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较之于线性噪声, 非线性噪声更广泛地存在于实际系统中, 但其研究远不能满足实际情况的需要. 针对作为非线性阻尼涨落噪声基本构成成分的二次阻尼涨落噪声, 本文考虑了周期信号与之共同作用下的线性谐振子, 关注这类具有基本意义的阻尼涨落噪声的非线性对系统共振行为的影响. 利用Shapiro-Loginov公式和Laplace变换推导了系统稳态响应振幅的解析表达式, 并分析了稳态响应振幅的共振行为, 且以数值仿真验证了理论分析的有效性. 研究发现: 系统稳态响应振幅关于非线性阻尼涨落噪声系数具有非单调依赖关系, 特别是非线性阻尼涨落噪声比线性阻尼涨落噪声更有助于增强系统对外部周期信号的响应程度; 而且, 非线性阻尼涨落噪声比线性阻尼涨落噪声使得稳态响应振幅关于噪声强度具有更为丰富的共振行为; 同时, 二次阻尼涨落噪声使得稳态响应振幅关于系统频率出现真正的共振现象; 而在这些现象和性质中, 非线性噪声项的非线性性质对共振行为起着关键的作用. 显然, 以二次阻尼涨落作为基本形式引入的非线性阻尼涨落噪声, 可以有助于提高微弱周期信号检测的灵敏度和实现对周期信号的频率估计.Although non-linear noise exists far more widely in actual systems than linear noise, the study on non-linear noise is far from meeting the needs of practical situations as yet. The phenomenon of stochastic resonance (SR) is a non-linear cooperative effect which is jointly produced by signal, noise, and system, obviously, it is closely related to the nature of the noise. As a result, the non-linear nature of the non-linear noise has an inevitable impact on the dynamic behavior of a system, so it is of great significance to study the non-linear noise's influence on the dynamic behavior of the system. The linear harmonic oscillator is the most basic model to describe different phenomena in nature, and the quadratic noise is the most basic non-linear noise. In this paper, we consider a linear harmonic oscillator driven by an external periodic force and a quadratic damping fluctuation. For the proposed model, we focus on the effect of non-linear nature of quadratic fluctuation on the system's resonant behavior. Firstly, by the use of the Shapiro-Loginov formula and the Laplace transform technique, the analytical expressions of the first moment and the steady response amplitude of the output signal are obtained. Secondly, by studying the impacts of noise parameters and system intrinsic frequency, the non-monotonic behaviors of the steady response amplitude are found. Finally, numerical simulations are presented to verify the effectiveness of the analytical result. According to the research, we have the following conclusions: (1) The steady response amplitude is a non-monotonic function of coefficients of the quadratic damping fluctuation. Furthermore, the non-linear damping fluctuation is easier to contribute the system's enhancing response to the external periodic signal than the linear fluctuation. (2) The evolution of the steady response amplitude versus noise intensity presents more resonant behaviors. One-peak SR phenomenon and double-peak SR phenomenon are observed at different values of coefficients of the quadratic noise, particularly, the SR phenomenon disappears at the positive quadratic coefficient of the quadratic noise. (3) The evolution of the steady response amplitude versus the system intrinsic frequency presents true resonance, i. e. the phenomenon of resonance appears when the external signal frequency is equal to the system intrinsic frequency. True resonance is not observed in the linear harmonic oscillator driven by a linear damping fluctuation as yet. In conclusion, all the researches show that the non-linear nature of non-linear noise plays a key role in system's resonant behavior, in addition, the non-linear damping fluctuation is conductive to the detection and frequency estimation of weak periodic signal.
[1] Benzi R, Sutera A, Vulpiani A 1981 J. Phys. A: Math. Gen. 14 L453
[2] Wiesenfeld K, Moss F 1995 Nature 373 33
[3] Gitterman M 2005 Physica A 352 309
[4] Benzi R 2010 Nonlinear Proc. Geophys. 17 431
[5] Gammaitoni L, Hnggi P, Jung P, Marchesoni F 2009 Eur. Phys. J. B 69 1
[6] McDonnell M D, Abbott D 2009 Plos Comput. Biol. 5 e1000348
[7] Wellens T, Shatokhin V, Buchleitner A 2004 Rep. Prog. Phys. 67 45
[8] Hnggi P, Jung P, Zerbe C, Moss F 1993 J. Stat. Phys. 70 25
[9] Gammaitoni L, Hnggi P, Jung P, Marchesoni F 1998 Rev. Mod. Phys. 70 223
[10] McNamara B, Wiesenfeld K 1989 Phys. Rev. A 39 4854
[11] Fox R F 1989 Phys. Rev. A 39 4148
[12] Fulinski A 1995 Phys. Rev. E 52 4523
[13] Katrin L, Romi M, Astrid R 2009 Phys. Rev. E 79 051128
[14] Berdichevsky V, Gitterman M 1996 Europhys. Lett. 36 161
[15] 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]
[16] Lin L F, Tian Y, Ma H 2014 Chin. Phys. B 23 080503
[17] Li D S, Li J H 2010 Commun. Theor. Phys. 53 298
[18] Gitterman M, Shapiro I 2011 J. Stat. Phys. 144 139
[19] Jiang S Q, Guo F, Zhou Y R, Gu T X 2007 International Conference on Communications, Circuits and Systems Fukuoka, Japan, July 11-13, 2007 p1044
[20] Ning L J, Xu W, Yao M L 2007 Chin. Phys. 16 2595
[21] Zhong S C, Yu T, Zhang L, Ma H 2015 Acta Phys. Sin. 64 020202 (in Chinese) [钟苏川, 蔚涛, 张路, 马洪 2015 物理学报 64 020202]
[22] Zhang L, Zhong S C, Peng H, Luo M K 2011 Chin. Phys. Lett. 28 090505
[23] Gitterman M 2004 Phys. Rev. E 69 041101
[24] Murray S I, Marlan O S, Willis E J 1974 Laser Physics (Rewood City: Addison-Wesley Publishing) p197
[25] Zhang L Y, Cao L, Wu D J 2008 Commun. Theor. Phys. 49 1310
[26] Sancho J M, San Miguel, Drr M D 1982 J. Stat. Phys. 28 291
[27] Sagues F, Migurel S M, Sacho J M 1984 Z. Phys. B 55 269
[28] Hector C, Fernando M, Enrique T 2006 Phys. Rev. E 74 022102
[29] Zhang L, Zhong S C, Peng H, Luo M K 2012 Acta Phys. Sin. 61 130503 (in Chinese) [张路, 钟苏川, 彭皓, 罗懋康 2012 物理学报 61 130503]
[30] Bena I, Broeck C V D, Kawai R, Lindenberg K 2002 Phys. Rev. E 66 045603
[31] Bena I 2006 Int. J. Mod. Phys. B 20 2825
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[1] Benzi R, Sutera A, Vulpiani A 1981 J. Phys. A: Math. Gen. 14 L453
[2] Wiesenfeld K, Moss F 1995 Nature 373 33
[3] Gitterman M 2005 Physica A 352 309
[4] Benzi R 2010 Nonlinear Proc. Geophys. 17 431
[5] Gammaitoni L, Hnggi P, Jung P, Marchesoni F 2009 Eur. Phys. J. B 69 1
[6] McDonnell M D, Abbott D 2009 Plos Comput. Biol. 5 e1000348
[7] Wellens T, Shatokhin V, Buchleitner A 2004 Rep. Prog. Phys. 67 45
[8] Hnggi P, Jung P, Zerbe C, Moss F 1993 J. Stat. Phys. 70 25
[9] Gammaitoni L, Hnggi P, Jung P, Marchesoni F 1998 Rev. Mod. Phys. 70 223
[10] McNamara B, Wiesenfeld K 1989 Phys. Rev. A 39 4854
[11] Fox R F 1989 Phys. Rev. A 39 4148
[12] Fulinski A 1995 Phys. Rev. E 52 4523
[13] Katrin L, Romi M, Astrid R 2009 Phys. Rev. E 79 051128
[14] Berdichevsky V, Gitterman M 1996 Europhys. Lett. 36 161
[15] 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]
[16] Lin L F, Tian Y, Ma H 2014 Chin. Phys. B 23 080503
[17] Li D S, Li J H 2010 Commun. Theor. Phys. 53 298
[18] Gitterman M, Shapiro I 2011 J. Stat. Phys. 144 139
[19] Jiang S Q, Guo F, Zhou Y R, Gu T X 2007 International Conference on Communications, Circuits and Systems Fukuoka, Japan, July 11-13, 2007 p1044
[20] Ning L J, Xu W, Yao M L 2007 Chin. Phys. 16 2595
[21] Zhong S C, Yu T, Zhang L, Ma H 2015 Acta Phys. Sin. 64 020202 (in Chinese) [钟苏川, 蔚涛, 张路, 马洪 2015 物理学报 64 020202]
[22] Zhang L, Zhong S C, Peng H, Luo M K 2011 Chin. Phys. Lett. 28 090505
[23] Gitterman M 2004 Phys. Rev. E 69 041101
[24] Murray S I, Marlan O S, Willis E J 1974 Laser Physics (Rewood City: Addison-Wesley Publishing) p197
[25] Zhang L Y, Cao L, Wu D J 2008 Commun. Theor. Phys. 49 1310
[26] Sancho J M, San Miguel, Drr M D 1982 J. Stat. Phys. 28 291
[27] Sagues F, Migurel S M, Sacho J M 1984 Z. Phys. B 55 269
[28] Hector C, Fernando M, Enrique T 2006 Phys. Rev. E 74 022102
[29] Zhang L, Zhong S C, Peng H, Luo M K 2012 Acta Phys. Sin. 61 130503 (in Chinese) [张路, 钟苏川, 彭皓, 罗懋康 2012 物理学报 61 130503]
[30] Bena I, Broeck C V D, Kawai R, Lindenberg K 2002 Phys. Rev. E 66 045603
[31] Bena I 2006 Int. J. Mod. Phys. B 20 2825
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