搜索

x

留言板

尊敬的读者、作者、审稿人, 关于本刊的投稿、审稿、编辑和出版的任何问题, 您可以本页添加留言。我们将尽快给您答复。谢谢您的支持!

姓名
邮箱
手机号码
标题
留言内容
验证码

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

引用本文:
Citation:

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

The resonant behavior of an over-damped linear fractional Langevin equation

Zhong Su-Chuan, Gao Shi-Long, Wei Kun, Ma Hong
PDF
导出引用
  • 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.
    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.
    • 基金项目: 国家自然科学基金(批准号: 11171238)资助的课题.
    • Funds: Project supported by the National Natural Science Foundation of China (Grant No. 11171238).
    [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 物理学报 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 物理学报 58 2895]

    [9]

    Ning L J, Xu W 2009 Acta Phys. Sin. 58 2889 (in Chinese) [宁丽娟, 徐伟 2009 物理学报 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

  • [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 物理学报 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 物理学报 58 2895]

    [9]

    Ning L J, Xu W 2009 Acta Phys. Sin. 58 2889 (in Chinese) [宁丽娟, 徐伟 2009 物理学报 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

  • [1] 彭皓, 任芮彬, 钟扬帆, 蔚涛. 三态噪声激励下分数阶耦合系统的随机共振现象. 物理学报, 2022, 71(3): 030502. doi: 10.7498/aps.71.20211272
    [2] 彭皓, 任芮彬, 蔚涛. 三态噪声激励下分数阶耦合系统的随机共振现象研究. 物理学报, 2021, (): . doi: 10.7498/aps.70.20211272
    [3] 谢勇, 刘若男. 过阻尼搓板势系统的随机共振. 物理学报, 2017, 66(12): 120501. doi: 10.7498/aps.66.120501
    [4] 马正木, 靳艳飞. 二值噪声激励下欠阻尼周期势系统的随机共振. 物理学报, 2015, 64(24): 240502. doi: 10.7498/aps.64.240502
    [5] 刘德浩, 任芮彬, 杨博, 罗懋康. 涨落作用下周期驱动的分数阶过阻尼棘轮模型的混沌输运现象. 物理学报, 2015, 64(22): 220501. doi: 10.7498/aps.64.220501
    [6] 靳艳飞, 李贝. 色关联的乘性和加性色噪声激励下分段非线性模型的随机共振. 物理学报, 2014, 63(21): 210501. doi: 10.7498/aps.63.210501
    [7] 田艳, 黄丽, 罗懋康. 噪声交叉关联强度的时间周期调制对线性过阻尼系统的随机共振的影响. 物理学报, 2013, 62(5): 050502. doi: 10.7498/aps.62.050502
    [8] 田祥友, 冷永刚, 范胜波. 一阶线性系统的调参随机共振研究. 物理学报, 2013, 62(2): 020505. doi: 10.7498/aps.62.020505
    [9] 屠浙, 彭皓, 王飞, 马洪. 色噪声参激和周期调制噪声外激联合驱动的分数阶线性振子的共振行为. 物理学报, 2013, 62(3): 030502. doi: 10.7498/aps.62.030502
    [10] 张广丽, 吕希路, 康艳梅. 稳定噪声环境下过阻尼系统中的参数诱导随机共振现象. 物理学报, 2012, 61(4): 040501. doi: 10.7498/aps.61.040501
    [11] 张路, 钟苏川, 彭皓, 罗懋康. 乘性二次噪声驱动的线性过阻尼振子的随机共振. 物理学报, 2012, 61(13): 130503. doi: 10.7498/aps.61.130503
    [12] 高仕龙, 钟苏川, 韦鹍, 马洪. 过阻尼分数阶Langevin方程及其随机共振. 物理学报, 2012, 61(10): 100502. doi: 10.7498/aps.61.100502
    [13] 张良英, 金国祥, 曹力. 具有频率噪声的单模激光线性模型随机共振. 物理学报, 2011, 60(4): 044207. doi: 10.7498/aps.60.044207
    [14] 张莉, 刘立, 曹力. 过阻尼谐振子的随机共振. 物理学报, 2010, 59(3): 1494-1498. doi: 10.7498/aps.59.1494
    [15] 靳艳飞, 胡海岩. 一类线性阻尼振子的随机共振研究. 物理学报, 2009, 58(5): 2895-2901. doi: 10.7498/aps.58.2895
    [16] 宁丽娟, 徐伟. 信号调制下分段噪声驱动的线性系统的随机共振. 物理学报, 2009, 58(5): 2889-2894. doi: 10.7498/aps.58.2889
    [17] 郭立敏, 徐 伟, 阮春蕾, 赵 燕. 二值噪声驱动下二阶线性系统的随机共振. 物理学报, 2008, 57(12): 7482-7486. doi: 10.7498/aps.57.7482
    [18] 徐 伟, 靳艳飞, 徐 猛, 李 伟. 偏置信号调制下色关联噪声驱动的线性系统的随机共振. 物理学报, 2005, 54(11): 5027-5033. doi: 10.7498/aps.54.5027
    [19] 靳艳飞, 徐 伟, 李 伟, 徐 猛. 具有周期信号调制噪声的线性模型的随机共振. 物理学报, 2005, 54(6): 2562-2567. doi: 10.7498/aps.54.2562
    [20] 张良英, 曹 力, 吴大进. 具有色关联的色噪声驱动下单模激光线性模型的随机共振. 物理学报, 2003, 52(5): 1174-1178. doi: 10.7498/aps.52.1174
计量
  • 文章访问数:  4517
  • PDF下载量:  853
  • 被引次数: 0
出版历程
  • 收稿日期:  2012-01-08
  • 修回日期:  2012-02-20
  • 刊出日期:  2012-09-05

线性过阻尼分数阶Langevin方程的共振行为

  • 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.

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.

English Abstract

参考文献 (24)

目录

    /

    返回文章
    返回