High-dimensional Kuramoto model limited on smooth curved surfaces

• A high-dimensional Kuramoto model on curved surfaces is established.
• A necessary and sufficient condition for equilibria is obtained.
• The linearized system around an equilibrium is derived.
• Results on stability and instability are obtained.
• Almost global synchronization is achieved for some graphs.

In this Letter, a high-dimensional Kuramoto model limited on smooth curved surfaces is established. Some synchronization phenomena of this new model are displayed by simulations. 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 for the high-dimensional Kuramoto model limited on ellipsoids with complete or tree graphs. Moreover, numerical simulations are given to validate the obtained theoretical results.