Adaptive pinning control of cluster synchronization in complex networks with Lurie-type nonlinear dynamics

• The cluster synchronization in networks with disturbances and delays is studied.
• An adaptive control strategy is presented for the networks with Lurie-type dynamics.
• The results in this paper do not require the global information of the networks.

In this paper, cluster synchronization problem is investigated for a kind of complex dynamical networks with Lurie-type non-linear dynamics. With the aid of a local update law, a pinning adaptive control strategy is proposed to solve the cluster synchronization problem of the networks. Some simple criteria for cluster synchronization are presented for the networks with external disturbances and time delays. Firstly, sufficient conditions are established to realize cluster synchronization of the networks without disturbances and delays. Then, the cluster synchronization problem of the networks with external disturbances is considered. Finally, the cluster synchronization in the networks with time-varying delays is further studied. Numerical examples are given to verify and illustrate the theoretical results.