On Exponential Synchronization Rates for High-Dimensional Kuramoto Models With Identical Oscillators and Digraphs
Shanshan Peng, Jinxing Zhang, Jiandong Zhu, Jianquan Lu, Xiaodi Li
Abstract
For a high-dimensional Kuramoto model with identical oscillators under a general digraph that has a directed spanning tree, although the exponential synchronization has been proved under some initial state constraints, exponential synchronization rates have not been described exactly until now. In this article, the supremum of exponential synchronization rates is precisely determined as the smallest real part of the nonzero 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.