Cluster synchronization in oscillatory networks

Chaotic synchronization has attracted a growing interest in physics with applications in many areas of science. In the context of coupled chaotic elements, many different types of synchronization have been studied in the past two decades. The most important ones are complete or identical synchronization (CS),(1-3) phase synchronization (PS),(4,5) lag synchronization (LS),(6) and generalized synchronization (GS).(7,8) In this paper, we focus on CS in ensembles of nonsymmetrically coupled chaotic cells. CS was first discovered and is the simplest form of synchronization in chaotic systems. It consists of a perfect hooking of the chaotic trajectories of two or many identical systems even though their initial conditions may be different. In arrays of coupled systems, CS can take the forms of global (full) and cluster (partial) synchronization. In global synchronization, all the elements of the ensemble are mutually synchronized. The phenomenon of cluster synchronization is observed when an ensemble of oscillators splits into groups of synchronized elements (for a review, see Refs. 9 and 10). An analytical and numerical study of the conditions for the existence and stability of cluster CS is presented.