Averaging and Cluster Synchronization of Kuramoto Oscillators

In this paper, we investigate cluster synchronization of heterogeneous Kuramoto oscillators, where multiple synchronized groups of oscillators coexist in a connected network. Motivated by recent studies on brain networks, we provide a framework to analyze stability of the cluster synchronization manifold via timescale separation. A condition known as almost equitable partitions is employed to characterize an invariant manifold of the Kuramoto dynamics. Relying on averaging methods, we show that fast inter-cluster oscillations help the oscillators to achieve synchronization within clusters. In order to prove the main results, we extend conventional averaging methods to accommodate our problem setting. Then, we derive a general stability condition for multi-cluster synchronization. To find a less conservative condition, two-cluster synchronization with a special structure is also considered. Our results are demonstrated through a numerical example of a network with three clusters.