions, and entropies	p. 223
Numerical estimation of correlation integrals, dimensions, entropies	p. 223
Classical time series analysis, correlation integrals, and predictability	p. 227
Classical time series analysis	p. 227
Optimal linear predictors	p. 228
Gaussian time series	p. 229
Determinism and Autocovariances	p. 229
Predictability and correlation integrals	p. 232
L'histoire se répète	p. 232
Local linear predictors	p. 234
Miscellaneous subjects	p. 235
Lyapunov exponents	p. 235
Estimation of Lyapunov exponents from a time series	p. 237
The Kantz-Diks test: Discriminating between time series and testing for reversibility	p. 237
Exercises	p. 239
Differential topology and measure theory	p. 243
Topology	p. 243
Differentiable manifolds	p. 246
Measure theory	p. 252
Miscellaneous KAM theory	p. 257
Introduction	p. 257
Classical (conservative) KAM theory	p. 258
Dissipative KAM theory	p. 260
On the KAM proof in the dissipative case	p. 261
Reformulation and some notation	p. 262
On the Newtonian iteration	p. 263
Historical remarks	p. 265
Exercises	p. 266
Miscellaneous bifurcations	p. 269
Local bifurcations of low codimension	p. 270
Saddle-node bifurcation	p. 271
Period doubling bifurcation	p. 271
Hopf bifurcation	p. 272
Hopf-Neimark-Sacker bifurcation	p. 273
The center-saddle bifurcation	p. 274
Quasi-periodic bifurcations	p. 276
The quasi-periodic center-saddle bifurcation	p. 276
The quasi-periodic Hopf bifurcation	p. 278
Transition to chaos	p. 282
Exercises	p. 286
Derivation of the Lorenz equations	p. 287
Geometry and flow of an incompressible fluid	p. 287
Heat transport and the influence of temperature	p. 289
Rayleigh stability analysis	p. 290
Restriction to a 3-dimensionaI state space	p. 292
Guide to the literature	p. 297
General references	p. 297
On ergodic theory	p. 298
On Hamiltonian dynamics	p. 299
On normal forms and bifurcations	p. 299
Bibliography	p. 301
Index	p. 311
Table of Contents provided by Ingram. All Rights Reserved.

Dynamical Systems and Chaos

At a Glance

Hardcover

Industry Reviews

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