线性过阻尼分数阶Langevin方程的共振行为

钟苏川; 高仕龙; 韦鹍; 马洪

doi:10.7498/aps.61.170501

摘要

通过将广义Langevin方程中的系统内噪声建模为分数阶高斯噪声,推导出分数阶Langevin方程, 其分数阶导数项阶数由系统内噪声的Hurst指数所确定.讨论了处于强噪声环境下的线性过阻尼分数阶 Langevin方程在周期信号激励下的共振行为,利用Shapiro-Loginov公式和Laplace变换, 推导了系统响应的一、二阶稳态矩和稳态响应振幅、方差的解析表达式.分析表明,适当参数下, 系统稳态响应振幅和方差随噪声的某些特征参数、周期激励信号的频率及系统部分参数的变化出现了广义的随机共振现象.

关键词:

Abstract

By choosing the internal noise as a fractional Gaussian noise, we obtain the fractional Langevin equation. We explore the phenomenon of stochastic resonance in an over-damped linear fractional Langevin equation subjected to an external sinusoidal forcing. The influence of fluctuations of environmental parameters on the dynamics of the system is modeled by a dichotomous noise. Using the Shapiro-Loginov formula and the Laplace transformation technique, we obtain the exact expressions of the first and second moment of the output signal, the mean particle displacement and the variance of the output signal in the long-time limit t→∞. Finally, the numerical simulation shows that the over-damped linear fractional Langevin equation reveals a lot of dynamic behaviors and the stochastic resonance (SR) in a wide sense can be found with internal noise and external noise.

Keywords:

作者及机构信息

1.
四川大学数学学院, 成都 610064;

2.
四川大学锦城学院, 成都 611731;

3.
乐山师范学院数信学院, 乐山 614000

基金项目: 国家自然科学基金(批准号: 11171238)资助的课题.

Authors and contacts

1.
College of Mathematics, Sichuan University, Chengdu 610065, China;

2.
Sichuan University Jincheng College, Sichuan University, Chengdu 611731, China;

3.
College of Mathematics and Information Science, Leshan Normal University, Leshan 614000, China

Funds: Project supported by the National Natural Science Foundation of China (Grant No. 11171238).

参考文献

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[2]	Gong Y B, Hou Z H, Xin H W 2004 Chemical Journal of Chinese Universities 25 1477 (in Chinese) [龚玉兵, 候中怀, 辛厚文 2004 高等学校化学学报 25 1477]
[3]	Guo Y F, Xu W 2008 Acta Phys. Sin. 57 6081 (in Chinese) [郭永峰, 徐伟 2008 57 6081]
[4]	Huang F, Liu F 2005 The Anziam Journal. 46 317
[5]	Liu F, Anh V, Turner I, Zhuang P 2003 J. Appl. Math. & Computing 13 233
[6]	Benson D A, Wheatcraft S W, Meerschaert M M 2000 Water Resources Research 36 1403
[7]	Gitterman M 2005 Physical A 352 309
[8]	Jin Y F, Hu H Y 2009 Acta Phys. Sin. 58 2895 (in Chinese) [靳艳飞, 胡海岩 2009 58 2895]
[9]	Ning L J, Xu W 2009 Acta Phys. Sin. 58 2889 (in Chinese) [宁丽娟, 徐伟 2009 58 2889]
[10]	Kou S C 2008 Institute of Mathematical Statistics in The Annals of Applied Statistics 2 7, 8
[11]	Kou S C, Xie X S 2004 Physical Review Letters 93 180603
[12]	Laas K, Mankin R, Reiter E 2011 International Journal of Mathematical Models and Methods in Applied Sciences 5 281, 283
[13]	Zhang J Q, Xin H W 2001 Progress in Chemistry 13 241 (in Chinese) [张季谦, 辛厚文 2001 化学进展 13 241]
[14]	Rao B P 2010 Statistical Inference for Fractional Diffusion Processes (1st Ed.) (India: A John Wiley and Sons) p7, 23
[15]	Mishura Y S 2008 Stochastic Calculus for Fractional Brownian Motion and Related Processes (1st Ed.) (German: Springer) p7
[16]	Mandelbrot B B, Van Ness J W 1968 Siam Review 10 427
[17]	Rekker A, Mankin R 2010 Wseas Transactions on Systems 9 207
[18]	Deng W H, Barkai E 2009 Physical Review E 79 01112
[19]	Podlubny I 1999 Fractional Differential Equations (1st Ed.) (San Diego: Academic Press) p79
[20]	Burov S, Barkai E 2008 Physical Review E 78 031112
[21]	Shapiro V E, Loginov V M 1978 Physica A 91 563
[22]	Oppenheim A V, Willsky A S, Nawab S H (Translated by Liu S T) 2005 Signals and Systems (9st Ed.) (Xian: Prentice Hall) pp128, 471, 497-500 (in Chinese) [奥本海姆. A. V. 著刘树棠译 2005 信号与系统(第九版)(西安:西安交通大学出版社) 第128, 471, 497-500页]
[23]	Soika E, Mankin R 2010 Advances in Biomedical Research 1790-5125 442
[24]	Sauga A, Mankin R, Ainsaar A 2010 Wseas Transactions on Systems 9 1021

施引文献

[1]	Bao J D 2009 Random Simulation Method of Classical and Quantum Dissipation System (1st Ed.) (Beijing: Science Press) p79, 84, 147 (in Chinese) [包景东 2009 经典和量子耗散系统的随机模拟方法(第一版)(北京:科学出版社) 第79, 84, 147页]
[2]	Gong Y B, Hou Z H, Xin H W 2004 Chemical Journal of Chinese Universities 25 1477 (in Chinese) [龚玉兵, 候中怀, 辛厚文 2004 高等学校化学学报 25 1477]
[3]	Guo Y F, Xu W 2008 Acta Phys. Sin. 57 6081 (in Chinese) [郭永峰, 徐伟 2008 57 6081]
[4]	Huang F, Liu F 2005 The Anziam Journal. 46 317
[5]	Liu F, Anh V, Turner I, Zhuang P 2003 J. Appl. Math. & Computing 13 233
[6]	Benson D A, Wheatcraft S W, Meerschaert M M 2000 Water Resources Research 36 1403
[7]	Gitterman M 2005 Physical A 352 309
[8]	Jin Y F, Hu H Y 2009 Acta Phys. Sin. 58 2895 (in Chinese) [靳艳飞, 胡海岩 2009 58 2895]
[9]	Ning L J, Xu W 2009 Acta Phys. Sin. 58 2889 (in Chinese) [宁丽娟, 徐伟 2009 58 2889]
[10]	Kou S C 2008 Institute of Mathematical Statistics in The Annals of Applied Statistics 2 7, 8
[11]	Kou S C, Xie X S 2004 Physical Review Letters 93 180603
[12]	Laas K, Mankin R, Reiter E 2011 International Journal of Mathematical Models and Methods in Applied Sciences 5 281, 283
[13]	Zhang J Q, Xin H W 2001 Progress in Chemistry 13 241 (in Chinese) [张季谦, 辛厚文 2001 化学进展 13 241]
[14]	Rao B P 2010 Statistical Inference for Fractional Diffusion Processes (1st Ed.) (India: A John Wiley and Sons) p7, 23
[15]	Mishura Y S 2008 Stochastic Calculus for Fractional Brownian Motion and Related Processes (1st Ed.) (German: Springer) p7
[16]	Mandelbrot B B, Van Ness J W 1968 Siam Review 10 427
[17]	Rekker A, Mankin R 2010 Wseas Transactions on Systems 9 207
[18]	Deng W H, Barkai E 2009 Physical Review E 79 01112
[19]	Podlubny I 1999 Fractional Differential Equations (1st Ed.) (San Diego: Academic Press) p79
[20]	Burov S, Barkai E 2008 Physical Review E 78 031112
[21]	Shapiro V E, Loginov V M 1978 Physica A 91 563
[22]	Oppenheim A V, Willsky A S, Nawab S H (Translated by Liu S T) 2005 Signals and Systems (9st Ed.) (Xian: Prentice Hall) pp128, 471, 497-500 (in Chinese) [奥本海姆. A. V. 著刘树棠译 2005 信号与系统(第九版)(西安:西安交通大学出版社) 第128, 471, 497-500页]
[23]	Soika E, Mankin R 2010 Advances in Biomedical Research 1790-5125 442
[24]	Sauga A, Mankin R, Ainsaar A 2010 Wseas Transactions on Systems 9 1021

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