Abstract

In this paper, we investigate the cluster synchronization problem with unbounded time-varying delays for complex networks by adding some external controllers. Previous related works mainly focused on bounded time-varying or constant time delays, which may not be consistent with the real world. Therefore, unbounded time-varying delays is considered in this paper, which can be regarded as the main difference between this paper and previous related works. We discuss the necessary condition for cluster synchronization with external controllers, which can be used to guide the design of external controllers. Then, by using the Lyapunov function method, we prove that cluster synchronization could be achieved under some sufficient criteria. Furthermore, the effects of time delays for the convergence of cluster synchronization are also discussed, which are named as the μ-cluster synchronization. Finally, numerical simulations are presented to show the validity of the obtained criteria.

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