Approximate solution to the stochastic Kuramoto model

We study Kuramoto phase oscillators with temporal fluctuations in the frequencies. The infinite-dimensional system can be reduced in a Gaussian approximation to two first-order differential equations. This yields a solution for the time-dependent order parameter, which characterizes the synchronization between the oscillators. The known critical coupling strength is exactly recovered by the Gaussian theory. Extensive numerical experiments further show that the analytical results are very accurate below and sufficiently above the critical value. We obtain the asymptotic order parameter in closed form, which suggests a tighter upper bound for the corresponding scaling. As a last point, we elaborate the Gaussian approximation in complex networks with distributed degrees.

Figure 1

Decay of stationary Fourier amplitudes with order $n$. For clarity, points are connected by lines. Gaussian theory is compared with the numerical truncation of Eq. (7) at $n=100$. Stationary values are taken at $t=500$ ($D=0.4$). Coupling strengths are $K=\{1.2K_c,2K_c,3.5K_c\}$. The inset shows the same data with different axes.

Figure 2

Time evolution of Kuramoto order parameter with initial condition $r\left(0\right)=r^g\left(0\right)=0.373$. (a1) compares theory and numerical experiments for different coupling strengths $K$ ($D=0.1$). The black dashed line shows the Gaussian theory $r^g\left(t\right)$, the blue dashed-dotted line the numerical integration of truncated Eqs. (7) at $n=30$, and the orange solid line the simulation of full dynamics (1) with initially Gaussian distributed phases. In (a2) a different noise intensity is used, $D=1.2$. A simulation of full dynamics is performed with system size $N=15000$ and time steps $0.05$ for $D=0.1$ and $0.005$ for $D=1.2$, respectively. (b) compares the accuracy between theory (dashed lines) and truncation of (7) at some $n$ (solid lines). (c) shows snapshots of phase distributions. For the theoretical lines the variance from Eq. (11) is used. The initial variance is ${\sigma }^2\left(0\right)=1.96$. For all snapshots, phases are shifted by the same amount so that mean values are located at $\varphi =\pi$. The parameters are the same as in (a1)'s uppermost curve.

Figure 3

Scaling of the asymptotic order parameter as a function of coupling strength divided by critical value $K/K_c$. In (a) theory (14) for $n=1$, $r_0^g$, is compared with numerical experiments [numerical integration of (1) with $N=5000$, $D=0.1$, integration step $0.05$, and time-averaged between $t=100$ and $t=200$; truncated system (7) is integrated until $t=1500$ and the final value for the order parameter is taken]. The inset shows square-root scaling slightly above critical coupling, which is not covered by the theory; the smallest value there $K-K_c=0.002$. In (b) theory is compared with the exact solution, Eq. (17), and with results from Ref. [20]. The inset shows a zoom-in. (c) compares levels of synchronization for various sources of disorder.