Synchronization Phenomena in Oscillator Networks: From Kuramoto and Chua to Chemical Oscillators

Abstract

This chapter addresses the problems of synchronization analysis in various types of oscillator networks. In particular, we derive sufficient conditions for emergence of multi-cluster formations in Kuramoto networks with dynamic coupling, prove the output-feedback synchronization of chaotic behavior in networks of Chua oscillators with nonlinear static coupling, and study the synchronization of complex spatiotemporal patterns in coupled infinite-dimensional reaction-diffusion models of chemical oscillators. The obtained results contribute towards a deeper understanding of the internal organization of oscillator networks, explain the prerequisites for the emergence of patterns of synchrony and justify their stability properties in terms of the dynamical characteristics of oscillators, parameters of couplings, and the interconnection topology of the network. The interplay of these three ingredients is required for the complex and dynamically rich behavior of the network. Theoretical results of the chapter are supplemented with numerical case studies.