Stochastic (in)stability of synchronisation of oscillators on networks

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.