# Stochastic Resonance in a Multistable System Driven by Gaussian Noise

Authors: Peiming Shi; Pei Li; Shujun An; Dongying Han
Journal: Discrete Dynamics in Nature and Society
 (2016)

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## Abstract

Stochastic resonance (SR) is investigated in a multistable system driven by Gaussian 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 parametersb, 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.


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## Body

## 1. 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 [2–6].There have been many theoretical developments of SR in conventional bistable systems [7–12]. Recently, there have appeared some extensions of SR, such as stochastic resonance in a harmonic oscillator [13], ghost stochastic resonance in the FitzHugh-Nagumo neuron model [14, 15], Transition in a Bistable Duffing System [16], time delay SR [17], trichotomous noise induced SR in a linear system [18], and superthreshold SR [19]. Literature [20–22] proposes a new model of multistable system. However, [7–22] did not study the SNR. In this paper, we use the model of multistable system driven by periodic signal and white noise which can realize the maximum utilization of noise and obtain better detection effects. So it is necessary to discuss the SNR of the multistable system.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 Section2, 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 b, c, and d on SNR are discussed. A discussion of the effects concludes the paper in Section 4.


## 2. 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:(1)
d
x
d
t
=
-
d
U
x
d
x
+
s
t
+
η
t
,where s
t
=
A
cos
⁡
2
π
f
t is the input signal, A is the periodic signal amplitude, f is the driving frequency, η
(
t
)
=
2
D
ε
t in which D is the noise intensity, and ε
t represents a Gaussian white noise with zero mean and unit variance. x
t is the multistable SR output signal. The potential function for the above multistable system can be denoted as [21, 22](2)
U
x
=
b
2
x
2
+
c
4
x
4
+
d
6
x
6
,where b, c, and d are system parameters. As shown in Figure 1, the potential function U
x is symmetrical and has three stable points (-
x
2
,
x
0 and x
2) and two unstable points (
-
x
1
,
x
1
):(3)
x
0
=
0
,
x
1
=
-
1
2
d
c
+
c
2
-
4
b
d
,
x
2
=
-
1
2
d
c
-
c
2
-
4
b
d
.Figure 1
The multistable potential functionU
(
x
).From (1) and (2), the Fokker-Planck equation [26] is given by(4)
∂
ρ
x
,
t
∂
t
=
-
∂
∂
x
-
b
x
-
c
x
3
-
d
x
5
+
A
cos
⁡
2
π
f
t
ρ
x
,
t
+
D
∂
2
∂
x
2
ρ
x
,
t
.Formula (4) contains nonlinear components, so it cannot obtain the steady state solution.When the input signal and noise intensity are very small,(5)
A
≪
1
,
D
≪
1
.The wholex area can be divided into three attraction domains; the first is the attraction domain of the steady-state solution x
=
-
-
1
/
2
d
(
c
-
c
2
-
4
b
d
), the second is the attraction domain of the steady-state solution x
=
0, and the last is the attraction domain of the steady-state solution x
=
-
1
/
2
d
(
c
-
c
2
-
4
b
d
). In the three attraction domains, the total probability of them contains, respectively [20],(6)
P
1
t
=
∫
-
∞
-
x
1
ρ
x
,
t
d
x
,
P
2
t
=
∫
-
x
1
x
1
ρ
x
,
t
d
x
,
P
3
t
=
∫
x
1
+
∞
ρ
x
,
t
d
x
.Obviously,P
1
(
t
)
+
P
2
(
t
)
+
P
3
(
t
)
=
1, when the frequency of input signal is very low(7)
f
≪
1
.In the condition of adiabatic approximation, we can get the master equation for the probability of exchange among the three quantities by simplifying (3):(8)
P
1
′
t
=
-
R
1
t
P
1
t
+
1
2
R
2
t
P
2
t
=
1
2
R
2
t
-
R
1
t
+
R
2
t
P
1
t
,
P
2
′
t
=
-
R
2
t
P
2
t
+
R
1
t
P
1
t
+
R
3
t
P
3
t
,
P
3
′
t
=
-
R
3
t
P
3
t
+
1
2
R
2
t
P
2
t
,where R
1,2
,
3
(
t
) are the escape rate [7]. They are considered as function of a weak periodic signal A
cos
⁡
(
2
π
f
t
), when A
≪
1, under the adiabatic approximation, the escape rate of R
1,2
,
3
(
t
) series expansion, ignoring the higher order terms, you can get the following expression:(9)
R
1
=
R
3
=
1
2
R
0
+
R
1
β
cos
⁡
2
π
f
t
+
R
2
β
2
cos
2
⁡
2
π
f
t
+
⋯
,
R
2
=
1
2
R
0
-
R
1
β
cos
⁡
2
π
f
t
+
R
2
β
2
cos
2
⁡
2
π
f
t
-
⋯
; then,(10)
R
1
+
R
2
=
R
0
+
R
2
β
2
cos
2
⁡
2
π
f
t
.Equations (8) can be solved as(11)
P
1
t
=
1
4
e
-
R
0
t
-
t
0
2
P
1
t
0
-
1
+
m
+
1
-
n
,
P
2
t
=
1
2
e
-
R
0
t
-
t
0
2
P
2
t
0
-
1
-
m
+
1
+
n
,
P
3
t
=
1
4
e
-
R
0
t
-
t
0
2
P
3
t
0
-
1
+
m
+
1
-
n
,where(12)
m
=
R
1
β
cos
⁡
2
π
f
t
0
-
θ
R
0
2
+
2
π
f
2
1
/
2
,
n
=
R
1
β
cos
⁡
2
π
f
t
-
θ
R
0
2
+
2
π
f
2
1
/
2
,
sin
⁡
θ
=
2
π
f
R
0
2
+
2
π
f
2
1
/
2
,
cos
⁡
θ
=
R
0
R
0
2
+
2
π
f
2
1
/
2
.When t
0
→
-
∞, P
1,2
,
3
(
t
) approaches P
1,2
,
3
s
(
t
):(13)
P
1
s
t
=
P
3
s
t
=
1
4
1
-
n
,
P
2
s
t
=
1
2
1
+
n
.LetP
i
t
+
τ
∣
j
,
t donate the probability to the system which is in j area at t moment when it is in i area at t
+
τ moment (
i
,
j
=
1
,
2
,
3
):(14)
P
1
t
+
τ
∣
1
,
t
=
1
4
e
-
R
0
τ
-
1
+
n
+
1
-
q
,
P
1
t
+
τ
∣
2
,
t
=
1
4
e
-
R
0
τ
1
+
n
+
1
-
q
,
P
2
t
+
τ
∣
2
,
t
=
1
2
e
-
R
0
τ
1
-
n
+
1
+
q
,
P
2
t
+
τ
∣
1
,
t
=
1
4
e
-
R
0
τ
-
1
-
n
+
1
+
q
,
P
2
t
+
τ
∣
3
,
t
=
1
4
e
-
R
0
τ
-
1
-
n
+
1
+
q
,
P
3
t
+
τ
∣
2
,
t
=
1
4
e
-
R
0
τ
1
+
n
+
1
-
q
,
P
3
t
+
τ
∣
3
,
t
=
1
4
e
-
R
0
τ
-
1
+
n
+
1
-
q
.In the progressive state, the correlation function of random variable is given by(15)
x
t
x
t
+
τ
=
lim
t
0
→
-
∞
⁡
∬
x
y
P
y
t
+
τ
∣
x
,
t
ρ
x
,
t
=
-
x
2
-
x
2
P
1
t
+
τ
∣
1
,
t
P
1
s
t
+
-
x
2
x
0
P
2
t
+
τ
∣
1
,
t
P
1
s
t
+
x
0
x
0
P
2
t
+
τ
∣
2
,
t
P
2
s
t
+
x
0
-
x
2
P
1
t
+
τ
∣
2
,
t
P
2
s
t
+
x
0
x
2
P
3
t
+
τ
∣
2
,
t
P
2
s
t
+
x
2
x
2
P
3
t
+
τ
∣
3
,
t
P
3
s
t
+
x
2
x
0
P
2
t
+
τ
∣
3
,
t
P
3
s
t
=
1
8
x
2
2
e
-
R
0
τ
-
1
+
2
n
-
n
2
+
1
-
q
-
n
+
n
q
+
1
4
x
0
2
e
-
R
0
τ
1
-
n
2
+
1
+
q
+
n
+
n
q
.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(16)
x
t
x
t
+
τ
Average
=
f
∫
0
1
/
f
x
t
x
t
+
τ
d
t
=
-
c
8
d
-
1
4
b
d
e
-
R
0
τ
-
1
-
R
1
2
β
2
2
R
0
2
+
2
π
f
2
+
1
+
R
1
2
β
2
cos
⁡
2
π
f
τ
2
R
0
2
+
2
π
f
2
+
-
1
8
d
c
+
c
2
-
4
b
d
e
-
R
0
τ
1
-
R
1
2
β
2
2
R
0
2
+
2
π
f
2
+
1
+
R
1
2
β
2
cos
⁡
2
π
f
τ
2
R
0
2
+
2
π
f
2
.Within the deduction made above, the output power spectral density of a multistable SR system can be obtained:(17)
S
w
=
∫
-
∞
+
∞
x
t
x
t
+
τ
Average
e
-
i
w
τ
d
τ
=
S
1
w
+
S
2
w
,where S
1
(
w
) and S
2
(
w
) 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: (18)
S
1
w
=
-
c
8
d
-
1
4
b
d
-
c
+
c
2
-
4
b
d
8
d
·
π
R
1
2
β
2
2
R
0
2
+
2
π
f
2
δ
w
-
2
π
f
,
S
2
w
=
c
8
d
+
1
4
b
d
-
c
+
c
2
-
4
b
d
8
d
·
2
R
0
R
0
2
+
w
2
.PutA
cos
⁡
(
2
π
f
t
) as constant processing; we can get the steady state solution of the available equation (4), the potential function of Φ
(
x
):(19)
Φ
x
=
b
2
x
2
+
c
4
x
4
+
d
6
x
6
-
A
x
cos
⁡
2
π
f
t
.The probability transition rate of type 1 can be obtained:(20)
R
1
t
=
U
″
-
x
1
U
″
-
x
2
1
/
2
2
π
exp
⁡
-
Φ
-
x
1
-
Φ
-
x
2
D
.Make x
1
=
0,(21)
x
0
=
-
-
1
2
d
c
+
c
2
-
4
b
d
,
x
2
=
-
1
2
d
c
-
c
2
-
4
b
d
-
-
1
2
d
c
+
c
2
-
4
b
d
,
R
0
=
2
·
R
1
t
A
cos
⁡
2
π
f
t
=
0
=
-
9
b
2
-
4
b
c
2
/
d
+
25
b
2
/
d
2
π
·
e
-
-
b
c
2
-
4
b
d
/
2
d
+
c
2
c
2
-
4
b
d
/
4
d
2
+
-
c
2
c
2
-
4
b
d
-
c
3
+
4
b
c
d
/
3
/
D
,
1
2
R
1
=
-
d
R
1
t
d
A
cos
⁡
2
π
f
t
A
cos
⁡
2
π
f
t
=
0
,
R
1
β
=
R
0
A
x
2
-
x
1
D
.To clearly describe the energy distribution of the system output, the SNR of the system output can be calculated as follows:(22)
SNR
=
∫
0
∞
S
1
w
d
w
S
2
w
=
2
π
f
=
-
c
/
8
d
-
1
/
4
b
/
d
-
c
+
c
2
-
4
b
d
/
8
d
c
/
8
d
+
1
/
4
b
/
d
-
c
+
c
2
-
4
b
d
/
8
d
·
π
R
0
A
2
-
c
/
d
-
2
b
/
d
4
D
2
.


## 3. The Effects of the Noise Intensity and System Parameters

In this section, we discuss the effect of each parameter on the system SNR.Figure2 shows the change trends of the SNR of a multistable SR method with b
=
0.52, c
=
-
0.31, and d
=
0.04 versus noise intensity D.Figure 2
SNR versus noise intensityD with b
=
0.52, c
=
-
0.31, and d
=
0.04.It can be seen from Figure2 that the change curve of the SNR is first increased and then decreased with the variation in noise intensity D; 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 intensityD with different system parameters b 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  b and the SNR is decreasing with the increase of b.Figure 3
SNR versus noise intensityD for different system parameters b: 0.4, 0.45, and 0.5. Other parameters are c
=
-
0.31 and d
=
0.04.Figure4 shows the curves of SNR versus noise intensity D with different system parameters c. With the increase of c, the whole curves are shifting to left and SNR is increasing.Figure 4
SNR versus noise intensityD for different system parameters c: −0.31, −0.3, and −0.29. Other parameters are b
=
0.52 and d
=
0.04.Figure5 shows the curves of SNR versus noise intensity D with different system parameters d. With the increase of d, the whole curves are shifting to the left and the SNR is increasing.Figure 5
SNR versus noise intensityD for different system parameters d: 0.03, 0.036, and 0.042. Other parameters are b
=
0.52 and c
=
-
0.31.


## 4. The Simulation

Take the same parameters as in Figure2 to detect the weak signal with the multistable stochastic resonance and then let D 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.Figure 6
The variation curve of the output signal amplitude with the addition of noiseD.Take the same parameters as in Figure3 to detect the weak signal with the multistable stochastic resonance. First, take b equal to 0.4 and let D take N 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 b 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.Figure 7
The variation curve of the output signal amplitude with the addition of noiseD under different b value.Take the same parameters as in Figure4 to detect the weak signal with the multistable stochastic resonance. First, take c equal to −0.31 and let D take N 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 c 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.Figure 8
The variation curve of the output signal amplitude with the addition of noiseD under different c value.Take the same parameters as in Figure5 to detect the weak signal with the multistable stochastic resonance. First, take d equal to 0.03 and let D take N 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 d 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.Figure 9
The variation curve of the output signal amplitude with the addition of noiseD under different d value.


## 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 parametersc and d, but it is decreasing with the increase of system parameters b. The SNR as a function of system parameters b, c, and d  will not be described in this paper.



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