Chaotic Dynamics

An Introduction Based on Classical Mechanics

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Paperback | 24 August 2006

At a Glance

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428 Pages

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24.41 x 16.99 x 2.21

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Published: 6th May 1996

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In the past few decades we have come to understand that even motions in simple systems can have complex and surprising properties. Chaotic Dynamics provides a clear introduction to these chaotic phenomena, based on geometrical interpretations and simple arguments, without the need for prior in-depth scientific and mathematical knowledge. Richly illustrated throughout, examples are taken from classical mechanics whose elementary laws are familiar to the reader. In order to emphasize the general features of chaos, the most important relations are also given in simple mathematical forms, independent of any mechanical interpretation. A broad range of potential applications are presented, ranging from everyday phenomena through engineering and environmental problems to astronomical aspects. Chaos occurs in a variety of scientific disciplines, and proves to be the rule, not the exception. This book is primarily intended for undergraduate students in science, engineering, and mathematics.

Industry Reviews

'... a well-organised, beautifully illustrated volume covering the main points of chaotic dynamical systems, using intuitive mechanical examples to motivate clear and lucid discussions ... It is certainly a book intended to teach or learn from ... Throughout the work self-contained text capsules appear, expanding on issues raised in the discussions, and containing excellent summaries, histories, explanations and more potential applications ... excellent layout and ordering ... I would happily lend this book to any student wishing to begin learning the subject of chaotic dynamics, but I would certainly also demand its return.' Rob Sturman, University of Leeds '... highly recommended as an introductory text ...' Journal of Statistical Physics 'Even though there exists a lot of rather recent book on this subject, this manual is advisable for its clear presentation of the topics and for the simple and rigourous use of mathematical tools. A merit of this book is the cornerstone starting point: classical mechanics, which is the fundamental model for any other dynamical science. The book is planned mainly for undergraduate students of science, engineering and computational mathematics, but in the reviewer's opinion it could be useful to anybody interested in complex dynamics.' Zentralblatt MATH 'The book is excellent and I shall recommends it to my students'. Alistair Borthwick, Department of Engineering Science, University of Oxford ' ... the book is thus very well suited to be an introductory text, and can also serve individual learning ... It contains especially useful material also for completing a course on theoretical mechanics with regards to chaotic systems.' Dr Arnd Backer, Physik Journal

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Non-Fiction Mathematics Applied Mathematics Chaos Theory Science Physics Classical Mathematics Fluid Mechanics Nonlinear Science Earth Sciences, Geography, Environment, Planning Earth Sciences

List of colour plates	p. ix
Preface	p. xiii
Acknowledgements	p. xv
How to read the book	p. xvi
The phenomenon: complex motion, unusual geometry	p. 1
Chaotic motion	p. 3
What is chaos?	p. 3
Examples of chaotic motion	p. 4
Phase space	p. 19
Definition of chaos; a summary	p. 21
How should chaotic motion be examined?	p. 22
Brief history of chaos	p. 23
Fractal objects	p. 24
What is a fractal?	p. 24
Types of fractals	p. 32
Fractal distributions	p. 40
Fractals and chaos	p. 45
Brief history of fractals	p. 47
Introductory concepts	p. 49
Regular motion	p. 51
Instability and stability	p. 51
Instability, randomness and chaos	p. 59
Stability analysis	p. 65
Emergence of instability	p. 67
How to determine manifolds numerically	p. 73
Stationary periodic motion: the limit cycle (skiing on a slope)	p. 76
General phase space	p. 79
Driven motion	p. 90
General properties	p. 90
Harmonically driven motion around a stable state	p. 95
Harmonically driven motion around an unstable state	p. 98
Kicked harmonic oscillator	p. 100
Fixed points and their stability in two-dimensional maps	p. 103
The area contraction rate	p. 105
General properties of maps related to differential equations	p. 106
The world of non-invertible maps	p. 108
In what systems can we expect chaotic behaviour?	p. 109
Investigation of chaotic motion	p. 111
Chaos in dissipative systems	p. 113
Baker map	p. 114
Kicked oscillators	p. 131
Henon-type maps	p. 147
Parameter dependence: the period-doubling cascade	p. 149
General properties of chaotic motion	p. 154
The trap of the 'butterfly effect'	p. 159
Determinism and chaos	p. 168
Summary of the properties of dissipative chaos	p. 171
What use is numerical simulation?	p. 172
Ball bouncing on a vibrating plate	p. 174
Continuous-time systems	p. 175
The water-wheel	p. 181
The Lorenz model	p. 187
Transient chaos in dissipative systems	p. 191
The open baker map	p. 193
Kicked oscillators	p. 199
How do we determine the saddle and its manifolds?	p. 201
General properties of chaotic transients	p. 202
Summary of the properties of transient chaos	p. 210
Significance of the unstable manifold	p. 211
The horseshoe map	p. 213
Parameter dependence: crisis	p. 214
Transient chaos in water-wheel dynamics	p. 217
Other types of crises, periodic windows	p. 219
Fractal basin boundaries	p. 221
Other aspects of chaotic transients	p. 225
Chaos in conservative systems	p. 227
Phase space of conservative systems	p. 227
The area preserving baker map	p. 230
The origin of the baker map	p. 233
Kicked rotator - the standard map	p. 234
Connection between maps and differential equations	p. 236
Chaotic diffusion	p. 239
Autonomous conservative systems	p. 242
General properties of conservative chaos	p. 250
Summary of the properties of conservative chaos	p. 259
Homogeneously chaotic systems	p. 260
Ergodicity and mixing	p. 261
Conservative chaos and irreversibility	p. 262
Chaotic scattering	p. 264
The scattering function	p. 265
Scattering on discs	p. 266
Scattering in other systems	p. 274
Chemical reactions as chaotic scattering	p. 276
Summary of the properties of chaotic scattering	p. 277
Applications of chaos	p. 279
Spacecraft and planets: the three-body problem	p. 279
Chaos in the Solar System	p. 284
Rotating rigid bodies: the spinning top	p. 285
Chaos in engineering practice	p. 292
Climate variability and climatic change: Lorenz's model of global atmospheric circulation	p. 293
Chaos in different sciences	p. 300
Controlling chaos	p. 303
Vortices, advection and pollution: chaos in fluid flows	p. 304
Environmental significance of chaotic advection	p. 315
Epilogue: outlook	p. 318
Turbulence and spatio-temporal chaos	p. 320
Appendix	p. 322
Deriving stroboscopic maps	p. 322
Writing equations in dimensionless forms	p. 325
Numerical solution of ordinary differential equations	p. 329
Sample programs	p. 332
Numerical determination of chaos parameters	p. 337
Solutions to the problems	p. 342
Bibliography	p. 370
Index	p. 387
Table of Contents provided by Ingram. All Rights Reserved.

Chaotic Dynamics

An Introduction Based on Classical Mechanics

At a Glance

New Edition

Paperback

Industry Reviews

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