Exponential synchronization of the high-dimensional Kuramoto model with identical oscillators under digraphs

TL;DR

This paper proves exponential synchronization for high-dimensional identical Kuramoto oscillators on directed graphs with spanning trees, using a matrix Riccati differential equation to analyze error dynamics.

Contribution

It introduces a novel MRDE approach to analyze exponential synchronization in high-dimensional Kuramoto models on digraphs with spanning trees.

Findings

Exponential synchronization achieved under certain digraph conditions.

MRDE effectively describes error dynamics in high-dimensional Kuramoto models.

Numerical simulations confirm theoretical results.

Abstract

For the Kuramoto model and its variations, it is difficult to analyze the exponential synchronization under the general digraphs due to the lack of symmetry. %due to the asymmetry of the adjacency matrices. In this paper, for the high-dimensional Kuramoto model of identical oscillators, a matrix Riccati differential equation (MRDE) is proposed to describe the error dynamics. Based on the MRDE, the exponential synchronization is proved by constructing a total error function for the case of digraphs admitting spanning trees. Finally, some numerical simulations are given to illustrate the obtained theoretical results.