Condensation of Synchronous Orbits on Complex Networks

In this paper we study frequency synchronization of Kuramoto oscillators. We find a typical phenomenon of condensed synchronous orbits on single-layer or duplex networks through statistical mechanics analysis and numerical simulations, where the distribution of synchronous orbits is in a bell-shaped form. Further, we investigate phase synchronization on single-layer and duplex networks with different distributions of inherent frequencies. We find that normally distributed inherent frequencies with low variances are more beneficial for phase synchronization, and separately distributed inherent frequencies can slow down the synchronization process. In the end, we investigate the influence of one layer's inherent frequencies on the other layer's phase synchronization through inter-layer couplings. Interestingly, we find that one layer's inherent frequencies with a highly condensed distribution can greatly improve phase synchronization on the other layer. The results shed new lights to our understanding of the nature of synchronization on single-layer as well as multilayer complex networks of coupled Kuramoto oscillators.