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Weak Chaos and Quasi-Regular Patterns : Cambridge Nonlinear Science Series - Georgin Moiseevich Zaslavskii

Weak Chaos and Quasi-Regular Patterns

Cambridge Nonlinear Science Series


Published: 26th April 1991
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This book presents the fundamentals of chaos theory in conservative systems and provides a systematic study of the theory of transitional states of physical systems that lie between deterministic and chaotic behavior. The authors begin with the general concepts of Hamiltonian dynamics, stabililty, and chaos, and then discuss the theory of stochastic layers and webs and the numerous applications of this theory, particularly to pattern symmetry. Throughout, they are meticulous in providing a detailed presentation of the material, which enables the reader to learn the necessary computational methods and to apply them to other problems. The inclusion of computer graphics will aid understanding and the final section of the book contains a collection of patterns in art and living nature that will fascinate.

'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

Prefacep. xi
General Concepts
Hamiltonian dynamicsp. 1
Hamiltonian systemsp. 2
The phase portraitp. 5
'Action-angle' variablesp. 7
The nonlinear pendulump. 10
Multidimensional motionp. 14
The Poincare mappingp. 19
Stability and chaosp. 21
Nonlinear resonancep. 21
Internal nonlinear resonancep. 26
The KAM theoryp. 28
Local instabilityp. 30
Mixingp. 32
Dynamic Order and Chaos
The stochastic layerp. 36
The stochastic layer of a nonlinear pendulum: mapping close to a separatrixp. 36
The stochastic layer of a nonlinear pendulum: width of the layerp. 40
Weak interaction of resonancesp. 44
The standard mappingp. 47
Stochastic layer of a nonlinear resonancep. 50
Non-trivial effects of discretizationp. 53
Chaotic spinning of satellitesp. 55
Stochastic layer to stochastic sea transitionp. 58
The border of global chaosp. 60
Percival's variational principlep. 63
Cantorip. 67
Hamiltonian intermittencyp. 69
The acceleration of relativistic particlesp. 80
The stochastic webp. 86
KAM-tori and Arnold diffusionp. 87
Weak chaos and the stochastic webp. 90
Invariant tori inside the web (web-tori) and the width of the webp. 95
The KAM-tori to web-tori transitionp. 103
The uniform webp. 109
Mapping with a twistp. 109
The periodic webp. 113
The aperiodic web and symmetry of plane tilingsp. 122
The web's skeleton and the width of the webp. 128
Patterns in the case of a particle's diffusionp. 135
The breaking up of the web in the case of relativistic particlesp. 138
Spatial Patterns
Two-dimensional patterns with quasi-symmetryp. 143
What types of patterns are there?p. 144
Dynamic generation of patternsp. 148
Quasi-symmetry, Fourier spectrum and local isomorphismp. 155
Singularities of the phase volume-energy dependency (Van Hove singularities)p. 163
Dynamic organization in phase spacep. 167
Two-dimensional hydrodynamic patterns with symmetry and quasi-symmetryp. 170
Two-dimensional steady-state vortex flows in an ideal liquidp. 172
Stability of steady-state plane flows with symmetrical structurep. 177
Chaos of stream linesp. 188
Stream lines in spacep. 188
Stream lines of the ABC-flowp. 192
Three-dimensional flows with symmetry and quasisymmetryp. 195
Stochastic layers and stochastic webs in hydrodynamicsp. 200
Helical steady-state flowsp. 206
The stochasticity of stream lines in a stationary Rayleigh-Benard convectionp. 208
Patterns in art and naturep. 211
Two-dimensional tilings in artp. 211
Phyllotaxisp. 224
Notesp. 232
Referencesp. 239
Indexp. 250
Table of Contents provided by Syndetics. All Rights Reserved.

ISBN: 9780521373173
ISBN-10: 0521373174
Series: Cambridge Nonlinear Science Series
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 268
Published: 26th April 1991
Publisher: Cambridge University Press
Country of Publication: GB
Dimensions (cm): 22.8 x 15.2  x 1.9
Weight (kg): 0.57