Cluster synchronization for networks of generic linear systems and nonlinear oscillators via intermittent pinning control

This paper is concerned with applying intermittent pinning controllers to synchronization of interacting clusters of linearly coupled heterogeneous linear systems and nonlinear oscillators under general coupling topology. The cluster synchronization is analyzed via a unified approach to convergence analysis. The key idea of the work is based on the observation that the given cluster synchronization patterns can be realized if the underlying topology of each extended cluster has a directed spanning tree and an algebraic condition is satisfied. Then structural conditions are developed to ensure the related algebraic condition. Specifically, for linear systems, the algebraic condition can be guaranteed if the strength of the intra-cluster couplings are sufficiently strong and meanwhile the work time of the pinning controller in each period is sufficiently long; for nonlinear oscillators, the work time of the pinning controller can be arbitrarily short as long as the intra-cluster coupling strength for each cluster is sufficiently strong. Furthermore, both the lower bounds for such coupling strengths and length of the work time of the pinning controller in each period are explicitly specified.