On the complete synchronization of the Kuramoto phase model (Q1957108)

The authors consider the Kuramoto system of coupled phase oscillators \[ \frac{d\theta_i}{dt} = \Omega_i + \frac{K}{N}\sum_{j=1}^N \sin (\theta_j-\theta_i), \quad \theta_i\in S^1, \quad i=1,\dots,N, \] with initial conditions \(\theta_i(0)=\theta_{i0}\). The main results present sufficient conditions on the initial configurations \(\theta_{i0}\), which lead asymptotically to the completely synchronized state, i.e., \(|\theta_i(t)-\theta_j(t)|\to 0\) for \(t\to\infty\). In particular, if the initial phases \(\theta_{i0}\) of identical oscillators (\(\Omega_i=\mathrm{const}\)) are distributed in a half of the circle, then they will converge to the completely synchronized state exponentially fast. In contrast, for nonidentical oscillators, exponentially fast synchronization occurs for a more restricted class of initial conditions.