Finite-size scaling in the Kuramoto model

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.

Figure 1

Fourth order Runge-Kutta simulation results of the finite size Kuramoto model. Oscillator frequencies are distributed over the interval $[-1,1]$ according to the midpoint rule, Eq. (37), and $N=100$ [(a) and (c)] and 5000 [(b) and (d)]. The locking threshold $K_L^N$ is determined numerically with an accuracy of ${10}^{-9}$. The Runge-Kutta integration step is 0.005 and the maximal number of iterations is $5\times {10}^7$. (a) and (b) show the square root behavior of ${\lambda }_2$ as a function of $K-K_L^N$. The red lines give our theoretical prediction, Eq. (33), with no fitting parameter, the prefactor of Eq. (33) being computed numerically for the best estimate of the locking threshold obtained. The blue lines give the large $K$ asymptotics ${\lambda }_2=-K$. (c) and (d) show the square root behavior of $r-r_L$ as a function of $K-K_L^N$. The red lines give our theoretical prediction, Eq. (36), with no fitting parameter, the prefactor of Eq. (36) being computed numerically for the best estimate of the locking threshold obtained. The plots also present the infinite-$N$ limit results for $r-r_L$ as a function of the distance $K-K_L^{\infty }$ obtained by solving numerically the self consistent equation for the order parameter (crosses), as well as the $2/3$ scaling exponent prediction of Ref. [15] (green line).

Figure 2

Same as in Fig. 1 but for oscillator frequencies distributed according to the end-point rule, Eq. (38).

Figure 3

Behavior of $r-r_L$ as a function of $K-K_L^N$ for different $N$. The coupling range above the locking threshold for which $r-r_L\sim {(K-K_L^N)}^{1/2}$ decreases with the oscillator number. The dashed line indicating a $\sqrt{K-K_L}$ behavior is a guide to the eye.

Figure 4

Range $\delta K$ over which $r-r_L$ deviates from our theoretical prediction Eq. (36) by less than $5\%$ or $10\%$ as a function of the number of oscillators. Solid lines are best power-law fits.

Figure 5

Second and third largest nonvanishing Lyapunov exponents at the locking threshold, for midpoint and end-point frequency distributions [(a) and (b), respectively]. The solid and dashed lines give the interval defined by Eq. (42).