Pinning control of cluster synchronization in regular networks

Abstract

As an efficient approach to controlling the collective dynamics in networked systems, pinning synchronization has been extensively studied in the literature. Different from the existing studies in which the target is normally chosen as the global synchronization state, here we investigate the pinning control of cluster synchronization in regular networks, including the globally connected and ring-structure networks. For globally connected networks, it is found that when a fraction of nodes is pinned, the group of pinning nodes will form a synchronous cluster, while the unpinning nodes are always in the desynchronization state. We conduct a theoretical analysis of the stability of the synchronous cluster, which shows that the critical pinning strength for controlling cluster is inversely proportionally to the uniform coupling strength, but is independent of the number of pinning nodes. For ring-structure networks, however, it is found that the properties of the synchronous clusters, including the sizes and partition of the clusters, are affected by both the number of pinning nodes and the pinning locations. Theoretical analysis on the stabilities of the clusters shows that different from globally connected networks, the introduction of pinnings to the ring-structure network will degenerate the network symmetry, which in turn determines the properties of the clusters.