Stochastic resonance in a trapping overdamped monostable system

The response of a trapping overdamped monostable system to a harmonic perturbation is analyzed, in the context of stochastic resonance phenomenon. We consider the dynamics of a Brownian particle moving in a piecewise linear potential with a white Gaussian noise source. Based on linear-response theory and Laplace transform technique, analytical expressions of signal-to-noise ratio (SNR) and signal power amplification (SPA) are obtained. We find that the SNR is a nonmonotonic function of the noise intensity, while the SPA is monotonic. Theoretical results are compared with numerical simulations.

Figure 1

(Color online) Trapping piecewise linear monostable potential $\Phi \left(x,t\right)$ [see Eqs. (2, 3)] for the following set of parameter values: $k_1=20$, $k_2=1$, and $L=0.1$. Amplitude and frequency of the periodical forcing are $A=0.3$ and $\omega =0.1$, respectively. The potential is shown at three different times: $t=0$ (black solid line), $t=T/4$ (red long dashed line), and $t=T/2$ (green medium dashed-dotted line), with $T=2\pi /{\omega }_0$. The typical parameter of the potential profile is $H=k_{1}L=2$.

Figure 2

(Color online) SNR [Eq. (5)] of the system output of Eq. (1) versus dimensionless noise intensity $q/H$, for the particle position $x\left(t\right)$ moving along the trapping monostable potential of Fig. 1. The amplitude and the frequency of the driving periodic field are $A=0.3$ and ${\omega }_0=0.1$, respectively. Curve 1 (green solid line) corresponds to $k_1=20$, $k_2=1$, $L=0.1$ (potential profile 1) and curve 2 (red solid line) corresponds to $k_1=25$, $k_2=3$, $L=0.2$ (potential profile 2). The solid lines are the theoretical results obtained from Eq. (5), while the dots are the results of simulations obtained by numerical integration of Eq. (1).

Figure 3

Output signal power and output noise power as functions of dimensionless input noise intensity $q/H$ for two potential profiles, as computed from numerical integration of Eq. (1). (a) Potential profile 1 ($k_1=20$, $k_2=1$, $L=0.1$); (b) potential profile 2 ($k_1=25$, $k_2=3$, $L=0.2$). The values of other parameters are the same as in Fig. 2.

Figure 4

(Color online) SPA [Eq. (9)] of the system output of Eq. (1) versus dimensionless noise intensity $q/H$, for the particle position $x\left(t\right)$ moving along the trapping monostable potential of Fig. 1. The amplitude and the frequency of the driving periodic field are $A=0.3$ and ${\omega }_0=0.1$, respectively. Curve 1 (green solid line) corresponds to $k_1=20$, $k_2=1$, $L=0.1$ (potential profile 1) and curve 2 (red solid line) corresponds to $k_1=25$, $k_2=3$, $L=0.2$ (potential profile 2). The solid lines are the theoretical results obtained from Eq. (9), while the dots are the results of simulations obtained by numerical integration of Eq. (1).

Figure 5

(Color online) The two shaded areas of the parameter space $\left(k_1,k_2\right)$ indicate the regions where maximum of SPA or SNR can be observed. The amplitude and the frequency of the driving periodic field are $A=0.3$ and ${\omega }_0=0.1$, respectively. In the monostable system of Eq. (3), the maximum of SPA is observed for $k_1<0.305k_2$ $\left(k_2>3.27k_1\right)$, corresponding to confining monostable potentials; while the maximum of SNR is observed for $k_1>6.25k_2$ $\left(k_2<0.16k_1\right)$, corresponding to trapping monostable potentials. In the central white area, which is for $0.16k_1<k_2<3.27k_1$, SNR and SPA have monotonic behavior as a function of the dimensionless noise intensity $q/H$.

Figure 6

PD for the position $x$ of the Brownian particle moving along the trapping monostable potential profile 1 of Fig. 1 ($k_1=20$, $k_2=1$, $L=0.1$). The PD is shown at time $t=t_{max}=628$, for three different values of the noise intensity, namely, $q/H=0.15,0.35,1$. The first and third noise intensities correspond to small values of SNR, while the second value gives the maximum of SNR (see green solid line and blue circle in Fig. 2). Amplitude and frequency of the input periodical signal are $A=0.3$ and ${\omega }_0=0.1$, respectively.

Figure 7

PD for the position $x$ of the Brownian particle moving along the trapping monostable potential profile 2 ($k_1=25$, $k_2=3$, $L=0.2$). The PD is shown at time $t=t_{max}=628$, for three different values of the noise intensity, namely, $q/H=0.1,0.45,1$. The first and third noise intensities correspond to small values of SNR, while the second value gives the maximum of SNR (see red solid line and black circle in Fig. 2). Amplitude and frequency of the input periodical signal are $A=0.3$ and ${\omega }_0=0.1$, respectively.