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A broad class of natural and man-made systems exhibits rich patterns of cluster synchronization in healthy and diseased states, where different groups of interconnected oscillators converge to cohesive yet distinct behaviors. To provide a rigorous characterization of cluster synchronization, we study networks of heterogeneous Kuramoto oscillators and we quantify how the intrinsic features of the oscillators and their interconnection parameters affect the formation and the stability of clustered configurations. Our analysis shows that cluster synchronization depends on a graded combination of strong intra-cluster and weak inter-cluster connections, similarity of the natural frequencies of the oscillators within each cluster, and heterogeneity of the natural frequencies of coupled oscillators belonging to different groups. The analysis leverages linear and nonlinear control theoretic tools, and it is numerically validated. 

In this paper we study cluster synchronization in a network of Kuramoto oscillators, where groups of oscillators evolve cohesively and at different frequencies from the neigh- boring oscillators. Synchronization is critical in a variety of systems, where it enables complex functionalities and behaviors. Synchronization over networks depends on the oscillators’ dynamics, the interaction topology, and coupling strengths, and the relationship between these different factors can be quite intricate. In this work we formally show that three network properties enable the emergence of cluster synchronization. Specifically, weak inter-cluster connections, strong intra-cluster connections, and sufficiently diverse natural frequencies among oscillators belonging to different groups. Our approach relies on system-theoretic tools, and is validated with numerical studies.