Exponential Cluster Phase Synchronization Conditions for Second-Order Kuramoto Oscillators

This brief delivers cluster phase synchronization conditions for the second-order Kuramoto network under the effects of damping and inertia. These conditions are derived via an energy function method, which are related to the natural frequencies, damping and inertial constants of oscillators, and intra- and inter-cluster couplings among them. We show that each cluster can achieve exponential synchronization in the phase cohesive region when the inter-cluster coupling is sufficiently weak or strong. The effectiveness of presented conditions is validated on numerical examples.