Effect of time delay on the onset of synchronization of the stochastic Kuramoto model

We consider the Kuramoto model of globally coupled phase oscillators with time-delayed interactions, that is subject to Ornstein-Uhlenbeck ( Gaussian) colored or non-Gaussian colored noise. We investigate numerically the interplay between the influences of the finite correlation time of noise tau and the time delay tau(d) on the onset of the synchronization process. The cases for identical and nonidentical oscillators have both been considered. Among the obtained results for identical oscillators is a large increase of the synchronization threshold as a function of time delay for the colored non-Gaussian noise compared to the case of the colored Gaussian noise at low noise correlation time t. However, the difference reduces remarkably for large noise correlation times. For the case of nonidentical oscillators, the incoherent state may become unstable around the maximum value of the threshold ( as a function of time delay) even at lower coupling strength values in the presence of colored noise as compared to the noiseless case. We have studied the dependence of the critical value of the coupling strength ( the threshold of synchronization) on given parameters of the stochastic Kuramoto model in great detail and presented results for possible cases of colored Gaussian and non-Gaussian noises.