Abstract
This article studies the cluster synchronization problem of coupled nonlinear systems with directed topology and competitive relationships. We assume that nodes within the same cluster have the same intrinsic dynamics, while the dynamics of nodes belonging to different clusters are different. In the same cluster, there only exist cooperative relationships, and there may have competitive relationships among nodes belonging to different clusters. Under the assumptions that the dynamic of each node satisfies one-sided Lipschitz condition, the digraph of each cluster is strongly connected, and the digraph between different clusters satisfies the cluster-input-equivalence condition, some sufficient conditions for cluster synchronization in the cases of linear coupling and nonlinear coupling are obtained respectively. The obtained conditions are presented as some algebraic conditions which are easy to solve. Finally, the obtained results are validated by two numerical simulations.
Highlights
The synchronization phenomenon often exists in biology, ecology, engineering, physics and social sciences
If the nodes within the same cluster are cooperative and the nodes between different clusters are competitive, the network will show some new synchronization phenomena, such as bipartite synchronization [9, 12,13,14], modulus synchronization [15], interval bipartite consensus [16].When cooperation and competition may exist simultaneously between different clusters, the network will show more collective behaviours, and the cluster synchronization is a common occurrence in this setting. [17, 18]
This paper focuses on analyzing under what conditions the coupling networks (1) and (2) can achieve cluster synchronization
Summary
The synchronization (consensus) phenomenon often exists in biology, ecology, engineering, physics and social sciences. In [4], cluster synchronization in linear systems with partial state coupling is considered via pinning control. By considering the leaderless cluster synchronization problem, the authors the authors in [19] established the group synchronization results for heterogeneous systems with linear and nonlinear coupling, respectively, and the authors provided some corresponding algebraic conditions. This article aims to establish the results of cluster synchronization analysis for coupling network with competitive relationships and directed topology in a leaderless framework. The contribution of this article can be summarized as follows: 1) For the network including competitive relationships and directed topology, we will further relax the in-degree balanced condition in [2,3,4,19,21] to make the cluster synchronous manifold hold.
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