Phase-locked Patterns of the Kuramoto Model on 3-Regular Graphs

TL;DR

This paper investigates the existence and characteristics of non-synchronized phase-locked solutions in the Kuramoto model on 3-regular sparse networks, revealing multiple attractors and large angle differences.

Contribution

It demonstrates that most 3-regular networks support multiple stable phase-locked states and analyzes their basin structures and angle differences.

Findings

Most networks support multiple attracting phase-locked solutions.

Large graphs often have solutions with links exceeding π/2 angle difference.

The basin depths and widths of these solutions are characterized.

Abstract

We consider the existence of non-synchronized fixed points to the Kuramoto model defined on sparse networks: specifically, networks where each vertex has degree exactly three. We show that "most" such networks support multiple attracting phase-locked solutions that are not synchronized, and study the depth and width of the basins of attraction of these phase-locked solutions. We also show that it is common in "large enough" graphs to find phase-locked solutions where one or more of the links has angle difference greater than $\pi/2$.