Invariant cone and synchronization state stability of the mean field models

TL;DR

This paper proves the stability of synchronized states in mean field oscillator models like Winfree and Kuramoto, using invariant cone methods, regardless of oscillator count or frequency distribution.

Contribution

It introduces the positive invariant cone approach to establish synchronization stability in mean field models independently of system size and frequency distribution.

Findings

Stability of synchronized states proven for Winfree model.

Method applicable to other mean field models like Kuramoto.

Results hold regardless of oscillator number and frequency distribution.

Abstract

In this article we prove the stability of mean field systems as the Winfree model in the synchronized state. The model is governed by the coupling strength parameter $\kappa$ and the natural frequency of each oscillator. The stability is proved independently of the number of os-cillators and the distribution of the natural frequencies. In order to prove the main result, we introduce the positive invariant cone and we start by studying the linearized system. The method can be applied to others mean field models as the Kuramoto model.