Half a century of the theory of synchronization

This review offers an overview of the development of the theory of coupled oscillators and synchronization over the past half century from the perspective of my own contributions. Among such contributions, the following three specific works will be focused on, serving as some key points to illustrate the field’s evolution. These include the derivation of (1) a simplest partial differential equation exhibiting spatio-temporal chaos (Kuramoto–Sivashinsky equation), (2) a solvable mathematical model describing synchronization phase transitions (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 from the phase reduction of the complex Ginzburg–Landau equation (or its variants), i.e. the equation derived with a coworker in 1974 from a certain reaction-diffusion model. A quick overview will be given 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 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.