Half a century of the theory of synchronization

TL;DR

This paper reviews the development of synchronization theory over fifty years, highlighting key models like the Kuramoto equations, chimera states, and the role of phase reduction methods in understanding coupled oscillators.

Contribution

It provides a retrospective analysis of major theoretical advances in synchronization, emphasizing the derivation of fundamental models and the influence of phase reduction techniques.

Findings

Derivation of the Kuramoto-Sivashinsky equation for spatio-temporal chaos

Development of the solvable Kuramoto model for phase transition

Discovery of chimera states showing coexistence of coherence and incoherence

Abstract

This review offers a retrospective of the development of the theory of coupled oscillators and synchronization over the past half century. Among the various works made by myself during this period, the following three specific works will be focused on, serving as some key points to illustrate the field's evolution. They are the derivation of (1) a simple partial differential equation exhibiting spatio-tempoeral chaos (Kuramoto-Sivashinsky equaiton), (2) a solvable mathematical model describing synchronization phase transition (Kuramoto model), and the discovery of (3) coexistence of coherence and incoherence in nonlocally coupled oscillators (chimera states). It is emphasized that all these works resulted fron the phase reduction of the complex Ginzburg-Landau equation (or its variants), the equation which was derived with a coworker in 1974 from a certain reaction-diffusion model. A quick overview will also be made on how the above three works influenced the subsequent development of the field of coupled oscillators and synchronization. Finally, a few comments will be made on how the methods of dynamical reduction, such as the center-manifold reduction and phase reduction, are crucial for exploring this field in depth. This article is a largely faithful reproduction of the content presented in my award lecture.