Cluster Synchronization of Fractional-Order Complex Community Networks in Quaternion-Valued Field

Abstract

Without separating the fractional-order quaternionvalued complex networks (FOQVCNs) into four real-valued systems, cluster synchronization is investigated in this paper. First, the FOQVCNs are divided into several clusters, and the dynamic behaviors of the clusters are different. For the case of different dynamics behaviors of the network nodes, hybrid adaptive control is proposed, which means that coupling strength and feedback control gain are simultaneously regulated with the evolution of states. Particularly, some new analytical approaches for fractional-order adaptive schemes, including the method of contradiction, the Mean Value Theorem are developed to establish synchronization criteria of the network system. Furthermore, asymptotical synchronization of FOQVCNs with identical nodes is also studied by adaptive feedback controller and adjusting coupling strength as a special case. At last, two fractional-order quaternion-valued complex community networks are divided to synchronize to the desired trajectory, respectively and numerical simulations are provided to illustrate the validity of theoretical results.