# Configurational stability for the Kuramoto-Sakaguchi model

## TL;DR

This paper investigates the stability of phase-locked states in the Kuramoto-Sakaguchi model, a variation of the Kuramoto model with added phase-lag, providing new stability criteria and numerical insights.

## Contribution

It offers new sufficient conditions for stability and instability of phase-locked configurations in the Kuramoto-Sakaguchi model, extending understanding beyond the gradient case.

## Key findings

- Provided a sufficient condition for stability of phase-locked states.
- Derived a sufficient condition for instability, including an unstable manifold count.
- Presented numerical results for small and large oscillator networks.

## Abstract

The Kuramoto--Sakaguchi model is a modification of the well-known Kuramoto model that adds a phase-lag paramater, or "frustration" to a network of phase-coupled oscillators. The Kuramoto model is a flow of gradient type, but adding a phase-lag breaks the gradient structure, significantly complicating the analysis of the model. We present several results determining the stability of phase-locked configurations: the first of these gives a sufficient condition for stability, and the second a sufficient condition for instability. (In fact, the instability criterion gives a count, modulo 2, of the dimension of the unstable manifold to a fixed point and having an odd count is a sufficient condition for instability of the fixed point.) We also present numerical results for both small and large collections of Kuramoto--Sakaguchi oscillators.