Finite-size scaling in the Kuramoto model

Physical Review. E
Tommaso ColettaPhilippe Jacquod

Abstract

We investigate the scaling properties of the order parameter and the largest nonvanishing Lyapunov exponent for the fully locked state in the Kuramoto model with a finite number N of oscillators. We show that, for any finite value of N, both quantities scale as (K-K_{L})^{1/2} with the coupling strength K sufficiently close to the locking threshold K_{L}. We confirm numerically these predictions for oscillator frequencies evenly spaced in the interval [-1,1] and additionally find that the coupling range δK over which this scaling is valid shrinks like δK∼N^{-α} with α≈1.5 as N→∞. Away from this interval, the order parameter exhibits the infinite-N behavior r-r_{L}∼(K-K_{L})^{2/3} proposed by Pazó [Phys. Rev. E 72, 046211 (2005)]PLEEE81539-375510.1103/PhysRevE.72.046211. We argue that the crossover between the two behaviors occurs because at the locking threshold, the upper bound of the continuous part of the spectrum of the fully locked state approaches zero as N increases. Our results clarify the convergence to the N→∞ limit in the Kuramoto model.

References

Jul 1, 1967·Journal of Theoretical Biology·A T Winfree
Jan 15, 1996·Physical Review Letters·K WiesenfeldS H Strogatz
Dec 31, 2005·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·Diego Pazó
May 16, 2007·Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics·D Dane QuinnSteven H Strogatz
Jan 16, 2013·Proceedings of the National Academy of Sciences of the United States of America·Florian DörflerFrancesco Bullo
Jul 15, 2016·Physical Review. E·Bertrand Ottino-Löffler, Steven H Strogatz

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Citations

Jan 21, 2021·Physical Review. E·Ryosuke YonedaYoshiyuki Y Yamaguchi

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