Figure 1

The common-noise-induced phase synchronization. (a) The phase difference between two oscillators shows an exponential decay fluctuating around the theoretical behavior (solid line) derived from Eq. (7). Different curves correspond to different realizations of the random driving force $\xi (t)$. Here, $\mathit{Z}·\mathbf{\xi }=\sin (\varphi )\xi$, $\omega =1$, and $D=0.1$. (b) The Lyapunov exponent of the above synchronizing solution is shown as a function of the common-noise intensity (circles). The solid line shows an analytical result. (c) The Lyapunov exponent is calculated for the synchronizing solution of the Stuart-Landau oscillator. Numerical results are shown for both original (solid circles) and reduced (open squares) oscillator systems. The solid line represents a theoretical result. The parameter values are set as $c_0=2$, $c_2=1$, $D_{11}=D_{22}=D$, and $D_{12}=D_{21}=0$.Reuse & Permissions

Figure 2

The intermittent phase slips induced by uncommon additive noise sources to the oscillators defined by $\omega =1$, $\mathit{Z}·\mathbf{\xi }=\sin (\varphi )\xi$, and $\mathit{Z}·\mathbf{\eta }=\sin (\varphi )\eta$. The noise intensities are $D=0.1$ and $d=0.001$. (a) The distribution of the phase difference over a sufficiently long time. (b) The time evolution of the phase difference exhibits intermittent phase slips. (c) The distribution of the inter-phase-slip intervals shows a power law decay with an exponent of $-3/2$ (fitted by a solid line) and an exponential cutoff resulted from the additive noise term in Eq. (9).Reuse & Permissions

Figure 3

The unstable phase synchronization of integrate-and-fire models stimulated by a common-noise source. The phase-dependent sensitivity of this oscillator is discontinuous, and the noise-driven synchronization is not ensured. $I=2$ and no uncommon noise, $d=0$. (a) The Lyapunov exponent $\lambda$ is plotted as a function of the common-noise strength $D$. (b) The phase difference shows significant jitters, diffusing due to intermittent phase slips. Here and below, $D=0.1$. (c) The time evolution of the two phases displays temporarily synchronizing ($t<170$) and desynchronizing ($t>170$) states. (d) The distribution of the inter-phase-slip intervals shows an exponential decay with an exponent of $-3/2$ (fitted by a solid line).Reuse & Permissions