A fast-slow dynamical systems theory for the Kuramoto type phase model (Q640038)

The paper studies the Kuramoto system of coupled phase oscillators \[ \theta_i'= \Omega_i + \frac{K}{N}\sum_{j=1}^{N}\sin (\theta_j-\theta_i),\quad i=1,\dots,N. \] The rigorously proven main result states that the complex order parameter defined by \[ R(t)=\frac{1}{N}\sum_{j=1}^{N} e^{i\theta_j(t)} \] asymptotically converges to a constant function as \(N\to\infty\). The convergence is proven in a weak sense.