Robust Cluster Synchronization in Dynamical Networks With Directed Switching Topology via Averaging Method

Cluster synchronization in a network of nonlinear systems with directed topology and competitive relationships

“…Due to the influence of different structures within the network, the phenomenon of complete synchronization is relatively rare in the real world, and cluster (group) synchronization is more general. Thus, in the recent years, the cluster (group) synchronization problem has drawn significant attention from researchers due to its more accurate depiction of the real network [1][2][3][4][5].…”

“…To enable the system realize cluster synchronization, the pinning control is often used [1][2][3][4][5]. In essence, pinning control can be thought of as adding a virtual leader to each cluster to drive the node states of each cluster onto the same trajectory.…”

“…In [4], cluster synchronization in linear systems with partial state coupling is considered via pinning control. Similar method was extended for general linear and nonlinear network with external disturbance [2]. The problem of leaderless cluster synchronization is studied in [19][20][21].…”

“…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. In [2][3][4]19,21], however, all theoretical analysis are based on a particular hypothesis that the sum of the input degrees of other cluster nodes received by each cluster node is zero (in-degree balanced condition), which is a simple case of the cluster-input-equivalence condition.…”

“…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.…”