Chemical Oscillations, Waves and Turbulence

Overview

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.
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Overview

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 — 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.

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Product Details

  • ISBN-13: 9780486428819
  • Publisher: Courier Corporation
  • Publication date: 8/5/2003
  • Series: Dover Books on Chemistry Series
  • Edition description: Reprinted First Dover Edition
  • Edition number: 1
  • Pages: 178
  • Sales rank: 1,393,734
  • Product dimensions: 6.14 (w) x 9.21 (h) x 0.38 (d)

Table of Contents

1. Introduction
    Part I Methods
    2. Reductive Perturbation Method
      2.1 Oscillators Versus Fields of Oscillators
      2.2 The Stuart-Landau Equation
      2.3 Onset of Oscillations in Distributed Systems
      2.4 The Ginzburg-Landau Equation
    3. Method of Phase Description I
      3.1 Systems of Weakly Coupled Oscillators
      3.2 One-Oscillator Problem
      3.3 Nonlinear Phase Diffusion Equation
      3.4 Representation by the Floquet Eigenvectors
      3.5 Case of the Ginzburg-Landau Equation
    4. Method of Phase Description II
      4.1 Systematic Perturbation Expansion
      4.2 Generalization of the Nonlinear Phase Diffusion Equation
      4.3 Dynamics of Slowly Varying Wavefronts
      4.4 Dynamics of Slowly Phase-Modulated Periodic Waves
    Part II Applications
    5. Mutual Entrainment
      5.1 Synchronization as a Mode of Self-Organization
      5.2 Phase Description of Entrainment
        5.2.1 One Oscillator Subject to Periodic Force
        5.2.2 A Pair of Oscillators with Different Frequencies
        5.2.3 Many Oscillators with Frequency Distribution
      5.3 Calculation of ? for a Simple Model
      5.4 Soluble Many-Oscillator Model Showing Synchronization-Desynchronization Transitions
      5.5 Oscillators Subject to Fluctuating Forces
        5.5.1 One Oscillator Subject to Stochastic Forces
        5.5.2 A Pair of Oscillators Subject to Stochastic Forces
        5.5.3 Many Oscillators Which are Statistically Identical
      5.6 Statistical Model Showing Synchronization-Desynchronization Transitions
      5.7 Bifurcation of Collective Oscillations
    6. Chemical Waves
      6.1 Synchronization in Distributed Systems
      6.2 Some Properties of the Nonlinear Phase Diffusion Equation
      6.3 Development of a Single Target Pattern
      6.4 Development of Multiple Target Patterns
      6.5 Phase Singularity and Breakdown of the Phase Description
      6.6 Rotating Wave Solution of the Ginzburg-Landau Equation
    7 Chemical Turbulence
      7.1 Universal Diffusion-Induced Turbulence
      7.2 Phase Turbulence Equation
      7.3 Wavefront Instability
      7.4 Phase Turbulence
      7.5 Amplitude Turbulence
      7.6 Turbulence Caused by Phase Singularities
  Appendix
  A. Plane Wave Solutions of the Ginzburg-Landau Equation
  B. The Hopf Bifurication for the Brusselator
  References
  Subject Index

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