Cluster Synchronization of a Nonlinear Network With Fixed and Switching Topologies

This article mainly addresses the synchronization problem of leaderless and leader–follower clusters in directed topologically coupled nonlinear systems. The relationship between nodes within each cluster is cooperative, and nodes belonging to different clusters may compete with each other. For each case of leaderless and leader-following, we consider both fixed and switching topologies. The vector field at each node satisfies the one-sided Lipschitz condition. For situations where there is no leader and leader to follow, we do not need to use the in-degree balanced condition that is often required in most existing literature. Under the leaderless framework, the cluster synchronization (CS) problem is transformed into a stability problem through variable transformation. Under the leader-following framework, by equipping each cluster with a virtual leader, we design a new class of leader-following protocols that can be used to achieve CS of coupled nonlinear systems. Finally, two numerical examples are provided to illustrate the validity of the obtained results.