Emergence of synchronization in Kuramoto model with frustration under general network topology

TL;DR

This paper investigates how synchronization emerges in a generalized Kuramoto model with frustration on directed networks with a spanning tree, providing conditions for asymptotic synchronization based on initial data and network structure.

Contribution

It introduces a hierarchical analysis approach to establish synchronization conditions for the Kuramoto model on complex directed networks with frustration.

Findings

Synchronization occurs when initial phases are confined in a half circle.

A hierarchical dissipation mechanism explains the convergence to synchronized states.

The results apply to general digraphs with a spanning tree structure.

Abstract

In this paper, we will study the emergent behavior of Kuramoto model with frustration on a general digraph containing a spanning tree. We provide a sufficient condition for the emergence of asymptotical synchronization if the initial data is confined in half circle. As lack of uniform coercivity in general digraph, we apply the node decomposition criteria in \cite{H-L-Z20} to capture a clear hierarchical structure, which successfully yields the dissipation mechanism of phase diameter and a small invariant set after finite time. Then the dissipation of frequency diameter will be clear, which eventually leads to the synchronization.