Synopses & Reviews
This highly respected, frequently cited book addresses two exciting fields: pattern formation and synchronization of oscillators. It systematically develops the dynamics of many-oscillator systems of dissipative type, with special emphasis on oscillating reaction-diffusion systems. The author applies the reductive perturbation method and the phase description method to the onset of collective rhythms, the formation of wave patterns, and diffusion-induced chemical turbulence.
This two-part treatment starts with a section on methods, defining and exploring the reductive perturbation method and#8212; oscillators versus fields of oscillators, the Stuart-Landau equation, onset of oscillations in distributed systems, and the Ginzburg-Landau equations. It further examines methods of phase description, including systems of weakly coupled oscillators, one-oscillator problems, nonlinear phase diffusion equations, and representation by the Floquet eigenvectors.
Additional methods include systematic perturbation expansion, generalization of the nonlinear phase diffusion equation, and the dynamics of both slowly varying wavefronts and slowly phase-modulated periodic waves. The second part illustrates applications, from mutual entrainment to chemical waves and chemical turbulence. The text concludes with a pair of convenient appendixes.
Synopsis
A fundamental and frequently cited book in two very exciting fields: pattern formation and synchronization of oscillators. Provides asymptotic methods that can be applied to the dynamics of self-oscillating fields of the reaction-diffusion type and of some related systems. Graduate level. 40 figures.
Synopsis
A fundamental and frequently cited book in two very exciting fields: pattern formation and synchronization of oscillators. Provides asymptotic methods that can be applied to the dynamics of self-oscillating fields of the reaction-diffusion type and of some related systems. Graduate level.
Contents: 1. Introduction. 2. Reductive Perturbation Method. 3. Method of Phase Description I. 4. Method of Phase Description II. 5. Mutual Entrainment. 6. Chemical Waves. 7. Chemical Turbulence. Appendix. References. Subject Index. Unabridged republication of the edition originally published by Springer-Verlag, New York, 1984. 40 Figures.
Synopsis
A fundamental and frequently cited book provides asymptotic methods applicable to the dynamics of self-oscillating fields of the reaction-diffusion type. Graduate level. 40 figures. 1984 edition.
Table of Contents
1. Introduction
and#160;and#160;and#160; Part I Methods
and#160;and#160;and#160; 2. Reductive Perturbation Method
and#160;and#160;and#160;and#160;and#160; 2.1 Oscillators Versus Fields of Oscillators
and#160;and#160;and#160;and#160;and#160; 2.2 The Stuart-Landau Equation
and#160;and#160;and#160;and#160;and#160; 2.3 Onset of Oscillations in Distributed Systems
and#160;and#160;and#160;and#160;and#160; 2.4 The Ginzburg-Landau Equation
and#160;and#160;and#160; 3. Method of Phase Description I
and#160;and#160;and#160;and#160;and#160; 3.1 Systems of Weakly Coupled Oscillators
and#160;and#160;and#160;and#160;and#160; 3.2 One-Oscillator Problem
and#160;and#160;and#160;and#160;and#160; 3.3 Nonlinear Phase Diffusion Equation
and#160;and#160;and#160;and#160;and#160; 3.4 Representation by the Floquet Eigenvectors
and#160;and#160;and#160;and#160;and#160; 3.5 Case of the Ginzburg-Landau Equation
and#160;and#160;and#160; 4. Method of Phase Description II
and#160;and#160;and#160;and#160;and#160; 4.1 Systematic Perturbation Expansion
and#160;and#160;and#160;and#160;and#160; 4.2 Generalization of the Nonlinear Phase Diffusion Equation
and#160;and#160;and#160;and#160;and#160; 4.3 Dynamics of Slowly Varying Wavefronts
and#160;and#160;and#160;and#160;and#160; 4.4 Dynamics of Slowly Phase-Modulated Periodic Waves
and#160;and#160;and#160; Part II Applications
and#160;and#160;and#160; 5. Mutual Entrainment
and#160;and#160;and#160;and#160;and#160; 5.1 Synchronization as a Mode of Self-Organization
and#160;and#160;and#160;and#160;and#160; 5.2 Phase Description of Entrainment
and#160;and#160;and#160;and#160;and#160;and#160;and#160; 5.2.1 One Oscillator Subject to Periodic Force
and#160;and#160;and#160;and#160;and#160;and#160;and#160; 5.2.2 A Pair of Oscillators with Different Frequencies
and#160;and#160;and#160;and#160;and#160;and#160;and#160; 5.2.3 Many Oscillators with Frequency Distribution
and#160;and#160;and#160;and#160;and#160; 5.3 Calculation of ? for a Simple Model
and#160;and#160;and#160;and#160;and#160; 5.4 Soluble Many-Oscillator Model Showing Synchronization-Desynchronization Transitions
and#160;and#160;and#160;and#160;and#160; 5.5 Oscillators Subject to Fluctuating Forces
and#160;and#160;and#160;and#160;and#160;and#160;and#160; 5.5.1 One Oscillator Subject to Stochastic Forces
and#160;and#160;and#160;and#160;and#160;and#160;and#160; 5.5.2 A Pair of Oscillators Subject to Stochastic Forces
and#160;and#160;and#160;and#160;and#160;and#160;and#160; 5.5.3 Many Oscillators Which are Statistically Identical
and#160;and#160;and#160;and#160;and#160; 5.6 Statistical Model Showing Synchronization-Desynchronization Transitions
and#160;and#160;and#160;and#160;and#160; 5.7 Bifurcation of Collective Oscillations
and#160;and#160;and#160; 6. Chemical Waves
and#160;and#160;and#160;and#160;and#160; 6.1 Synchronization in Distributed Systems
and#160;and#160;and#160;and#160;and#160; 6.2 Some Properties of the Nonlinear Phase Diffusion Equation
and#160;and#160;and#160;and#160;and#160; 6.3 Development of a Single Target Pattern
and#160;and#160;and#160;and#160;and#160; 6.4 Development of Multiple Target Patterns
and#160;and#160;and#160;and#160;and#160; 6.5 Phase Singularity and Breakdown of the Phase Description
and#160;and#160;and#160;and#160;and#160; 6.6 Rotating Wave Solution of the Ginzburg-Landau Equation
and#160;and#160;and#160; 7 Chemical Turbulence
and#160;and#160;and#160;and#160;and#160; 7.1 Universal Diffusion-Induced Turbulence
and#160;and#160;and#160;and#160;and#160; 7.2 Phase Turbulence Equation
and#160;and#160;and#160;and#160;and#160; 7.3 Wavefront Instability
and#160;and#160;and#160;and#160;and#160; 7.4 Phase Turbulence
and#160;and#160;and#160;and#160;and#160; 7.5 Amplitude Turbulence
and#160;and#160;and#160;and#160;and#160; 7.6 Turbulence Caused by Phase Singularities
and#160; Appendix
and#160; A. Plane Wave Solutions of the Ginzburg-Landau Equation
and#160; B. The Hopf Bifurication for the Brusselator
and#160; References
and#160; Subject Index