Pinning Cluster Synchronization of Coupled Nonidentical Harmonic Oscillators Under Directed Topology

This paper addresses cluster synchronization of coupled harmonic oscillators over directed fixed and switching topologies, where uncoupled harmonic oscillators in the same cluster have the identical node dynamics, while any pair of nodes in different clusters are essentially distinguishing according to their local dynamics. A pinning control protocol is proposed for such coupled harmonic oscillators system, and then some graphic topology conditions easy to verify are established for cluster synchronization. The main contributions of the present investigation include: (i) cluster synchronization problem of coupled nonidentical harmonic oscillators is addressed over directed topology; (ii) desynchronizing motion of harmonic oscillators from different clusters doesn't depend on negative coupling weights but leader of every cluster; (iii) this paper deals with what kind of harmonic oscillators ought to be pinned in order to reach cluster synchronization. Finally, numerical examples and simulations demonstrate the obtained theoretical results.