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

TL;DR

This paper precisely determines the exponential synchronization rate of high-dimensional Kuramoto models with identical oscillators on general digraphs with spanning trees, extending previous results and using a simpler method.

Contribution

It provides an exact synchronization rate for models on general digraphs with spanning trees, broadening the understanding beyond strongly connected balanced digraphs.

Findings

Exact synchronization rate equals the smallest non-zero real part of Laplacian eigenvalues.

Extends results to general digraphs with spanning trees, not just strongly connected balanced ones.

Uses a simpler, elementary method compared to differential geometry approaches.

Abstract

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.