For the high-dimensional Kuramoto model with identical oscillators under a general digraph that has a directed spanning tree, although exponential synchronization was proved under some initial state constraints, the exact exponential synchronization rate has not been revealed until now. In this paper, the exponential synchronization rate is precisely determined as the smallest non-zero real part of Laplacian eigenvalues of the digraph. Our obtained result extends the existing results from the special case of strongly connected balanced digraphs to the condition of general digraphs owning directed spanning trees, which is the weakest condition for synchronization from the aspect of network structure. Moreover, our adopted method is completely different from and much more elementary than the previous differential geometry method.
Download