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

## TL;DR

This paper investigates how finite-time correlated noise influences synchronization in the Kuramoto model, developing an approximation to predict critical coupling changes and revealing that correlated noise promotes synchronization.

## Contribution

The study introduces a unified colored noise approximation to analyze the impact of time-correlated noise on the Kuramoto model's synchronization threshold.

## Key findings

- Time-correlated noise decreases the critical coupling $k_c$.
- The approximation qualitatively matches numerical results near $k_c$.
- Order parameter curves scale on a master curve regardless of $ au$.

## 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 $k_c$.