Finite Size Scaling in the Kuramoto Model

TL;DR

This paper studies how the synchronization properties of the Kuramoto model scale with system size, revealing a finite-size scaling law near the locking threshold and clarifying the convergence to the infinite system limit.

Contribution

It provides a detailed finite-size scaling analysis of the order parameter and Lyapunov exponent in the Kuramoto model near the locking threshold, including numerical confirmation and spectral insights.

Findings

Both the order parameter and Lyapunov exponent scale as (K-K_L)^{1/2} near the threshold.

The coupling range for this scaling shrinks as N^{-1.5} with increasing system size.

The infinite-N behavior follows a (K-K_L)^{2/3} scaling away from the critical region.

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 $\alpha\approx1.5$ as $N \rightarrow \infty$. Away from this interval, the order parameter exhibits the infinite-$N$ behavior $r-r_L \sim (K-K_L)^{2/3}$ proposed by Paz\'o [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 $N \rightarrow \infty$ limit in the Kuramoto model.