Exponential synchronization rate of proportional D-dimensional Kuramoto oscillators

For the high-dimensional Kuramoto model with identical oscillators, the exponential synchronization has been proved and the exponential synchronization rate has been given. For the high-dimensional Kuramoto model with nonidentical oscillators, the sufficient and necessary conditions for complete phase synchronization have been obtained, but the problem of exponential synchronization has not been solved. In this paper, for the proportional D-dimensional Kuramoto model, the matrix differential equations describing the synchronization error dynamics are constructed. When the interconnection graph is a strongly connected digraph, the local exponential synchronization rate is obtained.