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.
Author(s): Georgin Moiseevich Zaslavskiî, R. Z. Sagdeev, D. A. Usikov, A. A. Chernikov, A. R. Sagdeeva
Series: Cambridge Nonlinear Science Series
Publisher: CUP
Year: 1991
Language: English
Pages: 266
Cover......Page 1
About......Page 2
Cambridge Nonlinear Science Series 1......Page 4
Weak chaos and quasi-regular patterns......Page 6
Copyright - ISBN: 0521438284......Page 7
Contents......Page 8
Preface......Page 12
1 Hamiltonian dynamics......Page 14
1.1 Hamiltonian systems......Page 15
1.2 The phase portrait......Page 18
1.3 'Action-angle' variables......Page 20
1.4 The nonlinear pendulum......Page 23
1.5 Multidimensional motion......Page 27
1.6 The Poincare mapping......Page 32
2.1 Nonlinear resonance......Page 34
2.2 Internal nonlinear resonance......Page 39
2.3 The KAM theory......Page 41
2.4 Local instability......Page 43
2.5 Mixing......Page 45
3.1 The stochastic layer of a nonlinear pendulum: mapping close to a separatrix......Page 49
3.2 The stochastic layer of a nonlinear pendulum: width of the layer......Page 53
3.3 Weak interaction of resonances......Page 57
3.4 The standard mapping......Page 60
3.5 Stochastic layer of a nonlinear resonance......Page 63
3.6 Non-trivial effects of discretization......Page 66
3.7 Chaotic spinning of satellites......Page 68
4 Stochastic layer to stochastic sea transition......Page 71
4.1 The border of global chaos......Page 73
4.2 Percival's variational principle......Page 76
4.3 Cantori......Page 80
4.4 Hamiltonian intermittency......Page 82
4.5 The acceleration of relativistic particles......Page 93
5 The stochastic web......Page 99
5.1 KAM-tori and Arnold diffusion......Page 100
5.2 Weak chaos and the stochastic web......Page 103
5.3 Invariant tori inside the web (web-tori) and the width of the web......Page 108
5.4 The KAM-tori to web-tori transition......Page 116
6.1 Mapping with a twist......Page 122
6.2 The periodic web......Page 126
6.3 The aperiodic web and symmetry of plane tilings......Page 135
6.4 The web's skeleton and the width of the web......Page 141
6.5 Patterns in the case of a particle's diffusion......Page 147
6.6 The breaking up of the web in the case of relativistic particles......Page 151
7 Two-dimensional patterns with quasi-symmetry......Page 156
7.1 What types of patterns are there?......Page 157
7.2 Dynamic generation of patterns......Page 161
7.3 Quasi-symmetry, Fourier spectrum and local isomorphism......Page 168
7.4 Singularities of the phase volume-energy dependency (Van Hove singularities)......Page 176
7.5 Dynamic organization in phase space......Page 180
8 Two-dimensional hydrodynamic patterns with symmetry and quasi-symmetry......Page 183
8.1 Two-dimensional steady-state vortex flows in an ideal liquid......Page 185
8.2 Stability of steady-state plane flows with symmetrical structure......Page 190
9.1 Stream lines in space......Page 201
9.2 Stream lines of the ABC-flow......Page 205
9.3 Three-dimensional flows with symmetry and quasi-symmetry......Page 208
9.4 Stochastic layers and stochastic webs in hydrodynamics......Page 213
9.5 Helical steady-state flows......Page 219
9.6 The stochasticity of stream lines in a stationary Ray-leigh-Benard convection......Page 221
10.1 Two-dimensional tilings in art......Page 224
10.2 Phyllotaxis......Page 237
Notes......Page 245
References......Page 252
Index......Page 264
