Partial synchronization and clustering in a system of diffusively coupled chaotic oscillators

We examine the problem of partial synchronization (or clustering) in diffusively coupled arrays of identical chaotic oscillators with periodic boundary conditions. The term partial synchronization denotes a dynamic state in which groups of oscillators synchronize with one another, but there is no synchronization among the groups. By combining numerical and analytical methods we prove the existence of partially synchronized states for systems of three and four oscillators. We determine the stable clustering structures and describe the dynamics within the clusters. Illustrative examples are presented for coupled Rössler systems. At the end of the paper, synchronization in larger arrays of chaotic oscillators is discussed.