Collective behavior of globally coupled Langevin equations with colored noise in the presence of stochastic resonance

The long-time collective behavior of globally coupled Langevin equations in a dichotomous fluctuating potential driven by a periodic source is investigated. By describing the collective behavior using the moments of the mean field and single-particle displacements, we study stochastic resonance and synchronization using the exact steady-state solutions and related stability criteria. Based on the simulation results and the criterion of the stationary regime, the notable differences between the stationary and nonstationary regimes are demonstrated. For the stationary regime, stochastic resonance with synchronization is discussed, and for the nonstationary regime, the volatility clustering phenomenon is observed.

Figure 1

The realization of system Eq. (1) and the corresponding noise realization with $a=8$, $A=1$, $\mathrm{\Omega }=1.5\pi$, $v=1$, and $\epsilon =1$ for different numbers of particles. Figures 1 and 1 are depicted with $N=16$ and ${\sigma }^2=a^2+0.5av$. Figures 1 and 1 are depicted with $N=64$ and ${\sigma }^2=a^2+1.5av$. In the top panels of Figs. 1 and 1, the colored solid lines depict the trajectories of the particles and the open circles depict the corresponding mean field. In the bottom panel of Figs. 1 and 1, the solid lines depict the corresponding noise realization of the top panels of Figs. 1 and 1, respectively. The realizations near the initial time ($t=0$) in the top panels of Figs. 1 and 1 are shown in Figs. 1 and 1, respectively. Different colored lines indicate different particle displacements.

Figure 2

The realization of the mean field in the nonstationary regime and the corresponding heavy-tailed distribution of the mean field. The parameters are $a=8$, $A=1$, $\mathrm{\Omega }=1.5\pi$, $\epsilon =1$, $v=1$, and ${\sigma }^2=a^2+1.5av$. Figure 2 shows ${10}^4$ realizations of the mean field. Figure 2 shows that the corresponding distribution of the realizations satisfied $S>0.1$ at $t=22$.

Figure 3

A plot of the phase diagram in the $v\text{−}\sigma$ plane and the output amplitude of the first moment of the mean field. The parameter region is plotted in Fig. 3 with $a=8,A=1,\mathrm{\Omega }=1.5\pi$. For the simulation result in Fig. 3, the value is defined by $Q=1-exp(-{\int }_0^T〈x^2〉dt/T)$, which takes a value from 0 (dark) to 1 (light). A lighter $Q$ value indicates a larger ${\int }_0^T〈x^2〉dt/T$, and $T=300$ was used for the simulation with time step 0.003. For the analytical result in Fig. 3, the red solid line is the critical line between the stationary regime and the nonstationary regime, and the red dotted line is the critical line for the emergence of SR in the stationary regime. In Fig. 3, the output amplitude of the first moment is plotted for $v\approx 0.88<{\mathrm{\Omega }}^2/a$ (blue), $v=2.73\approx {\mathrm{\Omega }}^2/a$ (black), and $v\approx 3.22>{\mathrm{\Omega }}^2/a$ (red); for the analytical results, the solid part means the output amplitude in the stationary regime, and the dotted part means the output amplitude in the nonstationary regime; the open circle lines stand for the corresponding numerically estimated results. All the estimated results are obtained from ${10}^4$ realizations.