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研究了在内噪声、外噪声（固有频率涨落噪声）及周期激励信号共同作用下具有指数型记忆阻尼的广义Langevin方程的共振行为. 首先将其转化为等价的三维马尔可夫线性系统，再利用Shapiro-Loginov公式和Laplace变换导出系统响应一阶矩和稳态响应振幅的解析表达式. 研究发现，当系统参数满足Routh-Hurwitz稳定条件时，稳态响应振幅随周期激励信号频率、记忆阻尼及外噪声参数的变化存在“真正”随机共振、传统随机共振和广义随机共振，且随机共振随着系统记忆时间的增加而减弱. 数值模拟计算结果表明系统响应功率谱与理论结果相符.
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关键词:
- 随机共振 /
- 广义Langevin方程 /
- 稳态响应振幅
The stochastic resonance is investigated in the generalized Langevin equation with exponential memory kernel subjected to the joint action of internal noise, external noise and external sinusoidal forcing. The system is converted into three-dimensional Markovian Langevin equations. Furthermore, using the Shapiro-Loginov formula and the Laplace transformation technique, the exact expressions of the first moment and the steady response amplitude are obtained. The research results show that with the variations of external sinusoidal force frequency and the parameters of memory kernel and external noise, the system presents bona-fide stochastic resonance, conventional stochastic resonance and stochastic resonance in a broad sense under the condition of Routh-Hurwitz stability. In addition, the stochastic resonance can be weakened as the memory time increases. Moreover, the numerical results of power spectrum of system are in agreement with the analytic results.[1] Kang Y M, Xu J X, Xie Y 2003 Acta Phys. Sin. 52 2712 (in Chinese) [康艳梅, 徐健学, 谢勇 2003 物理学报 52 2712]
[2] Tian Y, Huang L, Luo M K 2013 Acta Phys. Sin. 62 050502 (in Chinese) [田艳, 黄丽, 罗懋康 2013 物理学报 62 050502]
[3] Ning L J, Xu W 2007 Physica A 382 415
[4] Xu W, Jin Y F, Xu M, Li W 2005 Acta Phys. Sin. 54 5027 (in Chinese) [徐伟, 靳艳飞, 徐猛, 李伟2005 物理学报 54 5027]
[5] Berdichevsky V, Gitterman M 1999 Phys. Rev. E 60 1494
[6] Zhang L Y, Jin G X, Cao L, Wang Z Y 2012 Chin. Phys. B 21 120502
[7] Gitterman M 2012 Physica A 391 5343
[8] Gitterman M 2004 Phys. Rev. E 69 041101
[9] Zhang L Y, Jin G X, Cao L 2012 Acta Phys. Sin. 61 080502 (in Chinese) [张良英, 金国祥, 曹力 2012 物理学报 61 080502]
[10] Yu T, Zhang L, Luo M K 2013 Acta Phys. Sin. 62 120504 (in Chinese) [蔚涛, 张路, 罗懋康 2013 物理学报 62 120504]
[11] 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]
[12] Mankin R, Laas K, Sauga A 2011 Phys. Rev. E 83 061131
[13] Despósito M A, Viñales A D 2009 Phys. Rev. E 80 021111
[14] Viñales A D, Wang K G, Despósito M A 2009 Phys. Rev. E 80 011101
[15] Bao J D, Song Y L, Ji Q, Zhuo Y Z 2005 Phys. Rev. E 72 011113
[16] Siegle P, Goychuk I, Talkner P, Hänggi P 2010 Phys. Rev. E 81 011136
[17] Bao J D, Zhuo Y Z 2003 Phys. Rev. Lett. 91 138104
[18] Bao J D, Bai Z W 2005 Chin. Phys. Lett. 22 1845
[19] Zhong S C, Gao S L, Wei K, Ma H 2012 Acta Phys. Sin. 61 170501 (in Chinese) [钟苏川, 高仕龙, 韦鹍, 马洪 2012 物理学报 61 170501]
[20] Neiman A, Sung W 1996 Phys. Lett. A 223 341
[21] Shapiro V E, Loginov V M 1978 Physica A 91 563
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[1] Kang Y M, Xu J X, Xie Y 2003 Acta Phys. Sin. 52 2712 (in Chinese) [康艳梅, 徐健学, 谢勇 2003 物理学报 52 2712]
[2] Tian Y, Huang L, Luo M K 2013 Acta Phys. Sin. 62 050502 (in Chinese) [田艳, 黄丽, 罗懋康 2013 物理学报 62 050502]
[3] Ning L J, Xu W 2007 Physica A 382 415
[4] Xu W, Jin Y F, Xu M, Li W 2005 Acta Phys. Sin. 54 5027 (in Chinese) [徐伟, 靳艳飞, 徐猛, 李伟2005 物理学报 54 5027]
[5] Berdichevsky V, Gitterman M 1999 Phys. Rev. E 60 1494
[6] Zhang L Y, Jin G X, Cao L, Wang Z Y 2012 Chin. Phys. B 21 120502
[7] Gitterman M 2012 Physica A 391 5343
[8] Gitterman M 2004 Phys. Rev. E 69 041101
[9] Zhang L Y, Jin G X, Cao L 2012 Acta Phys. Sin. 61 080502 (in Chinese) [张良英, 金国祥, 曹力 2012 物理学报 61 080502]
[10] Yu T, Zhang L, Luo M K 2013 Acta Phys. Sin. 62 120504 (in Chinese) [蔚涛, 张路, 罗懋康 2013 物理学报 62 120504]
[11] 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]
[12] Mankin R, Laas K, Sauga A 2011 Phys. Rev. E 83 061131
[13] Despósito M A, Viñales A D 2009 Phys. Rev. E 80 021111
[14] Viñales A D, Wang K G, Despósito M A 2009 Phys. Rev. E 80 011101
[15] Bao J D, Song Y L, Ji Q, Zhuo Y Z 2005 Phys. Rev. E 72 011113
[16] Siegle P, Goychuk I, Talkner P, Hänggi P 2010 Phys. Rev. E 81 011136
[17] Bao J D, Zhuo Y Z 2003 Phys. Rev. Lett. 91 138104
[18] Bao J D, Bai Z W 2005 Chin. Phys. Lett. 22 1845
[19] Zhong S C, Gao S L, Wei K, Ma H 2012 Acta Phys. Sin. 61 170501 (in Chinese) [钟苏川, 高仕龙, 韦鹍, 马洪 2012 物理学报 61 170501]
[20] Neiman A, Sung W 1996 Phys. Lett. A 223 341
[21] Shapiro V E, Loginov V M 1978 Physica A 91 563
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