Kuramoto model with uniformly spaced frequencies:Finite-N asymptotics of the locking threshold

TL;DR

This paper derives asymptotic formulas for the finite-size corrections to the locking threshold in the Kuramoto model with evenly spaced frequencies, providing exact scaling laws and prefactors that match numerical simulations.

Contribution

It analytically determines the leading finite-N corrections to the Kuramoto model's locking threshold for evenly spaced frequencies, including exact prefactors.

Findings

Leading correction scales as N^{-3/2} or N^{-1} depending on frequency spacing rule.

Derived exact asymptotic formulas match numerical results.

Provides precise prefactors for finite-N corrections.

Abstract

We study phase locking in the Kuramoto model of coupled oscillators in the special case where the number of oscillators, $N$, is large but finite, and the oscillators' natural frequencies are evenly spaced on a given interval. In this case, stable phase-locked solutions are known to exist if and only if the frequency interval is narrower than a certain critical width, called the locking threshold. For infinite $N$, the exact value of the locking threshold was calculated 30 years ago; however, the leading corrections to it for finite $N$ have remained unsolved analytically. Here we derive an asymptotic formula for the locking threshold when $N \gg 1$. The leading correction to the infinite-$N$ result scales like either $N^{-3/2}$ or $N^{-1}$, depending on whether the frequencies are evenly spaced according to a midpoint rule or an endpoint rule. These scaling laws agree with numerical results obtained by Paz\'{o} [Phys. Rev. E 72, 046211 (2005)]. Moreover, our analysis yields the exact prefactors in the scaling laws, which also match the numerics.