By: Y. Kuramoto
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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