Coletta Tommaso, Delabays Robin, Jacquod Philippe
School of Engineering, University of Applied Sciences of Western Switzerland, CH-1951 Sion, Switzerland.
Section de Mathématiques, Université de Genève, CH-1211 Genève, Switzerland.
Phys Rev E. 2017 Apr;95(4-1):042207. doi: 10.1103/PhysRevE.95.042207. Epub 2017 Apr 13.
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.
我们研究了具有有限数量(N)个振子的Kuramoto模型中完全锁定状态下序参量和最大非零Lyapunov指数的标度性质。我们表明，对于(N)的任何有限值，当耦合强度(K)足够接近锁定阈值(K_{L})时，这两个量都按((K - K_{L})^{1/2})标度。我们对振子频率在区间([-1,1])内均匀分布的情况进行了数值验证，此外还发现，随着(N \to \infty)，这种标度有效的耦合范围(\delta K)像(\delta K \sim N^{-\alpha})一样收缩，其中(\alpha \approx 1.5)。在该区间之外，序参量表现出Pazó提出的无限(N)行为(r - r_{L} \sim (K - K_{L})^{2/3}) [《物理评论E》72, 046211 (2005)] PLEEE81539 - 375510.1103/PhysRevE.72.046211。我们认为这两种行为之间的交叉发生是因为在锁定阈值处，随着(N)增加，完全锁定状态谱的连续部分的上限趋近于零。我们的结果阐明了Kuramoto模型中向(N \to \infty)极限的收敛情况。
