  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 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 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 intra-cluster coupling is
sufficiently stronger than the inter-cluster coupling, the natural frequencies
of the oscillators in distinct clusters are sufficiently different, or, in the
case of two clusters, when the intra-cluster dynamics is homogeneous. We
illustrate and validate the effectiveness of our theoretical results via
numerical studies.
