# Performance Investigation of Stochastic Resonance in Three Types of Asymmetric Bistable System Driven by Trichotomous Noise

Authors: Si-Hai Zhao; Jiang-Ye Xu; Yu-Xiao Liu; Ze-Xing Zhao; Zhong-Shun Qin
Journal: Complexity
 (2020)

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## Abstract

This paper proposes a new system whose potential function is with three types of asymmetric potential wells, driven by trichotomous noise. Firstly, the three types of asymmetric bistable system are described in detail, and the changes of asymmetric bistable system potential function under different asymmetric factors are analyzed. Secondly, the effect of potential function parameters, asymmetric factorα, noise intensity D, and the probability of particle transition q is discussed, using numerical simulation. The detection effects of traditional symmetric SR and three types of asymmetric SR are observed and compared under the driving of trichotomous noise and periodic signals. The mean of signal-to-noise ratio gain is the indicator of the system's effectiveness on enhancing weak signal. The results indicate that it can make the detection effect of the asymmetric system better than that of the traditional bistable system by adjusting the parameters of the asymmetric stochastic resonance system and trichotomous noise.


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## Body

## 1. Introduction

Asymmetry is a ubiquitous phenomenon which has been a widespread concern for researchers for a long time due to it is potential applications in physical and natural systems, such as neural networks [1, 2], wireless communications [3], optical metamaterials [4], and medical imaging [5]. In addition, as demonstrated in a number of studies, the effect of stochastic resonance can be enhanced in the case that the potential function is not symmetric [6–10]. However, there are relatively few studies devoted to the asymmetric system SR driven by trichotomous noise.It is generally accepted that noise has a negative effect on the extraction and detection of weak signals. However, if the noise is used reasonably, the weak signal can be enhanced according to the stochastic resonance theory. Stochastic resonance is regarded as a method of weak signal detection [11–14] since it was firstly proposed to express the periodic alternations between the Earth’s ice age and the warm climate period in 1981 [15]. Later, various potential function models of stochastic resonance were proposed based on the model of traditional bistable stochastic resonance [16, 17], such as monostable models [18–20], tristable models [21–23], and two metastable models [24]. Then, the underdamping SR system [25–27] that is different from the traditional systems (overdamped system [28–30]) has been paid much attention by researchers. In addition, in the research of SR, the driving noise is various, including Gaussian white noise [31–33] and colored noise [34–37]. And it can be divided into multiplicative noise and additive noise [32, 38–40], according to the action mode of driving noise to the system.Much work so far focusing on stochastic resonance always takes the Gaussian white noise as the driving noise. However, it should be noted that most of the noise in a physical system is not gaussian white noise. Trichotomous noise is a function of the time variablet that is converted between the three values of a, b, c. Considerable research efforts have been devoted to SR phenomenon driven by trichotomous noise [41–44]. However, SR systems with asymmetric potential function driven by trichotomous noise have not been realized yet. Hence, the combination of trichotomous noise and asymmetric SR system presents a wide range of possibilities for the further development of weak signal detection.


## 2. Models and Methods

The Langevin equation of asymmetric stochastic resonance system is considered as follows:(1)∂xt∂t=−∂Ux∂x+Acos2πft+ξt,where Acos2πft is the system input signal to be measured, A is the amplitude of the input signal to be measured, f is the frequency of the input signal to be measured, ξt is the trichotomous noise, and Ux is the potential function of asymmetric systems. Ux is given as follows:(2)Uix=−a0x22+b0x44,x≥0,−a0Aix22+b0Bix44,x<0,where m,n,Ai,andBi are the structural parameters of asymmetric systems, and these parameters meet such inequalities: a0,b0,Ai,Bi>0 and i = 1, 2, 3.


### 2.1. Three Kinds of Asymmetric Stochastic Resonance Systems

#### 2.1.1. The Bistable System with Asymmetric Depth of Potential Well

WhenA1=B1=α, (α is asymmetric factor) and there is no input signal, the potential function contains two minima in the potential well: xmin±=±a0/b0 (stable points) and maxima in the potential barrier xmax=0 (unstable point). The widths of left and right potential well are both a0/b0. Then, the depths of the wells can be obtained as follows:(3)ΔUL=αa024b0,(4)ΔUR=a024b0.As can be seen from formulae (3) and (4), only the depth of the left well is controlled by the asymmetric factor α. Figure 1(a) shows the different depths of potential function caused by α when a0  = 1 and b0  = 1, where the depths of the left potential well increase with α increasing.Figure 1
(a) PotentialU1x for different values of α. (b) Potential U2x for different values of α. (c) Potential U3x for different values of α.
(a)(b)(c)


#### 2.1.2. The Bistable System with Asymmetric Width of Potential Well

WhenA2=1/α2, B2=1/α4, the potential function contains two stable points, xmin+=a0/b0 and xmin−=−αa0/b0, and an unstable point xmax=0. The depths of left and right potential well are both a02/4b0. Then, the widths of the wells can be obtained as follows:(5)ΔWL=αa0b0,(6)ΔWR=a0b0.As can be seen from formulae (5) and (6), the width of the left well is controlled by the asymmetric factor α. Figure 1(b) shows the different widths of the potential function caused by α when a0  = 1 and b0  = 1, where the widths of the left potential well increase with α increasing.


#### 2.1.3. The Bistable System with Both Asymmetric Width and Asymmetric Depth of Potential Well

WhenA3≠B3, A32≠B3, the bistable system has potential wells with both asymmetric width and depth. In this research, A3=1, B3=1/α4 (the combination of U1x and U2x) are selected as typical and the potential function contains two stable points, xmin+=a0/b0 and xmin−=−α2a0/b0, and an unstable point xmax=0. Then, the depths and widths of the wells can be obtained as follows:(7)ΔUL=α4a024b0,(8)ΔUR=a024b0,(9)ΔWL=α2a0b0,(10)ΔWR=a0b0.As can be seen from formulae (7)–(10), the left well is controlled by the asymmetric factor α. Figure 1(c) shows the differences of the potential function caused by α when a0  = 1 and a0  = 1, where the widths and depths of the left potential well increase with α increasing.


### 2.2. Method of Generating Trichotomous Noise

In (1), ξt is trichotomous noise, and the method of generating it will be introduced in this section. Trichotomous noise belongs to colored noise. The transitions state of trichotomous noise is between three values (a, b, and c) [42] and it is divided into symmetry trichotomous noise a=−c,b=0 and asymmetry trichotomous noise a≠−c,b≠0 according to the fluctuation values. The trichotomous process is a Poisson process, and the jump between the three values occurs with the constant probabilities:(11)Psa=Psc=q,Psb=1−2q.The transition probabilities between the three states can be obtained as follows:(12)Paa=Pa,t+τ|a,t=q−q−1e−vt,Pba=Pca=Pa,t+τ|b,t=Pa,t+τ|c,t=q1−e−vt,Pbb=Pb,t+τ|b,t=1−2q1−e−vt,Pab=Pcb=Pb,t+τ|a,t=Pb,t+τ|c,t=1−2q1−e−vt,Pcc=Pc,t+τ|c,t=q−q−1e−vt,Pac=Pbc=Pc,t+τ|a,t=Pc,t+τ|b,t=q1−e−vt.Here,τ>0, 0<q<1/2, v>0, τ>0, v=1/τcor, and τcor is the noise correlation time. The trichotomous process is completely determined by (11) and (12). Thus, the mean value ξt and correlation function ξt+τξt of the trichotomous noise in the steady state can be calculated as(13)ξt=aq+b1−2q+cq,(14)ξt+τξt=ξτs2+Dv2e−vτ.Here, the noise intensityD is defined as(15)D=∫0∞ξτξ0s−ξτs2dτ,=2va2q+b21−2q+c2q−aq+b1−2q+cq2.According to the description above, the numerical method is used to generate trichotomous noise. Firstly, the functionξt starts at a, ξ0=a. Then, The uniformly distributed random function Rt with value range [0, 1] is used to compare with the conditional probability given in (12) in order to determine the value of ξ0+Δt. If R0<Paa, the value of function ξ0+Δt is a, and if Paa<R0<Paa+Pab, the value of function ξ0+Δt is b; otherwise, the value of function ξ0+Δt is c. And then, if the value of function ξ0+Δt is b, R0+Δt will be used to determine the value of ξ0+2Δt. If R0+Δt<Pba, the value of function ξ0+2Δt is a, and if Pba<R0<Pba+Pba, the value of function ξ0+2Δt is b; otherwise, the value of function ξ0+2Δt is c. The case where the value of ξ0+Δt is c is similar to the case where the value of ξ0+Δt is b. By repeating the procedure, we can obtain the trichotomous noise ξt.Figures2(a) and 2(b) show the proﬁles of the asymmetric trichotomous noise with parameters a = 3, b = 1, c = −2. Figures 2(c) and 2(d) show the proﬁles of the symmetric trichotomous noise with parameters a = 2, b = 0, c = −2. By comparing Figures 2(a) and 2(b), it is considered that the residence time extends with the increase of noise intensity D. When q=1/2, it shows that the trichotomous noise reduces to dichotomous noise in Figure 2(d).Figure 2
(a, b) The asymmetric trichotomous noise with parametersa = 3, b = 1, c = −2. (a) D = 1, q = 0.3. (b) D = 4, q = 0.3. (c, d) The symmetric trichotomous noise with parameters a = 2, b = 0, c = −2. (c) D = 0.5, q = 0.3. (d) D = 1, q = 0.5.
(a)(b)(c)(d)


## 2.1. Three Kinds of Asymmetric Stochastic Resonance Systems

### 2.1.1. The Bistable System with Asymmetric Depth of Potential Well

WhenA1=B1=α, (α is asymmetric factor) and there is no input signal, the potential function contains two minima in the potential well: xmin±=±a0/b0 (stable points) and maxima in the potential barrier xmax=0 (unstable point). The widths of left and right potential well are both a0/b0. Then, the depths of the wells can be obtained as follows:(3)ΔUL=αa024b0,(4)ΔUR=a024b0.As can be seen from formulae (3) and (4), only the depth of the left well is controlled by the asymmetric factor α. Figure 1(a) shows the different depths of potential function caused by α when a0  = 1 and b0  = 1, where the depths of the left potential well increase with α increasing.Figure 1
(a) PotentialU1x for different values of α. (b) Potential U2x for different values of α. (c) Potential U3x for different values of α.
(a)(b)(c)


### 2.1.2. The Bistable System with Asymmetric Width of Potential Well

WhenA2=1/α2, B2=1/α4, the potential function contains two stable points, xmin+=a0/b0 and xmin−=−αa0/b0, and an unstable point xmax=0. The depths of left and right potential well are both a02/4b0. Then, the widths of the wells can be obtained as follows:(5)ΔWL=αa0b0,(6)ΔWR=a0b0.As can be seen from formulae (5) and (6), the width of the left well is controlled by the asymmetric factor α. Figure 1(b) shows the different widths of the potential function caused by α when a0  = 1 and b0  = 1, where the widths of the left potential well increase with α increasing.


### 2.1.3. The Bistable System with Both Asymmetric Width and Asymmetric Depth of Potential Well

WhenA3≠B3, A32≠B3, the bistable system has potential wells with both asymmetric width and depth. In this research, A3=1, B3=1/α4 (the combination of U1x and U2x) are selected as typical and the potential function contains two stable points, xmin+=a0/b0 and xmin−=−α2a0/b0, and an unstable point xmax=0. Then, the depths and widths of the wells can be obtained as follows:(7)ΔUL=α4a024b0,(8)ΔUR=a024b0,(9)ΔWL=α2a0b0,(10)ΔWR=a0b0.As can be seen from formulae (7)–(10), the left well is controlled by the asymmetric factor α. Figure 1(c) shows the differences of the potential function caused by α when a0  = 1 and a0  = 1, where the widths and depths of the left potential well increase with α increasing.


## 2.1.1. The Bistable System with Asymmetric Depth of Potential Well

WhenA1=B1=α, (α is asymmetric factor) and there is no input signal, the potential function contains two minima in the potential well: xmin±=±a0/b0 (stable points) and maxima in the potential barrier xmax=0 (unstable point). The widths of left and right potential well are both a0/b0. Then, the depths of the wells can be obtained as follows:(3)ΔUL=αa024b0,(4)ΔUR=a024b0.As can be seen from formulae (3) and (4), only the depth of the left well is controlled by the asymmetric factor α. Figure 1(a) shows the different depths of potential function caused by α when a0  = 1 and b0  = 1, where the depths of the left potential well increase with α increasing.Figure 1
(a) PotentialU1x for different values of α. (b) Potential U2x for different values of α. (c) Potential U3x for different values of α.
(a)(b)(c)


## 2.1.2. The Bistable System with Asymmetric Width of Potential Well

WhenA2=1/α2, B2=1/α4, the potential function contains two stable points, xmin+=a0/b0 and xmin−=−αa0/b0, and an unstable point xmax=0. The depths of left and right potential well are both a02/4b0. Then, the widths of the wells can be obtained as follows:(5)ΔWL=αa0b0,(6)ΔWR=a0b0.As can be seen from formulae (5) and (6), the width of the left well is controlled by the asymmetric factor α. Figure 1(b) shows the different widths of the potential function caused by α when a0  = 1 and b0  = 1, where the widths of the left potential well increase with α increasing.


## 2.1.3. The Bistable System with Both Asymmetric Width and Asymmetric Depth of Potential Well

WhenA3≠B3, A32≠B3, the bistable system has potential wells with both asymmetric width and depth. In this research, A3=1, B3=1/α4 (the combination of U1x and U2x) are selected as typical and the potential function contains two stable points, xmin+=a0/b0 and xmin−=−α2a0/b0, and an unstable point xmax=0. Then, the depths and widths of the wells can be obtained as follows:(7)ΔUL=α4a024b0,(8)ΔUR=a024b0,(9)ΔWL=α2a0b0,(10)ΔWR=a0b0.As can be seen from formulae (7)–(10), the left well is controlled by the asymmetric factor α. Figure 1(c) shows the differences of the potential function caused by α when a0  = 1 and a0  = 1, where the widths and depths of the left potential well increase with α increasing.


## 2.2. Method of Generating Trichotomous Noise

In (1), ξt is trichotomous noise, and the method of generating it will be introduced in this section. Trichotomous noise belongs to colored noise. The transitions state of trichotomous noise is between three values (a, b, and c) [42] and it is divided into symmetry trichotomous noise a=−c,b=0 and asymmetry trichotomous noise a≠−c,b≠0 according to the fluctuation values. The trichotomous process is a Poisson process, and the jump between the three values occurs with the constant probabilities:(11)Psa=Psc=q,Psb=1−2q.The transition probabilities between the three states can be obtained as follows:(12)Paa=Pa,t+τ|a,t=q−q−1e−vt,Pba=Pca=Pa,t+τ|b,t=Pa,t+τ|c,t=q1−e−vt,Pbb=Pb,t+τ|b,t=1−2q1−e−vt,Pab=Pcb=Pb,t+τ|a,t=Pb,t+τ|c,t=1−2q1−e−vt,Pcc=Pc,t+τ|c,t=q−q−1e−vt,Pac=Pbc=Pc,t+τ|a,t=Pc,t+τ|b,t=q1−e−vt.Here,τ>0, 0<q<1/2, v>0, τ>0, v=1/τcor, and τcor is the noise correlation time. The trichotomous process is completely determined by (11) and (12). Thus, the mean value ξt and correlation function ξt+τξt of the trichotomous noise in the steady state can be calculated as(13)ξt=aq+b1−2q+cq,(14)ξt+τξt=ξτs2+Dv2e−vτ.Here, the noise intensityD is defined as(15)D=∫0∞ξτξ0s−ξτs2dτ,=2va2q+b21−2q+c2q−aq+b1−2q+cq2.According to the description above, the numerical method is used to generate trichotomous noise. Firstly, the functionξt starts at a, ξ0=a. Then, The uniformly distributed random function Rt with value range [0, 1] is used to compare with the conditional probability given in (12) in order to determine the value of ξ0+Δt. If R0<Paa, the value of function ξ0+Δt is a, and if Paa<R0<Paa+Pab, the value of function ξ0+Δt is b; otherwise, the value of function ξ0+Δt is c. And then, if the value of function ξ0+Δt is b, R0+Δt will be used to determine the value of ξ0+2Δt. If R0+Δt<Pba, the value of function ξ0+2Δt is a, and if Pba<R0<Pba+Pba, the value of function ξ0+2Δt is b; otherwise, the value of function ξ0+2Δt is c. The case where the value of ξ0+Δt is c is similar to the case where the value of ξ0+Δt is b. By repeating the procedure, we can obtain the trichotomous noise ξt.Figures2(a) and 2(b) show the proﬁles of the asymmetric trichotomous noise with parameters a = 3, b = 1, c = −2. Figures 2(c) and 2(d) show the proﬁles of the symmetric trichotomous noise with parameters a = 2, b = 0, c = −2. By comparing Figures 2(a) and 2(b), it is considered that the residence time extends with the increase of noise intensity D. When q=1/2, it shows that the trichotomous noise reduces to dichotomous noise in Figure 2(d).Figure 2
(a, b) The asymmetric trichotomous noise with parametersa = 3, b = 1, c = −2. (a) D = 1, q = 0.3. (b) D = 4, q = 0.3. (c, d) The symmetric trichotomous noise with parameters a = 2, b = 0, c = −2. (c) D = 0.5, q = 0.3. (d) D = 1, q = 0.5.
(a)(b)(c)(d)


## 3. Performance Index

There are many kinds of indexes to measure stochastic resonance, such as signal-to-noise ratio (SNR) [45, 46] and signal-to-noise ratio gain (SNRG) [47], and mean first-passage time (MFPT) [46]. In practical engineering, people are more concerned about the gain of the output SNR compared with the input SNR, so the SNR gain is chosen in this paper, given as(16)SNRG=Soutf0/ξoutf0Sinf0/ξinf0,where Soutf0 and Sinf0 represent the output and input power of the signal at input frequency f0, and ξoutf0 and ξinf0 represent the output and input power of the noise at input frequency f0. Due to the randomness of the noise, it is usually necessary to conduct numerous experiments and take the average value of them to ensure the reliability of the experimental results. Therefore, the average value of signal-to-noise ratio gain is defined as(17)A−SNRG=1n∑j=0j=nSNRGi,where n is the time of simulation and n=20 in this paper. SNRIj is SNRG of the j-th simulation.


## 4. Stochastic Resonance of Asymmetric Bistable System Driven by Trichotomous Noise

The purpose of this research is to cover the impact of system parameters and refer to the stochastic resonance of the asymmetric bistable system driven by trichotomous noise system parameters. The input sinusoidal signal isst=Acos2πf0t, where A=0.3, f0=0.01Hz. The parameters of trichotomous noise distribution are set to be a=1, b=0, c=−1, q=0.3, and D=0.5. A-SNRG is chosen to be the indicator. Simulation time is n=50, for each simulation experiment, the calculation step is set as h=1/fs, where the sampling frequency is fs=5Hz, and the sampling number is N=4096. The optimal parameter pair a,b is discovered by using the adaptive algorithms in Figure 3. Figure 3 shows a three-dimensional plot of the traditional bistable adaptive optimization in which the asymmetric factor is α=1. Figure 4(a) shows the time-domain diagram of the noisy signal which contains the trichotomous noise and sinusoidal signal. It is obvious that the input signal is completely submerged by trichotomous noise. Figure 4(b) is the power spectrum diagram of the noisy signal. It could be seen from the figure that the frequency f0=0.01Hz of the input signal cannot be identified easily.Figure 3
Three-dimensional diagram of the traditional symmetric bistable system SR adaptive optimization driven by trichotomous noise.Figure 4
(a) The time-domain diagram of noisy signal and (b) the power spectrum diagram of the noisy signal.
(a)(b)Therefore, both passing the stochastic resonance system and the addition of noise are used to enhance the weak signal to be measured. Figure5(a) shows the time-domain diagram of the output signal through the traditional bistable system SR system and the power spectrum of it. Figures 5(b)–5(d) show the diagram of the output signal through the bistable system with asymmetric depth, asymmetric width, and both asymmetric depth and width of potential. The peak of the asymmetric system power spectrum at f0=0.01Hz is higher than the symmetric system. So, the conclusion is that the asymmetric system which contains trichotomous noise and the periodic signal has a better detection effect than the symmetric systems, by changing the depth and width of one potential well.Figure 5
(a) Output of the symmetric system SR model and its power spectrum. (b) Output of SR model with asymmetric depth of the potential wellu1x and its power spectrum. (c) Output of SR model with asymmetric width of potential well u2x and its power spectrum. (d) Output of SR model with asymmetric depth and width of potential well u3x and its power spectrum.
(a)(b)(c)(d)Next, the parameter of noise and potential function are explored under different values of the asymmetric factor in order to obtain better detection parameters.


### 4.1. Influences of Asymmetric Factorα for SR with Asymmetric Depth of Potential Well

The parameters of the input signal containA0, f0 and trichotomous noise contains a, b, c, consistent with the above results. The parameters of the potential function are a0=1.9, b0=2.3 from the adaptive algorithms. The influence of the noise intensity D is shown in Figure 6. The influence of a0 and b0 with the noise intensity fixed D=0.5 is shown in Figure 7. Then, the influence of q with the noise intensity fixed D=0.5 and a0=1.9, b0=2.3 is shown in Figure 8.Figure 6
(a) A-SNRG of SR with asymmetric depth of potential well as a function of noise intensiveD and asymmetric factor with different values of α. (b) A-SNRG of SR with asymmetric depth of potential well as a function of noise intensive D for different values of the asymmetric factor with α. (c) α>1 and (d) α<1.
(a)(b)(c)(d)Figure 7
A-SNRG of SR with asymmetric depth of potential well as a function of the potential function parameter. (a)a0 and (b) b0.
(a)(b)Figure 8
A-SNRG of SR with asymmetric depth of potential well as a function of potential function parameterq.Figure6(a) shows that with the increase of the noise intensity D, the curves of A-SNRG first increase and then decrease for different values of α and it means that stochastic resonance with asymmetric depth of the potential well is generated under the drive of trichotomous noise. To see more clearly, the values of the asymmetric factor are 0.6, 1.0, and 1.4 given in Figure 6(b), respectively. The maximum of A-SNRG increases with an increase in depth of the potential well, but the deeper the well, the greater the noise intensity needed to reach the peak and the reason for this is the particles need larger noise intensity D to cross the barrier. The value of A-SNRG when α>1 is larger than it when α<1. Figures 6(c) and 6(d) show the SR curves for α>1 and α<1, respectively. Figure 6(c) shows that the value of A-SNRG is the maximum for α=1.4 and starts to decrease from 1.4. When α=2, the system cannot produce SR. Figure 6(d) shows that the curve has no peaks which means the system cannot produce SR when α=0.4 and the value of A-SNRG is starting to increase. These four ﬁgures demonstrate that a larger depth of a single potential well has a positive effect on enhancing the A-SNRG in the case that SR can be produced.Under the condition that SR can be produced, in Figure7(a), with the increase of parameter α, the peaks of A-SNRG curves increase and the values a0 corresponding to the peak decrease. In Figure 7(b), with the increase of parameter α, the peaks of A-SNRG curves increase and the values b0 corresponding to the peak increase at the same time. That is to say, there is a pair of optimal system parameters which can enhance the effect of SR.In Figure8, when α<1, the value of A-SNRG increases with q increasing. When α>1, the value of A-SNRG increases first and then decreases with q increasing. That is to say, for the case of shallow potential well depth, the longer the residence time of particle is, the better the effect of stochastic resonance will be. However, when the potential well is deep, there is an optimal q for the stochastic resonance system and the value of q does not enhance the effect of the SR system significantly.


### 4.2. Influences of Asymmetric Factorα for SR with Asymmetric Width of Potential Well

The parameters of the input signal containA0, f0 and trichotomous noise contains a, b, c, q consistent with the above results. The parameters of the potential function are a0=1.9, b0=2.3 from the adaptive algorithms. The influence of the noise intensity D is shown in Figure 9. Then, the influence of a0 and b0 with the noise intensity fixed D=0.5 is shown in Figure 10. After that, the influence of q with the noise intensity fixed D=0.5, a0=1.9, b0=2.3 is shown in Figure 11.Figure 9
A-SNRG of SR with asymmetric width of potential well as a function of noise intensiveD for asymmetric factor with different values of α.Figure 10
A-SNRG of SR with asymmetric width of potential well as a function of a potential function parameter. (a)a0 and (b)b0.
(a)(b)Figure 11
A-SNRG of SR with asymmetric width of potential well as a function of potential function parameterq.Figure9 shows that, with the increase of the noise intensity D, the curves of A-SNRG first increase and then decrease for different values of α and it means that stochastic resonance with asymmetric width of the potential well is generated under the drive of trichotomous noise. The values of the asymmetric factor are 0.5, 0.8, 1.0, 1.2, and 1.5 given in Figure 9, respectively.The maximum of A-SNRG decreases with an increase in width of the potential well, and the wider the well is, the greater the noise intensity will be needed to reach the peak. Compared with Figure6(a), the changes in noise intensity which is needed to reach the peak for different depth of the potential well are larger than those of different widths. That is to say, the depth of the potential well has more effect on the SR systems than the width. Obviously, the A-SNRG for the small α is always larger than the big α and the result exhibits the system with a smaller asymmetric factor α which means the potential function with narrower width has a better detection effect. The value of A-SNRG when α>1 is much larger than it when α<1. So the smaller width of one potential well in an asymmetric bistable system has a positive effect on stochastic resonance.In Figure10(a), with the increase of parameter α, the peaks of A-SNRG curves increase and the values a0 corresponding to each peak decrease. And it indicates that the smaller the value of a0 is, the better the detection effect will be with a certain b0. In Figure 10(b), with the increase of parameter α, the peaks of A-SNRG curves increase and the values b0 corresponding to each peak increase at the same time. And it indicates that the greater the value of b0 is, the better the detection effect will be with a certain a0. That is to say, there is a pair of optimal system parameters which can enhance the effect of SR.In Figure11, the value of A-SNRG increases with q increasing for different values of α. It means that the longer the residence time of particle is, the better the effect of the stochastic resonance will be with different widths of a single well. Compared with Figure 8, the width of the potential well has no effect on the trend of the curve for q-A-SNRG.


### 4.3. Influences of Asymmetric Factorα for SR with Asymmetric Depth and Width of Potential Well

The parameters of the input signal containA0, f0 and trichotomous noise contains a, b, c consistent with the above results. The parameters of the potential function are a0=1.9, b0=2.3 from the adaptive algorithms. The influence of the noise intensity D is shown in Figure 12. Then, the influence of a0 and b0 with the noise intensity fixed D=0.5 is shown in Figure 13. After that, the influence of q with the noise intensity fixed D=0.5 and a0=1.9, b0=2.3 is shown in Figure 14.Figure 12
(a) A-SNRG of SR with asymmetric depth and width of potential well as a function of noise intensiveD and asymmetric factor with different values of α. (b) A-SNRG of SR with asymmetric depth and width of potential well as a function of noise intensive D for different values of asymmetric factor with α, (c) α<1 and (d) α>1.
(a)(b)(c)(d)Figure 13
A-SNRG of SR with asymmetric depth and width of the potential well, as a function of the potential function parameter. (a)a0 and (b)b0.
(a)(b)Figure 14
A-SNRG of SR with asymmetric depth and width of potential well as a function of potential function parameterq.Figure12(a) shows that when α∈0.8,1.1, with the increase of the noise intensity D, the curves of A-SNRG first increase and then decrease for different values of α and it means that stochastic resonance with asymmetric depth and width of the potential well is generated under the drive of trichotomous noise. To see more clearly, the values of the asymmetric factor are 0.8, 1.0, and 1.2 given in Figure 12(b), respectively. When α=0.8, the curve of A-SNRG shows a downward trend in all the values of D∈0,3 which means the system cannot produce SR. When D>1, the value of A-SNRG for α=1.2 is larger than that for α=1. In the case that other parameters are the same and with the appropriate noise intensity, the system with the asymmetric well has a more positive effect on the SR than the traditional one. The value of A-SNRG when α>1 is larger than that when α≤1 under the condition that SR can be produced in Figure 12(b), and it means a larger asymmetric factor may have a better detection effect. Next, SR curves for α<1 and α>1 will be analyzed more clearly and Figures 12(c) and 12(d) show the SR curves for α<1 and α>1, respectively. Figure 12(c) shows that the curves have no peaks which means the system cannot produce SR until α=0.9. Figure 12(d) shows that the value of A-SNRG is maximum for α=1.1 and it decreases from α=1.1. When α=1.5, the values of A-SNRG are all under 1, so the system cannot produce SR with α=1.5. These four ﬁgures demonstrate that there is an optimum asymmetric factor that has a better detection effect than the traditional system.Under the condition that SR can be produced, in Figure13(a), with the increase of parameter α, the peaks of A-SNRG curves decrease and the values a0 corresponding to the peak decrease at the same time. In Figure 13(b), with the increase of parameter α, the values b0 corresponding to the peak increase at the same time. The peaks of A-SNRG curves remain stable when α∈0.9,1.1 and then decrease. That is to say, when α is around 1, b0 is meaningless to enhance the effect of SR.In Figure14, when α<1, the value of A-SNRG increases with q increasing. when α>1, the value of A-SNRG increases first and then decreases with q increasing. According to Figures 8 and 11, the trend of the curve when α>1 is determined by the depth of the well only.


## 4.1. Influences of Asymmetric Factorα for SR with Asymmetric Depth of Potential Well

The parameters of the input signal containA0, f0 and trichotomous noise contains a, b, c, consistent with the above results. The parameters of the potential function are a0=1.9, b0=2.3 from the adaptive algorithms. The influence of the noise intensity D is shown in Figure 6. The influence of a0 and b0 with the noise intensity fixed D=0.5 is shown in Figure 7. Then, the influence of q with the noise intensity fixed D=0.5 and a0=1.9, b0=2.3 is shown in Figure 8.Figure 6
(a) A-SNRG of SR with asymmetric depth of potential well as a function of noise intensiveD and asymmetric factor with different values of α. (b) A-SNRG of SR with asymmetric depth of potential well as a function of noise intensive D for different values of the asymmetric factor with α. (c) α>1 and (d) α<1.
(a)(b)(c)(d)Figure 7
A-SNRG of SR with asymmetric depth of potential well as a function of the potential function parameter. (a)a0 and (b) b0.
(a)(b)Figure 8
A-SNRG of SR with asymmetric depth of potential well as a function of potential function parameterq.Figure6(a) shows that with the increase of the noise intensity D, the curves of A-SNRG first increase and then decrease for different values of α and it means that stochastic resonance with asymmetric depth of the potential well is generated under the drive of trichotomous noise. To see more clearly, the values of the asymmetric factor are 0.6, 1.0, and 1.4 given in Figure 6(b), respectively. The maximum of A-SNRG increases with an increase in depth of the potential well, but the deeper the well, the greater the noise intensity needed to reach the peak and the reason for this is the particles need larger noise intensity D to cross the barrier. The value of A-SNRG when α>1 is larger than it when α<1. Figures 6(c) and 6(d) show the SR curves for α>1 and α<1, respectively. Figure 6(c) shows that the value of A-SNRG is the maximum for α=1.4 and starts to decrease from 1.4. When α=2, the system cannot produce SR. Figure 6(d) shows that the curve has no peaks which means the system cannot produce SR when α=0.4 and the value of A-SNRG is starting to increase. These four ﬁgures demonstrate that a larger depth of a single potential well has a positive effect on enhancing the A-SNRG in the case that SR can be produced.Under the condition that SR can be produced, in Figure7(a), with the increase of parameter α, the peaks of A-SNRG curves increase and the values a0 corresponding to the peak decrease. In Figure 7(b), with the increase of parameter α, the peaks of A-SNRG curves increase and the values b0 corresponding to the peak increase at the same time. That is to say, there is a pair of optimal system parameters which can enhance the effect of SR.In Figure8, when α<1, the value of A-SNRG increases with q increasing. When α>1, the value of A-SNRG increases first and then decreases with q increasing. That is to say, for the case of shallow potential well depth, the longer the residence time of particle is, the better the effect of stochastic resonance will be. However, when the potential well is deep, there is an optimal q for the stochastic resonance system and the value of q does not enhance the effect of the SR system significantly.


## 4.2. Influences of Asymmetric Factorα for SR with Asymmetric Width of Potential Well

The parameters of the input signal containA0, f0 and trichotomous noise contains a, b, c, q consistent with the above results. The parameters of the potential function are a0=1.9, b0=2.3 from the adaptive algorithms. The influence of the noise intensity D is shown in Figure 9. Then, the influence of a0 and b0 with the noise intensity fixed D=0.5 is shown in Figure 10. After that, the influence of q with the noise intensity fixed D=0.5, a0=1.9, b0=2.3 is shown in Figure 11.Figure 9
A-SNRG of SR with asymmetric width of potential well as a function of noise intensiveD for asymmetric factor with different values of α.Figure 10
A-SNRG of SR with asymmetric width of potential well as a function of a potential function parameter. (a)a0 and (b)b0.
(a)(b)Figure 11
A-SNRG of SR with asymmetric width of potential well as a function of potential function parameterq.Figure9 shows that, with the increase of the noise intensity D, the curves of A-SNRG first increase and then decrease for different values of α and it means that stochastic resonance with asymmetric width of the potential well is generated under the drive of trichotomous noise. The values of the asymmetric factor are 0.5, 0.8, 1.0, 1.2, and 1.5 given in Figure 9, respectively.The maximum of A-SNRG decreases with an increase in width of the potential well, and the wider the well is, the greater the noise intensity will be needed to reach the peak. Compared with Figure6(a), the changes in noise intensity which is needed to reach the peak for different depth of the potential well are larger than those of different widths. That is to say, the depth of the potential well has more effect on the SR systems than the width. Obviously, the A-SNRG for the small α is always larger than the big α and the result exhibits the system with a smaller asymmetric factor α which means the potential function with narrower width has a better detection effect. The value of A-SNRG when α>1 is much larger than it when α<1. So the smaller width of one potential well in an asymmetric bistable system has a positive effect on stochastic resonance.In Figure10(a), with the increase of parameter α, the peaks of A-SNRG curves increase and the values a0 corresponding to each peak decrease. And it indicates that the smaller the value of a0 is, the better the detection effect will be with a certain b0. In Figure 10(b), with the increase of parameter α, the peaks of A-SNRG curves increase and the values b0 corresponding to each peak increase at the same time. And it indicates that the greater the value of b0 is, the better the detection effect will be with a certain a0. That is to say, there is a pair of optimal system parameters which can enhance the effect of SR.In Figure11, the value of A-SNRG increases with q increasing for different values of α. It means that the longer the residence time of particle is, the better the effect of the stochastic resonance will be with different widths of a single well. Compared with Figure 8, the width of the potential well has no effect on the trend of the curve for q-A-SNRG.


## 4.3. Influences of Asymmetric Factorα for SR with Asymmetric Depth and Width of Potential Well

The parameters of the input signal containA0, f0 and trichotomous noise contains a, b, c consistent with the above results. The parameters of the potential function are a0=1.9, b0=2.3 from the adaptive algorithms. The influence of the noise intensity D is shown in Figure 12. Then, the influence of a0 and b0 with the noise intensity fixed D=0.5 is shown in Figure 13. After that, the influence of q with the noise intensity fixed D=0.5 and a0=1.9, b0=2.3 is shown in Figure 14.Figure 12
(a) A-SNRG of SR with asymmetric depth and width of potential well as a function of noise intensiveD and asymmetric factor with different values of α. (b) A-SNRG of SR with asymmetric depth and width of potential well as a function of noise intensive D for different values of asymmetric factor with α, (c) α<1 and (d) α>1.
(a)(b)(c)(d)Figure 13
A-SNRG of SR with asymmetric depth and width of the potential well, as a function of the potential function parameter. (a)a0 and (b)b0.
(a)(b)Figure 14
A-SNRG of SR with asymmetric depth and width of potential well as a function of potential function parameterq.Figure12(a) shows that when α∈0.8,1.1, with the increase of the noise intensity D, the curves of A-SNRG first increase and then decrease for different values of α and it means that stochastic resonance with asymmetric depth and width of the potential well is generated under the drive of trichotomous noise. To see more clearly, the values of the asymmetric factor are 0.8, 1.0, and 1.2 given in Figure 12(b), respectively. When α=0.8, the curve of A-SNRG shows a downward trend in all the values of D∈0,3 which means the system cannot produce SR. When D>1, the value of A-SNRG for α=1.2 is larger than that for α=1. In the case that other parameters are the same and with the appropriate noise intensity, the system with the asymmetric well has a more positive effect on the SR than the traditional one. The value of A-SNRG when α>1 is larger than that when α≤1 under the condition that SR can be produced in Figure 12(b), and it means a larger asymmetric factor may have a better detection effect. Next, SR curves for α<1 and α>1 will be analyzed more clearly and Figures 12(c) and 12(d) show the SR curves for α<1 and α>1, respectively. Figure 12(c) shows that the curves have no peaks which means the system cannot produce SR until α=0.9. Figure 12(d) shows that the value of A-SNRG is maximum for α=1.1 and it decreases from α=1.1. When α=1.5, the values of A-SNRG are all under 1, so the system cannot produce SR with α=1.5. These four ﬁgures demonstrate that there is an optimum asymmetric factor that has a better detection effect than the traditional system.Under the condition that SR can be produced, in Figure13(a), with the increase of parameter α, the peaks of A-SNRG curves decrease and the values a0 corresponding to the peak decrease at the same time. In Figure 13(b), with the increase of parameter α, the values b0 corresponding to the peak increase at the same time. The peaks of A-SNRG curves remain stable when α∈0.9,1.1 and then decrease. That is to say, when α is around 1, b0 is meaningless to enhance the effect of SR.In Figure14, when α<1, the value of A-SNRG increases with q increasing. when α>1, the value of A-SNRG increases first and then decreases with q increasing. According to Figures 8 and 11, the trend of the curve when α>1 is determined by the depth of the well only.


## 5. Conclusion

In this paper, we have focused on the SR phenomenon in three types of the asymmetric bistable system driven by trichotomous noise. The following conclusions can be drawn from the above research: (1) when the system parameters of the potential function are different functions of an asymmetric factor, the depth and width of a single potential well can be changed. (2) When the noise intensityD is relatively large, a better detection effect can be obtained by making the depth of a single well deeper. It can improve the detection effect by making the width of the potential well smaller for different noise intensities D. (3) Parameters a0, b0 of the potential function always have the optimal value regardless of the form of asymmetry. (4) The effect of particle residence time on the stochastic resonance system is affected by the depth of the potential well.



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*Source: 8842122-2020-11-10.xml*