Emergence of synchronization in Kuramoto model with general digraph

TL;DR

This paper investigates how synchronization emerges in the Kuramoto model on complex directed networks with a spanning tree, using hierarchical decomposition and hypo-coercivity to prove exponential convergence under certain conditions.

Contribution

It introduces a novel approach applying node decomposition and hypo-coercivity to analyze synchronization in general digraphs, extending previous results to more complex network structures.

Findings

Synchronization occurs exponentially fast in large coupling regimes.

Finite-time phase diameter reduction is achieved.

Hierarchical structure analysis enables understanding of dissipation in digraphs.

Abstract

In this paper, we study the complete synchronization of the Kuramoto model with general network containing a spanning tree, when the initial phases are distributed in an open half circle. As lack of uniform coercivity in general digraph, in order to capture the dissipation structure on a general network, we apply the node decomposition criteria in \cite{H-L-Z20} to yield a hierarchical structure, which leads to the hypo-coercivity. This drives the phase diameter into a small region after finite time in a large coupling regime, and the uniform boundedness of the diameter eventually leads to the emergence of exponentially fast synchronization.