Linear stability of the incoherent solution and the transition formula for the Kuramoto-Daido model

TL;DR

This paper analyzes the linear stability of the incoherent state in the Kuramoto-Daido model, deriving the critical coupling strength for stability transition for arbitrary frequency distributions.

Contribution

It introduces a continuous model for the Kuramoto-Daido system and derives an explicit formula for the critical coupling strength $K_c$ for any frequency distribution.

Findings

Incoherent solution is stable when coupling strength is below $K_c$.

Incoherent solution becomes unstable when coupling strength exceeds $K_c$.

The transition point $K_c$ is explicitly derived for arbitrary frequency distributions.

Abstract

The Kuramoto-Daido model, which describes synchronization phenomena, is a system of ordinary differential equations on $N$-torus defined as coupled harmonic oscillators, whose natural frequencies are drawn from some distribution function. In this paper, the continuous model for the Kuramoto-Daido model is introduced and the linear stability of its trivial solution (incoherent solution) is investigated. Kuramoto's transition point $K_c$, at which the incoherent solution changes the stability, is derived for an arbitrary distribution function for natural frequencies. It is proved that if the coupling strength $K$ is smaller than $K_c$, the incoherent solution is asymptotically stable, while if $K$ is larger than $K_c$, it is unstable.