Pinning Synchronization of Fractional-Order Two-Layer Networks: From Inter-Layer Synchronization to Cluster Synchronization

This article is dedicated to the cluster synchronization problem of fractional-order two-layer networks (FOTLNs) in two steps. The first step is to implement the inter-layer synchronization of FOTLNs by designing a partial directed inter-layer communication mode and a novel feedback controller imposed on the first layer network. Especially, a new method of treating a layer network as a whole is used, which provides an important idea for the analysis of inter-layer synchronization of the FOTLNs. On the basis of the first step, by designing pinning control strategy employed on the second layer network, the cluster synchronization of the FOTLNs is solved, subsequently. Note that the cluster synchronization of the FOTLNs are guaranteed only when the designed two controllers are activated simultaneously, which can be extended to the investigation of the cluster synchronization of networks with the single layer, the multi-layers or the integer order. Two illustrative examples are provided lastly to confirm the theoretical analysis.