Search

We investigate the collective dynamics of a population of XY model-type oscillators, globally coupled via non-separable interactions that are randomly chosen from a positive or negative value, and subject to thermal noise controlled by temperature $T$ . For a finite ratio of positive versus negative coupling, we find that the system at $T = 0$ exhibits a discontinuous, first-order like phase transition from the incoherent to the fully coherent state. We determine the critical threshold for this synchronization transition using a linear stability analysis for the fully coherent state and a heuristic stability argument for the incoherent state. Our theoretical results are supported by extensive numerical simulations which clearly display a first order like transition. Remarkably, the synchronization threshold induced by the type of random coupling considered here is identical to the one found in studies which consider uniform input or output strengths for each oscillator node [Hong and Strogatz, Phys. Rev. E 84, 046202 (2011); H. Hong and S. H. Strogatz, Phys. Rev. Lett. 106, 054102 (2011)]. When thermal noise is present, $T > 0$, the transition from incoherence to the partial coherence is continuous and the critical threshold is now larger compared to the deterministic case, T = 0. We formulate an exact mean-field theory applicable to heterogeneous network structures, based on recent work for the stochastic Kuramoto model using graphon objects representing graphs in the mean-field limit. Applying stability results for the incoherent solution, we derive an exact formula for the synchronization threshold for $T > 0$. In the limit $T \rightarrow 0$ we retrieve the result for the deterministic case with $T = 0$.

We consider a mean-field model of coupled phase oscillators with quenched disorder in the natural frequencies and coupling strengths. A fraction $p$ of oscillators are positively coupled, attracting all others, while the remaining fraction $1-p$ are negatively coupled, repelling all others. The frequencies and couplings are deterministically chosen in a manner which correlates them, thereby correlating the two types of disorder in the model. We first explore the effect of this correlation on the system's phase coherence. We find that there is a a critical width $\gamma_c$ in the frequency distribution below which the system spontaneously synchronizes. Moreover, this $\gamma_c$ is independent of $p$. Hence, our model and the traditional Kuramoto model (recovered when $p=1$) have the same critical width $\gamma_c$. We next explore the critical behavior of the system by examining the finite-size scaling and the dynamic fluctuation of the traditional order parameter. We find that the model belongs to the same universality class as the Kuramoto model with deterministically (not randomly) chosen natural frequencies for the case of $p, Comment: 7 pages, 6 figures