Dynamics Of Pattern, The

By: Alexander B. Ezersky, Mikhail I. Rabinovich, Patrick D. Weidman

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Paperback | 2 November 2000

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Spirals, vortices, crystalline lattices, and other attractive patterns are prevalent in nature. How do such beautiful patterns appear from the initial chaos? What universal dynamical rules are responsible for their formation? What is the dynamical origin of spatial disorder in nonequilibrium media? Based on the many visual experiments in physics, hydrodynamics, chemistry and biology, this study seeks to answer these and related intriguing questions. The mathematical models presented for the dynamical theory of pattern formation are nonlinear partial differential equations. The corresponding theory is not so accessible to a wide audience. Consequently, the authors attempt to synthesise long and complex mathematical calculations to exhibit the underlying physics. The book should be useful to final year undergraduates, but is primarily aimed at graduate students, postdoctoral fellows, and others interested in the puzzling phenomena of pattern formation.

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Non-Fiction Computing & I.T.Computer Science Artificial Intelligence Computer Vision Mathematics Calculus & Mathematical Analysis Differential Calculus & Equations Applied Mathematics Nonlinear Science Science Physics

Chaos Theory Booktopia Publisher Services World Scientific Publishing

Preface	p. vii
Patterns: Prelude to a Dynamical Description	p. 1
Linear Stage of Pattern Formation	p. 15
Model Equations	p. 27
Swift--Hohenberg equation	p. 29
Newell--Whitehead--Segel equation	p. 35
Coupled amplitude equations	p. 38
Phase equations	p. 42
The Ginzburg--Landau Equation	p. 45
The dissipative Ginzburg--Landau equation	p. 46
Nerve membrane excitation and the CGL equation	p. 48
Optical dynamics and the CGL equation	p. 50
Simple patterns in the CGL equation	p. 52
Phase equations revisited	p. 55
Gallery of phenomena	p. 57
'Crystal' Formation	p. 63
Quasicrystals	p. 75
Octagons, decagons, and dodecagons	p. 78
A generalized Swift-Hohenberg model	p. 81
The 'turbulent' crystal	p. 82
Breaking of Order	p. 87
A simple model for domain walls	p. 89
Topological defects	p. 92
The birth of penta-hepta defects	p. 96
Dislocations and domain walls in Faraday ripples	p. 102
Localized Patterns	p. 107
Bistable media	p. 107
Dynamical disorder of structures	p. 114
Particle interaction	p. 115
Chaotic scattering	p. 118
Spirals	p. 129
Active spirals	p. 133
Spirals in the complex Ginzburg--Landau equation	p. 133
Spirals in the FitzHugh-Nagumo model	p. 136
Passive spirals	p. 142
Spirals in the Faraday experiment	p. 143
Spirals in Rayleigh-Benard convection	p. 145
Patterns in Oscillating Soap Films	p. 151
Introduction	p. 151
Observations	p. 152
Models for vorticity generation	p. 158
Marangoni wave model	p. 160
The role of air	p. 162
Patterns in Colonies of Microorganisms	p. 173
Dictyostelium discoideum	p. 174
Esherichia coli	p. 178
Bacillus subtilis	p. 184
Spatial Disorder	p. 189
Introductory remarks	p. 189
Characteristics of space series	p. 193
The Grassberger-Procaccia algorithm	p. 197
Qualitative description of developing disorder	p. 199
Dynamical dimension of defect-mediated turbulence	p. 202
Patterns in Chaotic Media	p. 205
Introductory remarks	p. 205
Chaotic synchronization	p. 208
Coexistence of regular patterns and chaos	p. 211
Coarse grain spatio-temporal patterns	p. 214
Coherent patterns on a chaotic checkerboard	p. 220
Epilogue: Living matter and dynamic forms	p. 225
Hallucinations	p. 227
Spatio-temporal patterns and information processing	p. 234
A Short Guide to Nonlinear Dynamics	p. 239
Dynamical systems	p. 239
Types of dynamical systems	p. 240
Equilibrium states	p. 241
Homoclinic and heteroclinic trajectories	p. 245
Limit cycles	p. 248
Quasiperiodic motion	p. 249
Bifurcations	p. 250
Bifurcations of equilibrium states	p. 252
Birth of periodic motions	p. 258
Change of stability in periodic motions	p. 260
Chaotic Oscillations	p. 261
Characteristics of chaos and the strange attractor	p. 261
Chaotic Hamiltonian systems	p. 263
Chaotic self-excited oscillations	p. 264
Synchronization of oscillations	p. 266
Dynamical chaos and turbulence	p. 271
Key Experiments in Pattern Formation	p. 279
Parametrically excited patterns	p. 279
Experiments with liquids	p. 279
Experiments with granular material	p. 287
Thermal convection	p. 292
Rayleigh-Benard convection	p. 292
Patterns in Rayleigh-Benard convection	p. 295
Benard-Marangoni convection	p. 297
Diffusive chemical reactions	p. 302
Turing patterns	p. 302
Oscillating chemical reactions	p. 306
Bibliography	p. 309
Index	p. 320
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Dynamics Of Pattern, The

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