Stable fixed points in the Kuramoto model

TL;DR

This paper establishes a necessary condition for stable fixed points in the Kuramoto model and demonstrates that in the complete network, the homogeneous model lacks non-zero stable fixed points, supporting existing conjectures.

Contribution

It introduces a necessary condition for stable fixed points and proves the absence of non-zero stable fixed points in the complete homogeneous Kuramoto model.

Findings

Homogeneous complete network has no non-zero stable fixed points.

Supports the conjecture that the zero fixed point attracts almost all states.

Provides a necessary condition for stability in the Kuramoto model.

Abstract

We develop a necessary condition for the existence of stable fixed points for the general network Kuramoto model, and use it to show that for the complete network the homogeneous model has no non-zero stable fixed point solution. This result provides further evidence that in the homogeneous case the zero fixed point has an attractor set consisting of the entire space minus a set of measure zero, a conjecture of Verwoerd and Mason (2007).