Clustering And Synchronization In An Array Of Phase Oscillators And Effect Of Time Delay On Their Synchronization In A Network

The clustering and synchronization in an array of repulsively coupled phase oscillators of Kuramoto model are investigated. If the cluster of the phase oscillators is considered as a node in a network, the effect of time delay existed in the coupling on the synchronization in the network is discussed.Firstly, the dynamics of a group of repulsively coupled and symmetrically distributed Kuramoto phase oscillators is investigated. The clustering and synchronization behavior of a one-dimensional array of repulsively coupled phase oscillators are analyzed through numerical simulations. The array of phase oscillators is divided into several clusters according to their symmetry of the coupling structure. The phases of the oscillators in the same cluster are synchronized, while there is no synchronization among oscillators if they belong to different clusters. If the dynamics of one oscillator is perturbed slightly, the synchronization will be destroyed and then recovered after a short period of transient process. However, the synchronized state will be completely destroyed if the perturbation is quite strong. Due to the advantages of this phenomenon, it can be used in digital communication of one-to several.Secondly, the effect of time delay between the coupling among oscillators on the synchronization is discussed when the Kuramoto phase oscillators are acted as a node in a network. An array of 800 oscillators in a WS type small-world network is analyzed through numerical simulations. There is a linear relationship of the delay time and the time needed for synchronization in the network. It is found that the longer the delay time is, the longer the synchronization time of the network is needed. The rewiring probability P needed in construction of the small-world network plays a key role to the slope of the linear relationship.