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.

Inhalt

1.- 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.- 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.- A. Plane Wave Solutions of the Ginzburg-Landau Equation.- B. The Hopf Bifurcation for the Brusselator.- References.