Abstract
We study phase locking in the Kuramoto model of coupled oscillators in the special case where the number of oscillators, , 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 , the exact value of the locking threshold was calculated 30 years ago; however, the leading corrections to it for finite have remained unsolved analytically. Here we derive an asymptotic formula for the locking threshold when . The leading correction to the infinite- result scales like either or , depending on whether the frequencies are evenly spaced according to a midpoint rule or an end-point rule. These scaling laws agree with numerical results obtained by Pazó [D. Pazó, Phys. Rev. E 72, 046211 (2005)]. Moreover, our analysis yields the exact prefactors in the scaling laws, which also match the numerics.
- Received 11 April 2016
DOI:https://doi.org/10.1103/PhysRevE.93.062220
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