## Chemical Oscillations, Waves, and TurbulenceTbis book is intended to provide a few asymptotic methods which can be applied to the dynamics of self-oscillating fields of the reaction-diffusion type and of some related systems. Such systems, forming cooperative fields of a large num of interacting similar subunits, are considered as typical synergetic systems. ber Because each local subunit itself represents an active dynamical system function ing only in far-from-equilibrium situations, the entire system is capable of showing a variety of curious pattern formations and turbulencelike behaviors quite unfamiliar in thermodynamic cooperative fields. I personally believe that the nonlinear dynamics, deterministic or statistical, of fields composed of similar active (Le., non-equilibrium) elements will form an extremely attractive branch of physics in the near future. For the study of non-equilibrium cooperative systems, some theoretical guid ing principle would be highly desirable. In this connection, this book pushes for ward a particular physical viewpoint based on the slaving principle. The dis covery of tbis principle in non-equilibrium phase transitions, especially in lasers, was due to Hermann Haken. The great utility of this concept will again be dem onstrated in tbis book for the fields of coupled nonlinear oscillators. |

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### Contents

1 | |

Method of Phase Description I | 22 |

Method of Phase Description II | 32 |

Applications | 61 |

Chemical Waves | 89 |

Chemical Turbulence | 111 |

Appendix | 141 |

149 | |

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### Common terms and phrases

amplitude assumed asymptotic behavior Belousov-Zhabotinsky reaction bifurcation theory calculated chaos chaotic Chap chemical turbulence chemical waves coefficients collective oscillations condition constant corresponding coupled critical defined degrees of freedom denote deviation diffusion instability distribution dynamics eigenvalue eigenvectors evolution equation expressed fact finite fluctuations function Ginzburg-Landau equation Haken homoclinic orbits Hopf bifurcation identical Kuramoto large number limit cycle oscillators linear mathematical method modes nonlinear phase diffusion Note nullclines obtained onset ordinary differential equations oscillatory pacemaker pair parameter values partial differential equation periodic perturbation perturbation theory phase description phase diffusion equation phase singularity phase transitions phase turbulence equation phase waves phenomena physical plane waves problem pulse quantities reaction-diffusion equations reaction-diffusion systems represents rotating waves Sect slaving principle solution space spatial stability steady Stochastic Stuart-Landau equation synchronization T-periodic target patterns theory tion uniform oscillations vector wave patterns wavefront Winfree zero