Noise-induced synchronization of a large population of globally coupled nonidentical oscillators

We study a large population of globally coupled phase oscillators subject to common white Gaussian noise and find analytically that the critical coupling strength between oscillators for synchronization transition decreases with an increase in the intensity of common noise. Thus, common noise promotes the onset of synchronization. Our prediction is confirmed by numerical simulations of the phase oscillators as well as of limit-cycle oscillators.

Figure 1

(Color online) Numerical results for the phase model given by Eq. (1). Crosses (black) and open circles (orange) represent numerical data for $D=0$ and $D=0.02$, respectively. (a) Snapshot of the phase distribution for $K=1.99$. (b) Distribution of $A$ for (b-1) $K=1.96$, (b-2) $K=1.99$, and (b-3) $K=2.1$. Lines on the points are fitting curves. Histograms and curves are normalized for the maximum of curves to be 1. Point-dashed line (blue) and dashed line (green) represent the numerically identified $A_{\text{max}}$ for $D=0$ and $D=0.02$, respectively.

Figure 2

(Color online) $A_{\text{max}}$ for the phase model as a function of $K$. Crosses (black), open circles (orange) and filled circles (red) indicate the numerically identified $A_{\text{max}}$ for $D=0$, $D=0.02$, and $D=0.04$, respectively. Error bars represent the variance of $A_{\text{max}}$ for ten trials with different initial conditions and different noise processes. Point-dashed line (black), dashed line (orange), and a solid line (red) represent Eq. (12) for $D=0$, $D=0.02$, and $D=0.04$, respectively.

Figure 3

(Color online) Distribution of the long-time averaged frequencies of the phase oscillators for $K=2.02$. Crosses (black) and open circles (orange) with connecting lines are the numerical results for $D=0$ and $D=0.02$, respectively.

Figure 4

(Color online) Numerical results for the limit-cycle oscillators given by Eq. (14). Legends for (a) and (b) are the same as those in Figs. 2, 3, respectively. (a) Order parameter $A_{\text{max}}$ as a function of $K$. Lines represent Eq. (12). (b) Distribution of the long-time averaged frequencies for $K=2.02$. We used the data from $t=5\times {10}^6$ to $t=10\times {10}^6$. $N=1000$, $ϵ=0.01$, and ${\omega }_i=0.1+ϵ\text{tan}\left\{i\frac{\pi }{N}-\left(N+1\right)\frac{\pi }{2N}\right\}$.