Kuramoto Model on Smooth Curved Surfaces of High-dimensional Spaces

In this paper, a high-dimensional Kuramoto model limited on smooth curved surfaces is established. A necessary and sufficient condition of equilibria is obtained and the linearized system around an equilibrium is derived. As the considered smooth curved surface is an ellipsoid, some dynamical properties including limit behavior and instability are obtained. Based on those results, almost global synchronization is achieved. Moreover, numerical simulations are given to validate the obtained theoretical results.