Cluster synchronization of community network with distributed time delays via impulsive control**Project supported by the National Natural Science Foundation of China (Grant No. 61463022), the Natural Science Foundation of Jiangxi Province, China (Grant No. 20161BAB201021), and the Natural Science Foundation of Jiangxi Educational Committee, China (Grant No. GJJ14273).

Abstract

Cluster synchronization is an important dynamical behavior in community networks and deserves further investigations. A community network with distributed time delays is investigated in this paper. For achieving cluster synchronization, an impulsive control scheme is introduced to design proper controllers and an adaptive strategy is adopted to make the impulsive controllers unified for different networks. Through taking advantage of the linear matrix inequality technique and constructing Lyapunov functions, some synchronization criteria with respect to the impulsive gains, instants, and system parameters without adaptive strategy are obtained and generalized to the adaptive case. Finally, numerical examples are presented to demonstrate the effectiveness of the theoretical results.