Stochastic resonance of fractional-order coupled system excited by trichotomous noise

Peng Hao; Ren Rui-Bin; Zhong Yang-Fan; Yu Tao

doi:10.7498/aps.71.20211272

Abstract

In order to describe the motion behavior of coupled particles with mass fluctuations in a viscous medium, we propose a corresponding model, namely a fractional-order coupled system excited by trichotomous noise. By using the Shapiro-Loginov formula and the Laplace transform, we find the statistical synchronization of the system, then obtain analytical expression of the system output amplitude gain. On this basis, this paper focuses on the key points, which are the coupled system, the fractional order system and the trichotomous noise, analyzes the influences of coupling coefficient, system order and noise steady-state probability on the generalized stochastic resonance phenomenon of system’s output amplitude gain, and gives some reasonable explanations. Specifically, first, as the coupling coefficient increases, the generalized stochastic resonance phenomenon of the output amplitude gain of the system first increases and then weakens until it converges. This phenomenon shows that the appropriate coupling strength can promote the generation of system resonance, thereby reflecting the importance of studying coupled systems. Second, with the order of the system increases, the generalized stochastic resonance phenomenon of the system’s output amplitude gain weakens gradually. When the system order value is 1, that is, when the system degenerates into an integer order system, the peak value of its output amplitude gain is smallest. This phenomenon shows that the fractional order system can obtain a larger output amplitude gain than the traditional integer order system. Third, the effect of the steady-state probability of noise on the output amplitude gain of the system changes with other related parameters. Under certain parameter conditions, trichotomous noise can not only make the output amplitude of the system larger than that of the system excited by dichotomous noise, but also change the resonance type of the system. Finally, the correctness of the above results is verified by numerical simulation.

Keywords:

Authors and contacts

1.
College of Mathematics, Southwest Jiaotong University, Chengdu 610064, China
2.
College of Mathematics, Sichuan University, Chengdu 610064, China

Corresponding author: Yu Tao, scuyutao@163.com

Funds: Project supported by the Young Scientists Fund of the National Natural Science Foundation of China (Grant No.11501385).

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References

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Cited By

图 1 随q变化的GSR在$\alpha\text{-}\varepsilon$平面的相图, 其中$ m=1 $, $ \gamma=0.1 $, $ \omega=1 $, $ \varOmega=1.4 $, $ \lambda=0.1 $, $ a=0.7 $

Figure 1. Phase diagram for GSR versus q in the $\alpha\text{-}\varepsilon$ plane, with $ m=1 $, $ \gamma=0.1 $, $ \omega=1 $, $ \varOmega=1.4 $, $ \lambda=0.1 $, $ a=0.7 $

DownLoad: Full-Size Img PowerPoint

图 2 系统输出幅值增益随噪声稳态转移概率q变化的GSR现象, 其中$ m=1 $, $ \gamma=0.1 $, $ \omega=1 $, $ \varOmega=1.4 $, $ \lambda=0.1 $, $ a=0.7 $　(a) $ \varepsilon=4 $; (b) $ \alpha=0.2 $

Figure 2. The OAG versus q with $ m=1 $, $ \gamma=0.1 $, $ \omega=1 $, $ \varOmega=1.4 $, $ \lambda=0.1 $, $ a=0.7 $: (a) $ \varepsilon=4 $; (b) $ \alpha=0.2 $.

DownLoad: Full-Size Img PowerPoint

图 3 分数阶阻尼项的频率特征

Figure 3. Frequency characteristics of the fractional damping term

DownLoad: Full-Size Img PowerPoint

图 4 随 ε变化的GSR现象在$q\text{-}\alpha$平面的相图, 其中$ m=1 $, $ \gamma=0.1 $, $ \omega=1 $, $ \varOmega=1.6 $, $ \lambda=0.1 $, $ a=0.4 $

Figure 4. Phase diagram for GSR versus ε in the $q\text{-}\alpha$ plane, with $ m=1 $, $ \gamma=0.1 $, $ \omega=1 $, $ \varOmega=1.6 $, $ \lambda=0.1 $, $ a=0.4 $

DownLoad: Full-Size Img PowerPoint

图 5 系统输出幅值增益随耦合系数 ε变化的GSR现象, 其中$ m=1 $, $ \gamma=0.1 $, $ \omega=1 $, $ \varOmega=1.6 $, $ \lambda=0.1 $, $ a=0.4 $ (a) $q= $$ 0.3$; (b) $ \alpha=0.1 $

Figure 5. The OAG versus ε with $ m=1 $, $ \gamma=0.1 $, $ \omega=1 $, $ \varOmega=1.6 $, $ \lambda=0.1 $, $ a=0.4 $: (a) $ q=0.3 $; (b) $ \alpha=0.1 $.

DownLoad: Full-Size Img PowerPoint

图 6 系统输出信号　(a) 时域图; (b) 频域图. $m = 1$, $\gamma = 0.1$, $\alpha = 0.5$, $\varepsilon = 1$, $\omega = 1$, $\varOmega = 0.4\pi$, $a = 0.2$, $\lambda = 0.1$, $q = 0.3$

Figure 6. System output signal: (a) Time domain diagram; (b) frequency domain diagram. $ m=1 $, $ \gamma=0.1 $, $ \alpha=0.5 $, $ \varepsilon=1 $, $ \omega=1 $, $ \varOmega=0.4\pi $, $ a=0.2 $, $ \lambda=0.1 $, $ q=0.3 $.

DownLoad: Full-Size Img PowerPoint

图 7 仿真误差图　(a) 仿真与理论对比图; (b) 仿真误差与仿真次数关系图

Figure 7. Simulation error graph: (a) Comparison between simulation and theory; (b) relationship between simulation error and simulation times

DownLoad: Full-Size Img PowerPoint

map

[1]	Benzi R, Sutera A, Vulpiani A 1981 J. Phys. A 14 L453 Google Scholar
[2]	Wiesenfeld K, Moss F 1995 Nature 373 33 Google Scholar
[3]	Gammaitoni L, Hänggi P, Jung P, Marchesoni F 1998 Rev. Mod. Phys. 70 223 Google Scholar
[4]	Hänggi P 2002 Chem. Phys. Chem. 3 285 Google Scholar
[5]	Zhang W, Xiang B R 2006 Talanta 70 267 Google Scholar
[6]	Chen H, Varshney P K, Kay S M, Michels J H 2007 IEEE Trans. Signal Process. 55 3172 Google Scholar
[7]	Soika E, Mankin R, Ainsaar A 2010 Phys. Rev. E 81 011141 Google Scholar
[8]	Gang H, Ditzinger T, Ning C Z, Haken H 1993 Phys. Rev. Lett. 71 807 Google Scholar
[9]	Pikovsky A, Zaikin A, Ma D L C 2002 Phys. Rev. Lett. 88 050601 Google Scholar
[10]	Tessone C J, Mirasso C R, Toral R, Gunton J D 2006 Phys. Rev. Lett. 97 194101 Google Scholar
[11]	冷永刚, 王太勇, 郭焱, 吴振勇 2007 56 30 Google Scholar Leng Y G, Wang T Y, Guo Y, Wu Z Y 2007 Acta Phys. Sin. 56 30 Google Scholar
[12]	林敏, 黄咏梅, 方利民 2008 57 2041 Google Scholar Lin M, Huang Y M, Fang L M 2008 Acta Phys. Sin. 57 2041 Google Scholar
[13]	张莉, 刘立, 曹力 2010 59 1494 Google Scholar Zhang L, Liu L, Cao L 2010 Acta Phys. Sin. 59 1494 Google Scholar
[14]	Berdichevsky V, Gitterman M 1999 Phys. Rev. E 60 1494 Google Scholar
[15]	Yu T, Zhang L, Luo M K 2013 Phys. Scr. 88 045008 Google Scholar
[16]	Zhong S C, Ma H, Peng H, Zhang L 2015 Nonlinear Dyn. 82 535 Google Scholar
[17]	Yang B, Zhang X, Zhang L 2016 Phys. Rev. E 94 022119
[18]	Lin L F, Yu L, Wang H Q 2017 Commun. Nonlinear Sci. Numer. Simul. 43 171 Google Scholar
[19]	Zhang L, Lai L, Peng H 2018 Phys. Rev. E 97 012147 Google Scholar
[20]	Yu T, Zhang L, Ji Y D, Lai L 2019 Commun. Nonlinear Sci. Numer. Simul. 72 26 Google Scholar
[21]	Lin L F, Wang H Q 2019 Nonlinear Dyn. 98 801 Google Scholar
[22]	姜磊, 赖莉, 蔚涛, 罗懋康 2021 70 130501 Google Scholar Jiang L, Lai L, Yu T, Luo M K 2021 Acta Phys. Sin. 70 130501 Google Scholar
[23]	Ausloos M, Lambiotte R 2006 Phys. Rev. E 73 011105 Google Scholar
[24]	Gitterman M, Klyatskin V I 2010 Phys. Rev. E 81 051139 Google Scholar
[25]	Gitterman M, Shapiro I 2011 J. Stat. Phys. 144 139 Google Scholar
[26]	蔚涛, 张路, 罗懋康 2013 62 120504 Google Scholar Yu T, Zhang L, Luo M K 2013 Acta Phys. Sin. 62 120504 Google Scholar
[27]	蔚涛, 罗懋康, 华云 2013 62 210503 Google Scholar Yu T, Luo M K, Hua Y 2013 Acta Phys. Sin. 62 210503 Google Scholar
[28]	Broeck C V D 1983 J. Stat. Phys. 31 467 Google Scholar
[29]	Zhong S C, Wei K, Gao S L, Ma H 2015 J. Stat. Phys. 159 195 Google Scholar
[30]	Mankin R, Ainsaar A, Reiter E 1999 Phys. Rev. E 60 1374 Google Scholar
[31]	Lang R L, Yang L, Qin H L, Di G H 2012 Nonlinear Dyn. 69 1423 Google Scholar
[32]	Peng H, Ren R B, Li P F, Yu T 2020 Phys. Scr. 95 075214 Google Scholar
[33]	Xu P F, Jin Y F 2020 Chaos, Solitons Fractals 138 109857 Google Scholar
[34]	Jin Y F, Wang H Q 2020 Chaos, Solitons Fractals 133 109633 Google Scholar
[35]	Jin Y F, Niu S Y 2016 Int. J. Mod. Phys. B 30 1650102 Google Scholar
[36]	Bagley R L, Torvik P J 1984 Appl. Mech. 51 294 Google Scholar
[37]	Lutz L 2001 Phys. Rev. E 64 051106 Google Scholar
[38]	He G T, Tian Y, Wang Y 2013 J. Stat. Mech. 9 26 Google Scholar
[39]	Jiang L, Lai L, Yu T, Luo M K 2021 J. Stat. Mech. 2021 063208 Google Scholar
[40]	Cubero D 2008 Phys. Rev. E 77 021112 Google Scholar
[41]	Tang Y, Zou W, Lu J, Kurths J 2012 Phys. Rev. E 85 046207 Google Scholar
[42]	Atsumi Y, Hata S, Nakao H 2013 Phys. Rev. E 2013 88 Google Scholar
[43]	Oyarzabal R S, Szezech Jr J D, Batista A M 2017 Commun. Nonlinear Sci. Numer. Simul. 54 62 Google Scholar
[44]	Zhang L, Xu L, Yu T, Lai L, Zhong S C 2021 Commun. Nonlinear Sci. Numer. Simul. 93 105499 Google Scholar
[45]	Shapiro V E, Loginov V M 1978 Phys. A 91 563 Google Scholar

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