Stability of cluster formations in adaptive Kuramoto networks

TL;DR

This paper analyzes the stability of multi-cluster formations in adaptive Kuramoto networks, deriving conditions based on intra-cluster topology and plasticity, with numerical validation of the theoretical results.

Contribution

It introduces new stability conditions for adaptive Kuramoto networks, emphasizing the role of inter- and intra-cluster structures in cluster formation and stability.

Findings

Stability depends on intra-cluster and inter-cluster connection structures.

Cluster existence relies on inter-cluster connections, not intra-cluster links.

Numerical examples validate the theoretical stability conditions.

Abstract

This paper studies stability properties of multi-cluster formations in Kuramoto networks with adaptive coupling. Sufficient conditions for the local asymptotic stability of the corresponding synchronization invariant toroidal manifold are derived and formulated in terms of the intra-cluster interconnection topology and plasticity parameters of the adaptive couplings. The proposed sufficient stability conditions qualitatively mimic certain counterpart results for Kuramoto networks with static coupling which require sufficiently strong and dense intra-cluster connections and sufficiently weak and sparse inter-cluster ones. Remarkably, the existence of cluster formations depends on the interconnection structure between nodes belonging to different clusters and does not require any coupling links between nodes that form a cluster. On the other hand, the stability properties of clusters depend on the interconnection structure inside the clusters. This dependence constitutes the main contribution of the paper. Also, two numerical examples are provided to validate the proposed theoretical findings.