[Retracted] Performance Investigation of Stochastic Resonance in Three Types of Asymmetric Bistable System Driven by Trichotomous Noise

Zhao, Si-Hai; Xu, Jiang-Ye; Liu, Yu-Xiao; Zhao, Ze-Xing; Qin, Zhong-Shun

doi:https://doi.org/10.1155/2020/8842122

Complexity

On this page

Abstract Introduction Methods Conclusion Data Availability Conflicts of Interest References Copyright Related Articles

Research Article Retraction

!

This article has been Retracted. To view the article details, please click the ‘Retraction’ tab above.

Special Issue

Complexity Problems Handled by Advanced Computer Simulation Technology in Smart Cities 2020

View this Special Issue

Research Article | Open Access

Volume 2020 | Article ID 8842122 | https://doi.org/10.1155/2020/8842122

[Retracted] Performance Investigation of Stochastic Resonance in Three Types of Asymmetric Bistable System Driven by Trichotomous Noise

Si-Hai Zhao,¹Jiang-Ye Xu,¹Yu-Xiao Liu,¹Ze-Xing Zhao,¹and Zhong-Shun Qin¹

Academic Editor: Zhihan Lv

Received16 Sept 2020

Revised16 Oct 2020

Accepted24 Oct 2020

Published10 Nov 2020

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 factor , 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.

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 variable t 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:where is the system input signal to be measured, is the amplitude of the input signal to be measured, is the frequency of the input signal to be measured, is the trichotomous noise, and is the potential function of asymmetric systems. is given as follows:where are the structural parameters of asymmetric systems, and these parameters meet such inequalities: 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

When , ( is asymmetric factor) and there is no input signal, the potential function contains two minima in the potential well: (stable points) and maxima in the potential barrier (unstable point). The widths of left and right potential well are both . Then, the depths of the wells can be obtained as follows:

As can be seen from formulae (3) and (4), only the depth of the left well is controlled by the asymmetric factor . Figure 1(a) shows the different depths of potential function caused by when = 1 and = 1, where the depths of the left potential well increase with increasing.

(a)

(b)

(c)

2.1.2. The Bistable System with Asymmetric Width of Potential Well

When , , the potential function contains two stable points, and , and an unstable point . The depths of left and right potential well are both . Then, the widths of the wells can be obtained as follows:

As can be seen from formulae (5) and (6), the width of the left well is controlled by the asymmetric factor . Figure 1(b) shows the different widths of the potential function caused by when = 1 and = 1, where the widths of the left potential well increase with increasing.

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

When , , the bistable system has potential wells with both asymmetric width and depth. In this research, , (the combination of and ) are selected as typical and the potential function contains two stable points, and , and an unstable point . Then, the depths and widths of the wells can be obtained as follows:

As can be seen from formulae (7)–(10), the left well is controlled by the asymmetric factor . Figure 1(c) shows the differences of the potential function caused by when = 1 and = 1, where the widths and depths of the left potential well increase with increasing.

2.2. Method of Generating Trichotomous Noise

In (1), 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 and asymmetry trichotomous noise according to the fluctuation values. The trichotomous process is a Poisson process, and the jump between the three values occurs with the constant probabilities:

The transition probabilities between the three states can be obtained as follows:

Here, , , , , , and is the noise correlation time. The trichotomous process is completely determined by (11) and (12). Thus, the mean value and correlation function of the trichotomous noise in the steady state can be calculated as

Here, the noise intensity D is defined as

According to the description above, the numerical method is used to generate trichotomous noise. Firstly, the function starts at a, . Then, The uniformly distributed random function with value range [0, 1] is used to compare with the conditional probability given in (12) in order to determine the value of . If , the value of function is a, and if , the value of function is b; otherwise, the value of function is c. And then, if the value of function is b, will be used to determine the value of . If , the value of function is a, and if , the value of function is b; otherwise, the value of function is c. The case where the value of is c is similar to the case where the value of is b. By repeating the procedure, we can obtain the trichotomous noise .

Figures 2(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 , it shows that the trichotomous noise reduces to dichotomous noise in Figure 2(d).

(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 aswhere and represent the output and input power of the signal at input frequency , and and represent the output and input power of the noise at input frequency . 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 aswhere is the time of simulation and in this paper. 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 is , where , . The parameters of trichotomous noise distribution are set to be , , , , and 5. A-SNRG is chosen to be the indicator. Simulation time is , for each simulation experiment, the calculation step is set as , where the sampling frequency is , and the sampling number is . The optimal parameter pair 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 . 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 of the input signal cannot be identified easily.

(a)

(b)

Therefore, both passing the stochastic resonance system and the addition of noise are used to enhance the weak signal to be measured. Figure 5(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 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.

(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 Factor for SR with Asymmetric Depth of Potential Well

The parameters of the input signal contain , and trichotomous noise contains a, b, c, consistent with the above results. The parameters of the potential function are , from the adaptive algorithms. The influence of the noise intensity D is shown in Figure 6. The influence of and with the noise intensity fixed is shown in Figure 7. Then, the influence of q with the noise intensity fixed and , is shown in Figure 8.

(a)

(b)

(c)

(d)

(a)

(b)

Figure 6(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 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 is larger than it when . Figures 6(c) and 6(d) show the SR curves for and , respectively. Figure 6(c) shows that the value of A-SNRG is the maximum for and starts to decrease from 1.4. When , the system cannot produce SR. Figure 6(d) shows that the curve has no peaks which means the system cannot produce SR when 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 Figure 7(a), with the increase of parameter , the peaks of A-SNRG curves increase and the values corresponding to the peak decrease. In Figure 7(b), with the increase of parameter , the peaks of A-SNRG curves increase and the values 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 Figure 8, when , the value of A-SNRG increases with q increasing. When , 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 Factor for SR with Asymmetric Width of Potential Well

The parameters of the input signal contain , and trichotomous noise contains a, b, c, q consistent with the above results. The parameters of the potential function are , from the adaptive algorithms. The influence of the noise intensity D is shown in Figure 9. Then, the influence of and with the noise intensity fixed is shown in Figure 10. After that, the influence of q with the noise intensity fixed , , is shown in Figure 11.

(a)

(b)

Figure 9 shows that, with the increase of the noise intensity D, the curves of A-SNRG first increase and then decrease for different values of 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 Figure 6(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 is always larger than the big and the result exhibits the system with a smaller asymmetric factor which means the potential function with narrower width has a better detection effect. The value of A-SNRG when is much larger than it when . So the smaller width of one potential well in an asymmetric bistable system has a positive effect on stochastic resonance.

In Figure 10(a), with the increase of parameter , the peaks of A-SNRG curves increase and the values corresponding to each peak decrease. And it indicates that the smaller the value of is, the better the detection effect will be with a certain . In Figure 10(b), with the increase of parameter , the peaks of A-SNRG curves increase and the values corresponding to each peak increase at the same time. And it indicates that the greater the value of is, the better the detection effect will be with a certain . That is to say, there is a pair of optimal system parameters which can enhance the effect of SR.

In Figure 11, the value of A-SNRG increases with q increasing for different values of . 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 Factor for SR with Asymmetric Depth and Width of Potential Well

The parameters of the input signal contain , and trichotomous noise contains a, b, c consistent with the above results. The parameters of the potential function are , from the adaptive algorithms. The influence of the noise intensity D is shown in Figure 12. Then, the influence of and with the noise intensity fixed is shown in Figure 13. After that, the influence of q with the noise intensity fixed and , is shown in Figure 14.

(a)

(b)

(c)

(d)

(a)

(b)

Figure 12(a) shows that when , with the increase of the noise intensity D, the curves of A-SNRG first increase and then decrease for different values of 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 , the curve of A-SNRG shows a downward trend in all the values of which means the system cannot produce SR. When , the value of A-SNRG for is larger than that for . 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 is larger than that when 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 and will be analyzed more clearly and Figures 12(c) and 12(d) show the SR curves for and , respectively. Figure 12(c) shows that the curves have no peaks which means the system cannot produce SR until . Figure 12(d) shows that the value of A-SNRG is maximum for and it decreases from . When , the values of A-SNRG are all under 1, so the system cannot produce SR with . 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 Figure 13(a), with the increase of parameter , the peaks of A-SNRG curves decrease and the values corresponding to the peak decrease at the same time. In Figure 13(b), with the increase of parameter , the values corresponding to the peak increase at the same time. The peaks of A-SNRG curves remain stable when and then decrease. That is to say, when is around 1, is meaningless to enhance the effect of SR.

In Figure 14, when , the value of A-SNRG increases with q increasing. when , 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 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 intensity D 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 , 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.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

References

H. Sompolinsky and I. Kanter, “Temporal association in asymmetric neural networks,” Physical Review Letters, vol. 57, no. 22, p. 2861, 1986.
View at: Publisher Site | Google Scholar
K. Gopalsamy and X.-z. He, “Stability in asymmetric hopfield nets with transmission delays,” Physica D: Nonlinear Phenomena, vol. 76, no. 4, p. 344, 1994.
View at: Publisher Site | Google Scholar
J. Lu, M. Huang, J. Yang, and P. Li, Wireless Personal Communications, Springer Netherlands, Dordrecht, Netherlands, 2020.
C. Wu, A. B. Khanikaev, R. Adato, N. Arju, A. Altug, and G. Shvets, “Fano-resonant asymmetric metamaterials for ultrasensitive spectroscopy and identification of molecular monolayers,” Nature Materials, vol. 11, no. 1, p. 69, 2011.
View at: Publisher Site | Google Scholar
E. A. Othman, A. N. Yusoff, M. Mohamad, H. Abdul Manan, A. I. Abd Hamid, and V. Giampietro, “Hemispheric lateralization of auditory working memory regions during stochastic resonance: an fMRI study,” Journal of Magnetic Resonance Imaging, vol. 51, no. 6, p. 1821, 2020.
View at: Publisher Site | Google Scholar
D. C. Mei, Z. L. Jia, and C. J. Wang, “Combined effects of asymmetry and noise correlation on the noise-enhanced stability phenomenon in a bistable system,” Physica Scripta, vol. 84, no. 4, Article ID 045012, 2011.
View at: Publisher Site | Google Scholar
F. Long, “Time-delay induced symmetry restoration and noise enhanced stability phenomena under correlated noises in an asymmetric bistable system,” Indian Journal of Physics, vol. 88, no. 10, p. 1111, 2014.
View at: Publisher Site | Google Scholar
Z.-L. Jia and D.-C. Mei, “The critical effects of time delay and noise correlation on stochastic resonance in an asymmetric bistable system,” Modern Physics Letters B, vol. 25, no. 16, p. 1377, 2011.
View at: Publisher Site | Google Scholar
C. Li, F. Long, and D. Mei, “Correlated effects of noise on symmetry of an asymmetric bistable system,” Frontiers in Physics, vol. 10, no. 2, 2015.
View at: Publisher Site | Google Scholar
F. Long, L. Du, and D. Mei, “Asymmetric effects on the associated relaxation time and the correlation function of a bistable system with correlated noises,” Physica Scripta, vol. 79, no. 4, Article ID 045007, 2009.
View at: Publisher Site | Google Scholar
J. J. Collins, C. C. Chow, and T. T. Imhoff, “Stochastic resonance without tuning,” Nature, vol. 376, no. 6537, p. 236, 1995.
View at: Publisher Site | Google Scholar
L. Gammaitoni, F. Marchesoni, and S. Santucci, “Stochastic resonance as a bona fide resonance,” Physical Review Letters, vol. 74, no. 7, p. 1052, 1995.
View at: Publisher Site | Google Scholar
M. I. Dykman and P. V. E. Mcclintock, “What can stochastic resonance do?” Nature, vol. 391, no. 6665, p. 344, 1998.
View at: Publisher Site | Google Scholar
F. Moss and K. Wiesenfeld, “Stochastic resonance and the benefits of noise: from ice ages to crayfish and SQUIDs,” Nature, vol. 373, pp. 33–36, 1995.
View at: Publisher Site | Google Scholar
R. Benzi, A. Sutera, and A. Vulpiani, “The mechanism of stochastic resonance,” Journal of Physics A: Mathematical and General, vol. 14, no. 11, pp. L453–L457, 1981.
View at: Publisher Site | Google Scholar
B. Xu, F. Duan, R. Bao, and J. Li, “Stochastic resonance with tuning system parameters: the application of bistable systems in signal processing,” Chaos, Solitons & Fractals, vol. 13, no. 4, pp. 633–644, 2002.
View at: Publisher Site | Google Scholar
G. Hu, G. Nicolis, and C. Nicolis, “Periodically forced Fokker-Planck equation and stochastic resonance,” Physical Review A, vol. 42, no. 4, pp. 2030–2041, 1990.
View at: Publisher Site | Google Scholar
M. Evstigneev, P. Reimann, V. Pankov, and R. H. Prince, “Stochastic resonance in monostable overdamped systems,” Europhysics Letters (EPL), vol. 65, no. 1, p. 7, 2004.
View at: Publisher Site | Google Scholar
J. M. G. Vilar and J. M. Rubí, “Divergent signal-to-noise ratio and stochastic resonance in monostable systems,” Physical Review Letters, vol. 77, no. 14, p. 2863, 1996.
View at: Publisher Site | Google Scholar
A. N. Grigorenko, S. I. Nikitin, and G. V. Roschepkin, “Stochastic resonance at higher harmonics in monostable systems,” Physical Review E, vol. 56, no. 5, pp. R4907–R4910, 1997.
View at: Publisher Site | Google Scholar
S. Lu, Q. He, H. Zhang, S. Zhang, and F. Kong, “Note: Signal amplification and filtering with a tristable stochastic resonance cantilever,” Review of Scientific Instruments, vol. 84, no. 2, Article ID 026110, 2013.
View at: Publisher Site | Google Scholar
P. Shi, X. Su, and D. Han, “Stochastic resonance in tristable system induced by dichotomous noise,” Modern Physics Letters B, vol. 30, no. 31, p. 1650377, 2016.
View at: Publisher Site | Google Scholar
S. Lu, Z. Dai, Y. Liu, G. Liu, H. Yang, and F. Wang, “Decreasing bit error rate by using enhanced tristable logical stochastic resonance in heavy noise condition,” Chinese Journal of Physics, vol. 58, pp. 179–189, 2019.
View at: Google Scholar
L. Valerio, “Stochastic resonance for nonequilibrium systems,” Physical Review E, vol. 100, p. 62124, 2019.
View at: Google Scholar
F. Guo, Y. R. Zhou, and Y. Zhang, “Stochastic resonance in a time-delayed bistable system subjected to dichotomous noise and white noise,” Chinese Journal of Physics, vol. 48, no. 2, pp. 294–303, 2010.
View at: Google Scholar
A. M. D. Paor and B. Cogan, “Windspeed-dependent underdamping and its cure in the self-excited series-wound aerogenerator,” Applied Energy, vol. 42, no. 4, pp. 253–267, 1992.
View at: Publisher Site | Google Scholar
T. Kawamura, Y. Imamura, H. Motoshiromizu, and M. Abe, “Evaluation of overdamping behavior of a vibrating conductive cylinder in a strong magnetic field,” IEEE Transactions on Magnetics, vol. 52, no. 10, p. 1, 2016.
View at: Publisher Site | Google Scholar
N. V. Agudov, A. V. Krichigin, D. Valenti, and B. Spagnolo, “Stochastic resonance in a trapping overdamped monostable system,” Physical Review E, vol. 81, no. 5 Pt 1, Article ID 051123, 2010.
View at: Publisher Site | Google Scholar
L. Liu, L. Cao, and L. Zhang, “Stochastic resonance in an overdamped harmonic oscillator,” Acta Physica Sinica, vol. 59, no. 3, pp. 1494–1498, 2010.
View at: Google Scholar
P. Landa, V. Ushakov, and J. Kurths, “Rigorous theory of stochastic resonance in overdamped bistable oscillators for weak signals,” Chaos, Solitons & Fractals, vol. 30, no. 3, pp. 574–578, 2006.
View at: Publisher Site | Google Scholar
P. Ushakov, S. C. Zhong, T. Zhe, and M. Hong, “Stochastic resonance of over-damped bistable system driven by chirp signal and Gaussian white noise,” Acta Physica Sinica, vol. 62, no. 8, Article ID 080501, 2013.
View at: Publisher Site | Google Scholar
B. Zhou and W. Xu, “Stochastic resonance in an asymmetric bistable system driven by multiplicative colored noise and additive white noise,” Chaos, Solitons & Fractals, vol. 38, no. 4, p. 1146, 2008.
View at: Publisher Site | Google Scholar
A. V. Barzykin and K. Seki, “Stochastic resonance driven by gaussian multiplicative noise,” Europhysics Letters (EPL), vol. 40, no. 2, p. 117, 1997.
View at: Publisher Site | Google Scholar
P. H. Nggi, P. Jung, C. Zerbe, and F. Moss, “Can colored noise improve stochastic resonance?” Journal of Statistical Physics, vol. 70, p. 25, 1993.
View at: Google Scholar
B. Xu, J. Li, F. Duan, and J. Zheng, “Effects of colored noise on multi-frequency signal processing via stochastic resonance with tuning system parameters,” Chaos, Solitons & Fractals, vol. 16, no. 1, p. 93, 2003.
View at: Publisher Site | Google Scholar
X. Gu, “Stochastic resonance driven by time-delayed feedback in a bistable system with colored noise,” European Physical Journal D, vol. 66, p. 67, 2012.
View at: Publisher Site | Google Scholar
V. Sorokin and I. Blekhman, “On the stochastic resonance phenomenon in parametrically excited systems,” European Journal of Applied Mathematics, vol. 30, no. 5, pp. 986–1003, 2018.
View at: Publisher Site | Google Scholar
K. K. Wang, H. Ye, Y. J. Wang, and S. H. Li, “Impact of time delay and a multiplicative periodic signal on stochastic resonance and steady states shift for a stochastic insect outbreak system subjected to Gaussian noises,” Chinese Journal of Physics, vol. 56, pp. 2204–2217, 2018.
View at: Google Scholar
G. Zhang, Y. J. Zhang, T. Q. Zhang, and R. Mdsohel, “Stochastic resonance in an asymmetric bistable system driven by multiplicative and additive Gaussian noise and its application in bearing fault detection,” Chinese Journal of Physics, vol. 56, pp. 1173–1186, 2018.
View at: Google Scholar
P. Shi, W. Zhang, D. Yuan, H. Xia, D. Han, and R. Fu, “Stochastic resonance in a time polo-delayed asymmetry bistable system driven by multiplicative white noise and additive color noise,” Chaos Solitons & Fractals, vol. 108, pp. 8–14, 2018.
View at: Publisher Site | Google Scholar
H. Zhang, T. Yang, Y. Xu, and W. Xu, “Parameter dependence of stochastic resonance in the FitzHugh-Nagumo neuron model driven by trichotomous noise,” European Physical Journal B, vol. 88, p. 125, 2015.
View at: Publisher Site | Google Scholar
B. Zhou and D. Lin, “Stochastic resonance in a time-delayed bistable system driven by trichotomous noise,” Indian Journal of Physics, vol. 91, no. 3, p. 299, 2017.
View at: Publisher Site | Google Scholar
Q. Qi and B. Zhou, “Stochastic resonance in an underdamped periodic potential system with symmetric trichotomous noise,” Indian Journal of Physics, vol. 94, no. 1, pp. 81–86, 2019.
View at: Publisher Site | Google Scholar
L. Lin, H. Wang, X. Huang, and Y. Wen, “Generalized stochastic resonance for a fractional harmonic oscillator with bias-signal-modulated trichotomous noise,” International Journal of Modern Physics B, vol. 32, Article ID 1850072, p. 23, 2017.
View at: Publisher Site | Google Scholar
W. Zhao, L. Wang, and J. Fan, “Theory and method for weak signal detection in engineering practice based on stochastic resonance,” International Journal of Modern Physics B, vol. 31, no. 28, Article ID 1750212, 2017.
View at: Publisher Site | Google Scholar
P.-M. Shi, Q. Li, and D.-Y. Han, “Stochastic resonance and MFPT in an asymmetric bistable system driven by correlated multiplicative colored noise and additive white noise,” International Journal of Modern Physics B, vol. 31, no. 14, Article ID 1750113, 2017.
View at: Publisher Site | Google Scholar
A. A. Saha and G. V. Anand, “Design of detectors based on stochastic resonance,” Signal Processing, vol. 83, no. 6, pp. 1193–1212, 2003.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2020 Si-Hai Zhao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies