Stochastic (in)stability of synchronisation of oscillators on networks

TL;DR

This paper analyzes how correlated noise affects the stability of oscillator synchronization on networks using the Kuramoto model, deriving thresholds for noise tolerance and validating with numerical simulations.

Contribution

It introduces an analytical framework for assessing stochastic stability of synchronized oscillators on networks, using Fokker-Planck equations and saddle point approximation.

Findings

Derived analytical thresholds for noise parameters affecting stability

Established a lower bound on noise tolerance using Mean First Passage Time

Validated theoretical results with numerical simulations on complex networks

Abstract

We consider the influence of correlated noise on the stability of synchronisation of oscillators on a general network using the Kuramoto model for coupled phases $\theta_i$. Near the fixed point $\theta_i \approx \theta_j \ \forall i,j$ the impact of the noise is analysed through the Fokker-Planck equation. We deem the stochastic system to be `weakly unstable' if the Mean First Passage Time for the system to drift outside the fixed point basin of attraction is less than the time for which the noise is sustained. We argue that a Mean First Passage Time, computed near the phase synchronised fixed point, gives a useful lower bound on the tolerance of the system to noise. Applying the saddle point approximation, we analytically derive general thresholds for the noise parameters for weak stochastic stability. We illustrate this by numerically solving the full Kuramoto model in the presence of noise for an example complex network.