Stability Conditions for Cluster Synchronization in Networks of Heterogeneous Kuramoto Oscillators

In this paper, we study cluster synchronization in networks of oscillators with heterogenous Kuramoto dynamics, where multiple groups of oscillators with identical phases coexist in a connected network. Cluster synchronization is at the basis of several biological and technological processes; yet, the underlying mechanisms to enable the cluster synchronization of Kuramoto oscillators have remained elusive. In this paper, we derive quantitative conditions on the network weights, cluster configuration, and oscillators' natural frequency that ensure the asymptotic stability of the cluster synchronization manifold; that is, the ability to recover the desired cluster synchronization configuration following a perturbation of the oscillators' states. Qualitatively, our results show that cluster synchronization is stable when the intracluster coupling is sufficiently stronger than the intercluster coupling, the natural frequencies of the oscillators in distinct clusters are sufficiently different, or, in the case of two clusters, when the intracluster dynamics is homogeneous. We validate the effectiveness of our theoretical results via numerical studies.