A stochastic approximation for the finite-size Kuramoto–Sakaguchi model

We perform a stochastic model reduction of the Kuramoto–Sakaguchi model for finitely many coupled phase oscillators with phase frustration. Whereas in the thermodynamic limit coupled oscillators exhibit stationary states and a constant order parameter, finite-size networks exhibit persistent temporal fluctuations of the order parameter. These fluctuations are caused by the interaction of the synchronised oscillators with the non-entrained oscillators. We present numerical results suggesting that the collective effect of the non-entrained oscillators on the synchronised cluster can be approximated by a Gaussian process. This allows for an effective closed evolution equation for the synchronised oscillators driven by a Gaussian process which we approximate by a two-dimensional Ornstein–Uhlenbeck process. Our reduction reproduces the stochastic fluctuations of the order parameter and leads to a simple stochastic differential equation for the order parameter.