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研究了由高斯白噪声和色噪声作用下的非线性动力学系统的不稳定态演化问题. 在弱噪声极限下, 运用本征值本征矢理论得到了非定态解ρ(x, t)的近似表达式; 分析了色噪声自关联时间τ, 强度α对ρ(x, t)以及对一、二阶矩的影响. 数值模拟发现: 1)t在一定范围内, ρ(x, t)是变量x和t的单调函数, 且随τ的增大而增大, 反之, 随α的增大而减小; 2)一阶矩是τ和α的单调函数, 但二阶矩却是非单调函数, 在参数影响下发生了相变现象.
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关键词:
- 奥恩斯坦-乌伦贝克过程 /
- 本征值 /
- 本征矢 /
- 非定态解
In this paper, the unstable state evolution problem of the non-linear dynamical system driven by Gaussian white and colored noise is investigated. Using the eigenvalue and eigenvector theory, the expression of the approximate time-dependent solution (ρ(x, t)) is derived. The effects of parameters on ρ(x, t), mean and normalized variance are also analyzed. Numerical simulations show that 1) ρ(x, t) is a monotonic function of t and x under the certain limits of t, which increases with τ increasing, but decreases with α increasing; it is very remarkable for large τ and large α; 2) the mean of the state variable x is positive, which increases with τ increasing, but decreases with α increasing; the normalized variance of the state variable x is a non-monotonic function of the α and τ. Therefore, a phase transition phenomenon is found in this system.-
Keywords:
- Ornstein-Uhlenbeck process /
- eigenvalue /
- eigenvector /
- the time-dependent solution
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[2] Jung P, Hanggi P 1988 J. Opt. Soc. Am. B 5 979
[3] Hu G 1944 Stochastic Force and Nonliear Systems (Shanghai: Shanghai Scientific and Technological Education Publishing House) (in Chinese) [胡岗 1944 随机力与非线性系统 (上海: 上海科技教育出版社)]
[4] Guo F, Luo X D, Li S F, Zhou Y R 2010 Chin. Phys. B 19 080504
[5] San Miguel M, Sancho J M 1981 Z. Phys. B: Condens. Master. 43 361
[6] Jia Y, Jia J R 1995 Phys. Rev. A 53
[7] Liang G Y, Cao L, Wu D J 2004 Physica E 335 371
[8] Ke S Z, Cao L, Wu D J 1999 J. Huazhong Univ. Sci. 27 98 (in Chinese) [柯圣志, 曹力, 吴大进 1999 华中理工大学学报 27 98]
[9] Luo X Q, Zhu S Q 2002 Acta Phys. Sin. 51 977 (in Chinese) [罗晓琴, 朱士群 2002 物理学报 51 977]
[10] Dong X J 2007 Acta Phys. Sin. 56 5618 (in Chinese) [董小娟 2007 物理学报 56 5618]
[11] Wang B, Shao J H, Wu X Q 2009 Acta Phys. Sin. 58 1377 (in Chinese) [王兵, 卲继红, 吴秀清 2009 物理学报 58 1377]
[12] Dan W, Zhu S Q 2007 Phys. Rev. Lett. A 363 202
[13] Zhang J J, Jin Y F 2011 Acta Phys. Sin. 60 120501 (in Chinese) [张静静, 靳艳飞 2011 物理学报 60 120501]
[14] Wang C J 2012 Acta Phys. Sin. 61 010503 (in Chinese) [王参军 2012 物理学报 61 010503]
[15] Zhang G T, Huang J J 2012 Acta Phys. Sin. 61 140205 (in Chinese) [张国亭, 黄俊杰 2012 物理学报 61 140205]
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[1] Risken H 1985 The Fokker-Planck Equation: Methods of Solution and Applications (Berlin: Springer-Verlag)
[2] Jung P, Hanggi P 1988 J. Opt. Soc. Am. B 5 979
[3] Hu G 1944 Stochastic Force and Nonliear Systems (Shanghai: Shanghai Scientific and Technological Education Publishing House) (in Chinese) [胡岗 1944 随机力与非线性系统 (上海: 上海科技教育出版社)]
[4] Guo F, Luo X D, Li S F, Zhou Y R 2010 Chin. Phys. B 19 080504
[5] San Miguel M, Sancho J M 1981 Z. Phys. B: Condens. Master. 43 361
[6] Jia Y, Jia J R 1995 Phys. Rev. A 53
[7] Liang G Y, Cao L, Wu D J 2004 Physica E 335 371
[8] Ke S Z, Cao L, Wu D J 1999 J. Huazhong Univ. Sci. 27 98 (in Chinese) [柯圣志, 曹力, 吴大进 1999 华中理工大学学报 27 98]
[9] Luo X Q, Zhu S Q 2002 Acta Phys. Sin. 51 977 (in Chinese) [罗晓琴, 朱士群 2002 物理学报 51 977]
[10] Dong X J 2007 Acta Phys. Sin. 56 5618 (in Chinese) [董小娟 2007 物理学报 56 5618]
[11] Wang B, Shao J H, Wu X Q 2009 Acta Phys. Sin. 58 1377 (in Chinese) [王兵, 卲继红, 吴秀清 2009 物理学报 58 1377]
[12] Dan W, Zhu S Q 2007 Phys. Rev. Lett. A 363 202
[13] Zhang J J, Jin Y F 2011 Acta Phys. Sin. 60 120501 (in Chinese) [张静静, 靳艳飞 2011 物理学报 60 120501]
[14] Wang C J 2012 Acta Phys. Sin. 61 010503 (in Chinese) [王参军 2012 物理学报 61 010503]
[15] Zhang G T, Huang J J 2012 Acta Phys. Sin. 61 140205 (in Chinese) [张国亭, 黄俊杰 2012 物理学报 61 140205]
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