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For the high-dimensional Kuramoto model with identical oscillators under ageneral digraph that has a directed spanning tree, although exponentialsynchronization was proved under some initial state constraints, the exactexponential synchronization rate has not been revealed until now. In thispaper, the exponential synchronization rate is precisely determined as thesmallest non-zero real part of Laplacian eigenvalues of the digraph. Ourobtained result extends the existing results from the special case of stronglyconnected balanced digraphs to the condition of general digraphs owningdirected spanning trees, which is the weakest condition for synchronizationfrom the aspect of network structure. Moreover, our adopted method iscompletely different from and much more elementary than the previousdifferential geometry method.

Exact exponential synchronization rate of high-dimensional Kuramoto models with identical oscillators and digraphs