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arXiv · 1512.02321

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

Bertrand Ottino-Loffler·Steven Strogatz
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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 Pazó [Phys. Rev. E 72, 046211 (2005)]. Moreover, our analysis yields the exact prefactors in the scaling laws, which also match the numerics.

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Bertrand Ottino-Loffler, Steven Strogatz. 2016-04-08. Kuramoto model with uniformly spaced frequencies:Finite-N asymptotics of the locking threshold. https://doi.org/10.1103/physreve.93.062220

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