Cyclic synchronous patterns in coupled discontinuous maps.

Cyclic collective behaviors are commonly observed in biological and neuronal systems, yet the dynamical origins remain unclear. Here, by models of coupled discontinuous map lattices, we investigate the cyclic collective behaviors by means of cluster synchronization. Specifically, we study the synchronization behaviors in lattices of coupled periodic piecewise-linear maps and find that in the nonsynchronous regime the maps can be synchronized into different clusters and, as the system evolves, the synchronous clusters compete with each other and present the recurring process of cluster expanding, shrinking, and switching, i.e., showing the cyclic synchronous patterns. The dynamical mechanisms of cyclic synchronous patterns are explored, and the crucial roles of basin distribution are revealed. Moreover, due to the discontinuity feature of the map, the cyclic patterns are found to be very sensitive to the system initial conditions and parameters, based on which we further propose an efficient method for controlling the cyclic synchronous patterns.