Finite Size Scaling in the Kuramoto Model

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 $\delta K$ over which this scaling is valid shrinks like $\delta K\sim N^{-\alpha }$ with $\alpha \approx 1.5$ as $N\to \infty$. Away from this interval, the order parameter exhibits the infinite-$N$ behavior $r-r_L\sim (K-K_L{)}^{2/3}$ proposed by Pazó [Phys. Rev. E 72, 046211 (2005)]. 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 $N\to \infty$ limit in the Kuramoto model.

arXiv comment7 pages, 5 figures

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Cited by in corpus (7)

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• Low Dimensional Dynamics of the Kuramoto Model with Rational Frequency Distributions
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• Universality of critical dynamics on a complex network
• Data-driven stochastic modeling of coarse-grained dynamics with finite-size effects using Langevin regression