Continuous limit of the moments system for the globally coupled phase oscillators

The Kuramoto model, which describes synchronization phenomena, is a system of ordinary differential equations on $N$-torus defined as coupled harmonic oscillators. The order parameter is often used to measure the degree of synchronization. In this paper, a few properties of the continuous model for the Kuramoto model are investigated. In particular, the moments systems are introduced for both of the Kuramoto model and its continuous model. By using them, it is proved that the order parameter of the $N$-dimensional Kuramoto model converges to that of the continuous model as $N\to \infty$.