The stability of fixed points for a Kuramoto model with Hebbian interactions

TL;DR

This paper analyzes the stability of fixed points in a modified Kuramoto model with dynamic, Hebbian-like coupling, showing that their stability can be directly related to the classical fixed-coupling Kuramoto model, especially in all-to-all networks.

Contribution

It establishes a direct relationship between the fixed points and stability of the dynamic-coupling Kuramoto model and the classical fixed-coupling model, simplifying stability analysis.

Findings

Fixed points and stability can be expressed in terms of classical Kuramoto model.

For all-to-all networks, the problem reduces to a known classical Kuramoto problem.

The stability analysis leverages existing solutions for the classical model.

Abstract

We consider a variation of the Kuramoto model with dynamic coupling, where the coupling strengths are allowed to evolve in response to the phase difference between the oscillators, a model first considered by Ha, Noh and Park. In particular we study the stability of fixed points for this model. We demonstrate a somewhat surprising fact: namely that the fixed points of this model, as well as their stability, can be completely expressed in terms of the fixed points and stability of the analogous classical Kuramoto problem where the coupling strengths are fixed to a constant (the same for all edges). In particular for the "all-to-all" network, where the underlying graph is the complete graph, the problem reduces to the problem of understanding the fixed points and stability of the all-to-all Kuramoto model with equal edge weights, a problem that has been completely solved.