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.
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Bode meets Kuramoto: Synchronized Clusters in Oscillatory Networks
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.
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- Award ID(s):
- 1631112
- NSF-PAR ID:
- 10105283
- Date Published:
- Journal Name:
- American Control Conference
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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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.more » « less
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Abstract Cluster synchronization in networks of coupled oscillators is the subject of broad interest from the scientific community, with applications ranging from neural to social and animal networks and technological systems. Most of these networks are directed, with flows of information or energy that propagate unidirectionally from given nodes to other nodes. Nevertheless, most of the work on cluster synchronization has focused on undirected networks. Here we characterize cluster synchronization in general directed networks. Our first observation is that, in directed networks, a cluster A of nodes might be one-way dependent on another cluster B: in this case, A may remain synchronized provided that B is stable, but the opposite does not hold. The main contribution of this paper is a method to transform the cluster stability problem in an irreducible form. In this way, we decompose the original problem into subproblems of the lowest dimension, which allows us to immediately detect inter-dependencies among clusters. We apply our analysis to two examples of interest, a human network of violin players executing a musical piece for which directed interactions may be either activated or deactivated by the musicians, and a multilayer neural network with directed layer-to-layer connections.
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Synchronization is crucial for the correct functi- onality of many natural and man-made complex systems. In this work we characterize the formation of synchronization patterns in networks of Kuramoto oscillators. Specifically, we reveal conditions on the network weights, structure and on the oscillators’ natural frequencies that allow the phases of a group of oscillators to evolve cohesively, yet independently from the phases of oscillators in different clusters. Our conditions are applicable to general directed and weighted networks of heterogeneous oscillators. Surprisingly, although the oscillators exhibit nonlinear dynamics, our approach relies entirely on tools from linear algebra and graph theory. Further, we develop a control mechanism to determine the smallest (as measured by the Frobenius norm) network perturbation to ensure the formation of a desired synchronization pattern. Our procedure allows us to constrain the set of edges that can be modified, thus enforcing the sparsity structure of the network perturbation. The results are validated through a set of numerical examples.more » « less
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