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Über dieses Buch

Tbis 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.

Inhaltsverzeichnis

Frontmatter

Introduction

1. Introduction

Abstract
Mathematically, a reaction-diffusion system is obtained by adding some diffusion terms to a set of ordinary differential equations which are first order in time. The reaction-diffusion model is literally an appropriate model for studying the dynamics of chemically reacting and diffusing systems. Actually, the scope of this model is much wider. For instance, in the field of biology, the propagation of the action potential in nerves and nervelike tissues is known to obey this type of equation, and some mathematical ecologists employ reaction-diffusion models for explaining various ecological patterns observed in nature. In some thermodynamic phase transitions, too, the evolution of the local order parameter is governed by reaction-diffusion-type equations if we ignore the fluctuating forces.
Yoshiki Kuramoto

Methods

2. Reductive Perturbation Method

Abstract
Small-amplitude oscillations near the Hopf bifurcation point are generally governed by a simple evolution equation. If such oscillators form a field through diffusion-coupling, the governing equation is a simple partial differential equation called the Ginzburg-Landau equation.
Yoshiki Kuramoto

3. Method of Phase Description I

Abstract
Weakly perturbed or weakly interacting finite-amplitude oscillations form a particular class of systems whose dynamics finds an extremely simplified description through the method presented here.
Yoshiki Kuramoto

4. Method of Phase Description II

Abstract
A deeper meaning of the phase description method of Chap. 3 manifests itself as we cast it into a more systematic formulation. This will widen the scope of the method, thereby encompassing problems not necessarily inherent in oscillator systems.
Yoshiki Kuramoto

Applications

5. Mutual Entrainment

Abstract
Collective oscillations in oscillator aggregates arise from the mutual entrainment among the constituent oscillators. This phenomenon seems to be of central importance in the self-organization in nature. The method of phase description I proves to be a convenient tool for approaching this problem.
Yoshiki Kuramoto

6. Chemical Waves

Abstract
The phase description can explain expanding target patterns in reaction-diffusion systems. The same method, however, breaks down for rotating spiral waves because of a phase singularity involved. The Ginzburg-Landau equation is then invoked.
Yoshiki Kuramoto

7. Chemical Turbulence

Abstract
Reaction-diffusion systems are expected to show spatio-temporal chaos in various circumstances. A few specific cases will be discussed. They include the turbulization of uniform oscillations, of propagating wave fronts and of rotating spiral waves.
Yoshiki Kuramoto

Backmatter

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