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.
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Exact and Approximate Stability Conditions for Cluster Synchronization of Kuramoto Oscillators
In this paper we derive exact and approximate conditions for the (local) stability of the cluster synchronization manifold for sparsely interconnected oscillators with heterogeneous and weighted Kuramoto dynamics. Cluster synchronization, which emerges when the oscillators can be partitioned in a way that their phases remain identical over time within each group, is critically important for normal and abnormal behaviors in technological and biological systems ranging from the power grid to the human brain. Yet, despite its importance, cluster synchronization has received limited attention, so that the fundamental mechanisms regulating cluster synchronization in important classes of oscillatory networks are still unknown. In this paper we provide the first conditions for the stability of the cluster synchronization manifold for general weighted networks of heterogeneous oscillators with Kuramoto dynamics. In particular, we discuss how existing results are inapplicable or insufficient to characterize the stability of cluster synchronization for oscillators with Kuramoto dynamics, provide rigorous quantitative conditions that reveal how the network weights and oscillators' natural frequencies regulate cluster synchronization, and offer examples to quantify the tightness of our conditions. Further, we develop approximate conditions that, despite their heuristic nature, are numerically shown to tightly capture the transition to stability of the cluster synchronization manifold.
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- Award ID(s):
- 1631112
- NSF-PAR ID:
- 10196089
- Date Published:
- Journal Name:
- American Control Conference
- Page Range / eLocation ID:
- 205 to 210
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Conference Paper:
- https://doi.org/10.23919/ACC.2019.8814837
