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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.
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Keywords:
- fractional Langevin equations /
- over-damped linear oscillator /
- fractional Gaussian noise /
- stochastic resonance
[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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[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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