Stochastic Resonance in a Multistable System Driven by Gaussian Noise

Stochastic resonance (SR) is investigated in amultistable system driven byGaussian white noise. Using adiabatic elimination theory and three-state theory, the signal-to-noise ratio (SNR) is derived.We find the effects of the noise intensity and the resonance system parameters b, c, and d on the SNR; the results show that SNR is a nonmonotonic function of the noise intensity; therefore, a multistable SR is found in this system, and the value of the peak changes with changing the system parameters.


Introduction
Stochastic resonance (SR) is first introduced by Benzi et al. [1] in 1981.In the past decades, SR has received considerable attention in the field of meteorology, and the topic has flourished in physics and neuroscience and weak signal detection [2][3][4][5][6].
In order to describe SR, McNamara and Wiesenfeld [7] introduced the signal-to-noise ratio, which is often used as an indicator of signal processing performance.Numerous studies have been developed to explain SR in continuous time using tools of statistical physics.
Literature [25] studied a solution of Kramers turnover problem for the case of two symmetric deep wells connected through a single shallow well; literature [26] analysed the occurrence of vibrational resonance in a damped quantic oscillator with double-well and triple-well potentials driven by both low-frequency force and high-frequency force; the splitting of the Kramers escape rate in an overdamped system with a triple-well potential was studied in [27].
The paper is organized as follows.In Section 2, we present the model for the multistable system.Then, the expression of the signal-to-noise ratio is derived.In Section 3, the effects of noise intensity and the resonance system parameters , , and  on SNR are discussed.A discussion of the effects concludes the paper in Section 4.

SNR of Multistable SR
The model of multistable SR is a multistable nonlinear system driven by periodic signal and white noise.The equation can be written as follows: where () =  cos(2) is the input signal,  is the periodic signal amplitude,  is the driving frequency, () = √ 2() in which  is the noise intensity, and () represents a Gaussian white noise with zero mean and unit variance.() is the multistable SR output signal.The potential function for the above multistable system can be denoted as [21,22] where , , and  are system parameters.As shown in Figure 1, the potential function () is symmetrical and has three stable points (− 2 ,  0 and  2 ) and two unstable points (− 1 ,  1 ): From ( 1) and ( 2), the Fokker-Planck equation [26] is given by Formula (4) contains nonlinear components, so it cannot obtain the steady state solution.
When the input signal and noise intensity are very small, The whole  area can be divided into three attraction domains; the first is the attraction domain of the steady-state solution  = − √ (−1/2)( − √  2 − 4), the second is the attraction domain of the steady-state solution  = 0, and the last is the attraction domain of the steady-state solution  = √ (−1/2)( − √  2 − 4).In the three attraction domains, the total probability of them contains, respectively [20], Obviously,  1 () +  2 () +  3 () = 1, when the frequency of input signal is very low In the condition of adiabatic approximation, we can get the master equation for the probability of exchange among the three quantities by simplifying (3): where  1,2,3 () are the escape rate [7].They are considered as function of a weak periodic signal  cos(2), when  ≪ 1, under the adiabatic approximation, the escape rate of  1,2,3 () series expansion, ignoring the higher order terms, you can get the following expression: then, Equations ( 8) can be solved as where When  0 → −∞,  1,2,3 () approaches   1,2,3 (): Let   ( +  | , ) donate the probability to the system which is in  area at  moment when it is in  area at  +  moment (,  = 1, 2, 3): In the progressive state, the correlation function of random variable is given by The correlation function is not only related with the time interval but also related with the start value of the time.So we take the average value of the correlation function Within the deduction made above, the output power spectral density of a multistable SR system can be obtained: where  1 () and  2 () are the power spectral densities of the output signal and the output noise, which are derived from the periodic input signal and the noise, respectively, as follows: Put  cos(2) as constant processing; we can get the steady state solution of the available equation ( 4), the potential function of Φ(): The probability transition rate of type 1 can be obtained: Make cos(2)=0 , To clearly describe the energy distribution of the system output, the SNR of the system output can be calculated as follows:

The Effects of the Noise Intensity and System Parameters
In this section, we discuss the effect of each parameter on the system SNR.
Figure 2 shows the change trends of the SNR of a multistable SR method with  = 0.52,  = −0.31,and  = 0.04 versus noise intensity .
It can be seen from Figure 2 that the change curve of the SNR is first increased and then decreased with the variation in noise intensity ; therefore, there exists an optimal noise for the maximum SNR.This typical phenomenon is a signature of multistable SR.Noise plays a role in the SNR within certain range of scale.
The SNR as a function of noise intensity  with different system parameters  is shown in Figure 3.It is seen that the positions of the higher peaks and the lower peaks are both shifting to the left with the increase of  and the SNR is decreasing with the increase of .
Figure 4 shows the curves of SNR versus noise intensity  with different system parameters .With the increase of , the whole curves are shifting to left and SNR is increasing.
Figure 5 shows the curves of SNR versus noise intensity  with different system parameters .With the increase of , the whole curves are shifting to the left and the SNR is increasing.

The Simulation
Take the same parameters as in Figure 2 to detect the weak signal with the multistable stochastic resonance and then let  take different values; and the amplitude of the corresponding characteristic frequency is recorded; finally, the curve of amplitude versus the noise is made.It can be seen that the simulation result in Figure 6 is consistent with the analysis in Figure 2.
Take the same parameters as in Figure 3 to detect the weak signal with the multistable stochastic resonance.First, take  equal to 0.4 and let  take  different values; then, the amplitude of the corresponding characteristic frequency is recorded and the curve of amplitude versus the noise is finally made.Second, take  equal to 0.45 and 0.5 and repeat the above operation, respectively.It can be seen that the simulation result in Figure 7 is consistent with the analysis in Figure 3.
Take the same parameters as in Figure 4 to detect the weak signal with the multistable stochastic resonance.First, take  equal to −0.31 and let  take  different values; then, the amplitude of the corresponding characteristic frequency is recorded and the curve of amplitude versus the noise is finally made.Second, take  equal to −0.3 and −0.29 and repeat the above operation, respectively.It can be seen that the simulation result in Figure 8 is consistent with the analysis in Figure 4.
Take the same parameters as in Figure 5 to detect the weak signal with the multistable stochastic resonance.First, take  equal to 0.03 and let  take  different values; then, the amplitude of the corresponding characteristic frequency is recorded and the curve of amplitude versus the noise is finally made.Second, take  equal to 0.036 and 0.042 and repeat the above operation, respectively.It can be seen that the simulation result in Figure 9 is consistent with the analysis in Figure 5.

Conclusion
In the paper, we first derive the expression of the multistable system SNR.Through the research about the effects of Gauss noise and system parameters on the multistable system SNR, we can draw the following conclusions: (1) the SNR expression is applicable to arbitrary signal amplitude; (2) the curve of the SNR versus noise intensity is nonmonotonic, which is a typical phenomenon of multistable SR; (3) the SNR peak is increasing gradually with the increase of system parameters  and , but it is decreasing with the increase of system parameters .The SNR as a function of system parameters , , and  will not be described in this paper.

Figure 7 :Figure 8 :
Figure 7: The variation curve of the output signal amplitude with the addition of noise  under different  value.

Figure 9 :
Figure 9: The variation curve of the output signal amplitude with the addition of noise  under different  value.