On synchronization of the double sphere Kuramoto model with connected undirected graphs

M.A. Lohe first proposed the double sphere Kuramoto model (On the double sphere model of synchronization, Physica D: Nonlinear Phenomena, 412:132642, 2020), which is a more general model than the high-dimensional Kuramoto model defined on a sphere. For the double sphere Kuramoto model with a complete graph, some theoretic results on synchronization were obtained. But for the case of general graphs, there is no report yet. In this paper, for the double sphere Kuramoto model composed of identical oscillators interacting through a connected undirected graph with a symmetric nonnegative adjacency matrix, the synchronization is proved by using LaSalle’s invariance principle. Moreover, the exponential synchronization is achieved, and the supremum of exponential synchronization rates is exactly obtained.