Time-delayed Kuramoto model in the Watts-Strogatz small-world networks

We study the synchronization of small-world networks of identical coupled phase oscillators through the Kuramoto interaction and uniform time delay. For a given intrinsic frequency and coupling constant, we observe synchronization enhancement in a range of time delays and discontinuous transition from the partially synchronized state with defect patterns to a glassy phase, characterized by a distribution of randomly frozen phase-locked oscillators. By further increasing the time delay, this phase undergoes a discontinuous transition to another partially synchronized state. We found the bimodal frequency distributions and hysteresis loops as indicators of the discontinuous nature of these transitions. Moreover, we found the existence of Chimera states at the onset of transitions.