Exact and Approximate Stability Conditions for Cluster Synchronization of Kuramoto Oscillators

In this paper we derive exact and approximate conditions for the (local) stability of the cluster synchronization manifold for sparsely interconnected oscillators with heterogeneous and weighted Kuramoto dynamics. Cluster synchronization, which emerges when the oscillators can be partitioned in a way that their phases remain identical over time within each group, is critically important for normal and abnormal behaviors in technological and biological systems ranging from the power grid to the human brain. Yet, despite its importance, cluster synchronization has received limited attention, so that the fundamental mechanisms regulating cluster synchronization in important classes of oscillatory networks are still unknown. In this paper we provide the first conditions for the stability of the cluster synchronization manifold for general weighted networks of heterogeneous oscillators with Kuramoto dynamics. In particular, we discuss how existing results are inapplicable or insufficient to characterize the stability of cluster synchronization for oscillators with Kuramoto dynamics, provide rigorous quantitative conditions that reveal how the network weights and oscillators' natural frequencies regulate cluster synchronization, and offer examples to quantify the tightness of our conditions. Further, we develop approximate conditions that, despite their heuristic nature, are numerically shown to tightly capture the transition to stability of the cluster synchronization manifold.