Cluster Synchronization of Heterogeneous Kuramoto Oscillators over Directed and Undirected Network Topologies

This paper studies the cluster synchronization of heterogeneous Kuramoto oscillators over both directed and undirected network topologies. It is assumed that there exist only cooperations among oscillators in the same cluster, while there may exist competitions between oscillators belonging to different clusters. A method based on Lyapunov functional is used to analyze the stability of frequency synchronization and some sufficient conditions are derived. It is shown that under a certain coupling structure, frequencies of the oscillators achieve cluster synchronization when the coupling strength within the intra-duster is strong enough, in which the lower bound of the coupling strength is clearly specified. Finally, some numerical simulations further visualize the cluster synchronization of ten oscillators in two clusters.