Cluster synchronization on multiple sub-networks of complex networks with nonidentical nodes via pinning control

In this paper, cluster synchronization on multiple sub-networks of complex networks with nonidentical nodes, stochastic disturbances and time-varying delays is investigated. Based on the leader-follower model, an improved network structure model for realizing the cluster synchronization on multiple sub-networks of complex networks is presented. In this improved network model, the complex networks are divided into multiple pairs of matching sub-networks, each of which consists of a leaders' sub-network and a followers' sub-network, such that the dynamics of the nodes belonging to the same pair of matching sub-networks are identical, while the ones belonging to different pairs of unmatched sub-networks are nonidentical. Furthermore, the nodes in a sub-network may be inevitably influenced by some stochastic disturbances and time-varying delays. In this new setting, the aim is to design some suitable pinning controllers on the chosen nodes of each followers' sub-network, such that the proposed method for realizing the cluster synchronization has good immunity to the influence of these stochastic factors. By using the Lyapunov stability theory and stochastic differential equation theory, some cluster synchronization criteria, and a pinning scheme that the nodes with very large or low degrees are good candidates for applying pinning controllers, are established such that all the nodes in each sub-network are exponentially synchronized to the average state of their matching leaders. Then, the attack and robustness of the pinning scheme are discussed. Finally, some simulation examples are presented to verify the theoretical analysis.