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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.
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Keywords:
- stochastic resonance /
- fractional system /
- coupled system /
- trichotomous noise
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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 $ 
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 $ .
图 3 分数阶阻尼项的频率特征
Figure 3. Frequency characteristics of the fractional damping term
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 $ 
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 $ .
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 $ .
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
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[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 物理学报 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 物理学报 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 物理学报 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 物理学报 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 物理学报 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 物理学报 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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