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将线性随机振动系统中通常的简谐势阱推广为更一般的幂函数型势阱，得到幂函数型单势阱非线性随机振动系统. 利用随机情形下的二阶Runge-Kutta算法研究了噪声强度、势阱参数和周期激励参数对系统稳态响应的一阶矩振幅和系统响应的稳态方差的影响. 对决定势阱形状的势阱参数之一b 历经bb > 2以及相当于简谐势阱的b=2等全部情况的研究表明：随噪声强度D的变化，系统稳态响应的一阶矩振幅可以在bb=2 简谐势阱以及b >2的情况，则无该现象发生；随势阱参数的变化，系统稳态响应的一阶矩振幅以及系统响应的稳态方差也可以发生非单调变化.
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关键词:
- 单势阱系统 /
- 随机振动 /
- 随机共振 /
- 随机Runge-Kutta算法
To generalize the harmonic potential of the linear random vibration system, a more general power type potential is presented, and the corresponding power function type nonlinear single-well random vibration system is obtained. The first moment of the system steady-state response and the stationary variance of the system response, which are influenced by noise strength, parameters of the potential and the periodic excitation, are studied by using the second order stochastic Runge-Kutta algorithm. The parameter b, which determines the shape of the potential, goes through b b > 2 and b=2 (harmonic potential), and it is shown that varying the noise strength, if b b=2 (harmonic potential) or b > 2, this phenomenon does not occur; varying the parameters of the potential, the first moment of the system steady-state response and the stationary variance of the system response can also be non-monotonic.-
Keywords:
- single-well system /
- random vibrations /
- stochastic resonance /
- stochastic Runge-Kutta algorithm
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[2] Einstein A 1905 Annalen der Physik 17 549
[3] Paez T L, Consulting T P, Colorado D 2012 Sound Vib. 46 52
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[5] Chen H, Varshney P K, Kay S M, Michels J H 2007 IEEE Trans. Sig. Process. 55 3172
[6] Gammaitoni L, Hänggi P, Jung P, Marchesoni F 1998 Rev. Mod. Phys. 70 223
[7] Gitterman M 2005 Physica A 352 309
[8] Zhang L, Zhong S C, Peng H, Luo M K 2012 Acta Phys. Sin. 61 130503 (in Chinese) [张路, 钟苏川, 彭浩, 罗懋康 2012 物理学报 61 130503]
[9] Zhao W L, Wang J, Wang L 2013 Chaos 23 033117
[10] Heinsalu E, Patriarca M, Marchesoni F 2009 Eur. Phys. J. B 69 19
[11] Li J L, Zeng L Z 2011 Chin. Phys. B 20 010503
[12] Agudov N V, Krichigin A V, Valenti D, Spagnolo B 2010 Phys. Rev. E 81 051123
[13] Grigorenko A N, Nikitin S I, Roschepkin G V 1997 Phys. Rev. E 56 4907
[14] Tian X Y, Leng Y G, Fan S B 2013 Acta Phys. Sin. 62 020505 (in Chinese) [田祥友, 冷永刚, 范胜波 2013 物理学报 62 020505]
[15] Zhang W, Xiang B R 2006 Talanta 70 267
[16] Gilbarg D, Trudinger N 2001 Elliptic Partial Differential Equations of Second Order (Berlin: Springer) pp149,152
[17] Lu Z H, Lin J H, Hu G 1993 Acta Phys. Sin. 42 1556 (in Chinese) [卢志恒, 林建恒, 胡岗 1993 物理学报 42 1556]
[18] Honeycutt 1992 Phys. Rev. A 45 62
[19] Bao J D 2009 Random Simulation Method of Classical and Quantum Dissipation System (Beijing: Science Press) p113 (in Chinese) [包景东 2009 经典和量子耗散系统的随机模拟方法 (北京: 科学出版社) 第113页]
[20] Li R H, Liu B 2009 The Numerical Solution of Differential Equations (4th ed.) (Beijing: Higher Education Press) pp33-37 (in Chinese) [李荣华, 刘播 2009 微分方程数值解法 (第四版) (北京: 高等教育出版社) 第33–37页]
[21] Rumelin W 1982 SIAM J. Numer. Anal. 19 604
[22] Cortes J C, Jodar L, Villafuerte L 2007 Math. Comput. Model. 45 757
[23] Pettersson R 1992 Stoch. Anal. Appl. 10 603
[24] Zhang W N, Du Z D, Xu B 2006 Ordinary Differential Equations (Beijing: Higher Education Press) pp89-108 (in Chinese) [张伟年, 杜正东, 徐冰 2006 常微分方程) (北京: 高等教育出版社) 第89–108页]
[25] Mitaim S, Kosko B 1998 Proc. IEEE 86 2152
-
[1] Zhu W Q 1998 Random Vibration (Beijing: Science Press) p1 (in Chinese) [朱位秋 1998 随机振动 (北京: 科学出版社) 第1页]
[2] Einstein A 1905 Annalen der Physik 17 549
[3] Paez T L, Consulting T P, Colorado D 2012 Sound Vib. 46 52
[4] Benzi R, Sutera A, Vulpiani A 1981 J. Phys. A: Math. Gen. 14 L453
[5] Chen H, Varshney P K, Kay S M, Michels J H 2007 IEEE Trans. Sig. Process. 55 3172
[6] Gammaitoni L, Hänggi P, Jung P, Marchesoni F 1998 Rev. Mod. Phys. 70 223
[7] Gitterman M 2005 Physica A 352 309
[8] Zhang L, Zhong S C, Peng H, Luo M K 2012 Acta Phys. Sin. 61 130503 (in Chinese) [张路, 钟苏川, 彭浩, 罗懋康 2012 物理学报 61 130503]
[9] Zhao W L, Wang J, Wang L 2013 Chaos 23 033117
[10] Heinsalu E, Patriarca M, Marchesoni F 2009 Eur. Phys. J. B 69 19
[11] Li J L, Zeng L Z 2011 Chin. Phys. B 20 010503
[12] Agudov N V, Krichigin A V, Valenti D, Spagnolo B 2010 Phys. Rev. E 81 051123
[13] Grigorenko A N, Nikitin S I, Roschepkin G V 1997 Phys. Rev. E 56 4907
[14] Tian X Y, Leng Y G, Fan S B 2013 Acta Phys. Sin. 62 020505 (in Chinese) [田祥友, 冷永刚, 范胜波 2013 物理学报 62 020505]
[15] Zhang W, Xiang B R 2006 Talanta 70 267
[16] Gilbarg D, Trudinger N 2001 Elliptic Partial Differential Equations of Second Order (Berlin: Springer) pp149,152
[17] Lu Z H, Lin J H, Hu G 1993 Acta Phys. Sin. 42 1556 (in Chinese) [卢志恒, 林建恒, 胡岗 1993 物理学报 42 1556]
[18] Honeycutt 1992 Phys. Rev. A 45 62
[19] Bao J D 2009 Random Simulation Method of Classical and Quantum Dissipation System (Beijing: Science Press) p113 (in Chinese) [包景东 2009 经典和量子耗散系统的随机模拟方法 (北京: 科学出版社) 第113页]
[20] Li R H, Liu B 2009 The Numerical Solution of Differential Equations (4th ed.) (Beijing: Higher Education Press) pp33-37 (in Chinese) [李荣华, 刘播 2009 微分方程数值解法 (第四版) (北京: 高等教育出版社) 第33–37页]
[21] Rumelin W 1982 SIAM J. Numer. Anal. 19 604
[22] Cortes J C, Jodar L, Villafuerte L 2007 Math. Comput. Model. 45 757
[23] Pettersson R 1992 Stoch. Anal. Appl. 10 603
[24] Zhang W N, Du Z D, Xu B 2006 Ordinary Differential Equations (Beijing: Higher Education Press) pp89-108 (in Chinese) [张伟年, 杜正东, 徐冰 2006 常微分方程) (北京: 高等教育出版社) 第89–108页]
[25] Mitaim S, Kosko B 1998 Proc. IEEE 86 2152
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