Phase synchronization analysis of bridge oscillators between clustered networks

Recent works aim to establish necessary and sufficient conditions to guarantee phase synchronization between clusters of oscillators, usually assuming knowledge of the intra-cluster connections, that is, connections among oscillators of the same cluster. In this context, this paper takes a different approach in studying the stability of the synchronous manifold between clusters. By focusing on the inter-cluster relations between the bridge oscillators, a simplified problem is considered where intra-cluster effects are described as perturbations. Based on Lyapunov's direct method, a framework is put forward to derive sufficient conditions for the ultimately boundedness of the phase difference between the bridge oscillators. This analysis does not rely on full information on the adjacency matrix describing the specific connections among oscillators within each cluster, an information that is not always available. The established theoretical conditions are compared to numerical simulations in two examples: (i) two interconnected clusters of Kuramoto oscillators, and (ii) a benchmark model of a power grid. Results indicate that the method is effective and that its conservativeness depends on the available network information. This framework can be generalized to different networks and oscillators.