The effect of time-correlated noise on the Kuramoto model studied via the unified colored noise approximation

Abstract

Many natural and social phenomena are characterized by synchronization. The Kuramoto model, taking into account the basic ingredients for observing synchronized states, allows to study mathematically synchronization in a simplified but nontrivial picture. Here we study how a noise that is correlated on a finite time-scale $\tau$ impacts the ability of the Kuramoto model to achieve synchronization. We develop an approximated theory that allows to compute the critical coupling constant $k_c$ as a function of the correlation time $\tau$. We obtain that that $k_c(\tau)$ decreases as $\tau$ increases indicating that time-correlated noise promotes synchronization. Moreover, we show that theory describes qualitatively well the degree of synchronization near $k_c$ obtained numerically. Finally, we show that, independently on the value of $\tau$, the curves of the order parameter versus $k$ scale on the same master curve even at values of $k$ very far from $k_c$.