Weak Chaos and Quasi-Regular Patterns

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Hardcover | 10 June 1991

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268 Pages

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22.86 x 15.24 x 1.91

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This book, the first in the Cambridge Nonlinear Science Series, presents the fundamentals of chaos theory in conservative systems, providing a systematic study of the theory of transitional states of physical systems which lie between deterministic and chaotic behaviour. The authors' treatment of transitions to chaos, the theory of stochastic layers and webs, and the numerous applications of this theory, particularly to pattern symmetry, will make the book of importance to scientists from many disciplines.The authors have been meticulous in providing a detailed presentation of the material, enabling the reader to learn the necessary computational methods and to apply them in other problems. The inclusion of a significant amount of computer graphics is also an important aid to understanding. The final section of the book contains a fascinating collection of patterns in art and living nature. The book will be of interest to graduate students and researchers in physics and mathematics who are interested in problems of chaos, irreversibility, statistical mechanics and theories of spatial patterns and symmetries. The perhaps unconventional links between chaos theory and other topics make the book particularly interesting.

Industry Reviews

'This book contains information that will benefit both students and researchers in many areas of physics, mathematics and other sciences who are working on problems in chaos, statistical mechanics, spatial patterns, symmetries, etc.' Contemporary Physics

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

Preface	p. xi
General Concepts
Hamiltonian dynamics	p. 1
Hamiltonian systems	p. 2
The phase portrait	p. 5
'Action-angle' variables	p. 7
The nonlinear pendulum	p. 10
Multidimensional motion	p. 14
The Poincare mapping	p. 19
Stability and chaos	p. 21
Nonlinear resonance	p. 21
Internal nonlinear resonance	p. 26
The KAM theory	p. 28
Local instability	p. 30
Mixing	p. 32
Dynamic Order and Chaos
The stochastic layer	p. 36
The stochastic layer of a nonlinear pendulum: mapping close to a separatrix	p. 36
The stochastic layer of a nonlinear pendulum: width of the layer	p. 40
Weak interaction of resonances	p. 44
The standard mapping	p. 47
Stochastic layer of a nonlinear resonance	p. 50
Non-trivial effects of discretization	p. 53
Chaotic spinning of satellites	p. 55
Stochastic layer to stochastic sea transition	p. 58
The border of global chaos	p. 60
Percival's variational principle	p. 63
Cantori	p. 67
Hamiltonian intermittency	p. 69
The acceleration of relativistic particles	p. 80
The stochastic web	p. 86
KAM-tori and Arnold diffusion	p. 87
Weak chaos and the stochastic web	p. 90
Invariant tori inside the web (web-tori) and the width of the web	p. 95
The KAM-tori to web-tori transition	p. 103
The uniform web	p. 109
Mapping with a twist	p. 109
The periodic web	p. 113
The aperiodic web and symmetry of plane tilings	p. 122
The web's skeleton and the width of the web	p. 128
Patterns in the case of a particle's diffusion	p. 135
The breaking up of the web in the case of relativistic particles	p. 138
Spatial Patterns
Two-dimensional patterns with quasi-symmetry	p. 143
What types of patterns are there?	p. 144
Dynamic generation of patterns	p. 148
Quasi-symmetry, Fourier spectrum and local isomorphism	p. 155
Singularities of the phase volume-energy dependency (Van Hove singularities)	p. 163
Dynamic organization in phase space	p. 167
Two-dimensional hydrodynamic patterns with symmetry and quasi-symmetry	p. 170
Two-dimensional steady-state vortex flows in an ideal liquid	p. 172
Stability of steady-state plane flows with symmetrical structure	p. 177
Chaos of stream lines	p. 188
Stream lines in space	p. 188
Stream lines of the ABC-flow	p. 192
Three-dimensional flows with symmetry and quasisymmetry	p. 195
Stochastic layers and stochastic webs in hydrodynamics	p. 200
Helical steady-state flows	p. 206
The stochasticity of stream lines in a stationary Rayleigh-Benard convection	p. 208
Miscellanea
Patterns in art and nature	p. 211
Two-dimensional tilings in art	p. 211
Phyllotaxis	p. 224
Notes	p. 232
References	p. 239
Index	p. 250
Table of Contents provided by Syndetics. All Rights Reserved.

Weak Chaos and Quasi-Regular Patterns

At a Glance

Hardcover

Industry Reviews

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