## The Dynamics of PatternsSpirals, 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 invaluable book answers those 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 have made every attempt to synthesize long and complex mathematical calculations to exhibit the underlying physics. The book will be useful for 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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### Contents

Prelude to a Dynamical Description | 1 |

Linear Stage of Pattern Formation | 15 |

Model Equations | 27 |

The GinzburgLandau Equation | 45 |

Crystal Formation | 63 |

Quasicrystals | 75 |

Breaking of Order | 87 |

Localized Patterns | 107 |

Patterns in Chaotic Media | 205 |

herein with H Abarbanel V Afraimovich I Aranson A GaponovGrekhov | 213 |

Living matter and dynamic forms | 225 |

Appendix A A Short Guide to Nonlinear Dynamics | 239 |

Bifurcations | 250 |

Chaotic Oscillations | 261 |

Dynamical chaos and turbulence | 271 |

Appendix B Key Experiments in Pattern Formation | 279 |

Spirals | 129 |

Patterns in Oscillating Soap Films | 151 |

Patterns in Colonies of Microorganisms | 173 |

Spatial Disorder | 189 |

Thermal convection | 292 |

Diffusive chemical reactions | 302 |

309 | |

### Other editions - View all

The Dynamics of Patterns M. I. Rabinovich,A. B. Ezersky,Patrick D. Weidman No preview available - 2000 |

### Common terms and phrases

Afenchenko amplitude bacteria behavior bifurcation boundary capillary waves cell CGL equation chaos chaotic Chapter chemical coefficients complex coordinate corresponding coupling crystal depends described by Eqs diffusion dimension dislocations dissipative domain wall dynamical system elements equilibrium evolution example existence experiments Faraday finite flow fluid frequency function gradient granular hexagonal lattice horizontal initial conditions instability interaction limit cycle linear liquid layer Lyapunov exponents modes neural neurons nonlinear observed order parameter oscillations oscillatory oscillons parametrically excited particles pattern formation penta-hepta defect periodic motion perturbations phase portrait phase space problem qualitative quasicrystal quasiperiodic motion Rabinovich Rayleigh-Benard convection regime region rolls rotation SH equation shown in Fig soap film solution of Eq space series spatial disorder spatio-temporal patterns spiral waves stable stationary structures supercriticality symmetry synchronization Taken theory topological charges topological defects trajectories turbulence Turing two-dimensional values variation velocity vibrating viscous vortex vortices wave vectors wavelength wavenumber