Cluster Synchronization on CDNs with Proportional Delay: Impulsive Effect Method

This chapter aims to study the cluster synchronization for a kind of dynamical complex networks contain nonidentical nonlinear Lur’e systems. Different from some previous works, this chapter mainly considers the proportional delay, which has the features of unbounded and time-varying. It has great side effects on the network synchronization. Based on the topological structure of nonidentically coupled Lur’e network, an effective impulsive pinning controller is proposed and imposed on the Lur’e system with directional path to other clusters. Considering different functions that the impulsive effects play, some sufficient criteria for the cluster synchronization of the nonidentically coupled Lur’e dynamical networks are derived by applying the extended parameters variation formula, the impulsive comparison principle, as well as the concept of average impulsive interval. At the same time, the exponential convergence rate is successfully calculated for different functions of impulse effects. At the end of this chapter, three numerical simulations are carried out to verify the rationality of the theoretical results and the control protocol.