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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.
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Keywords:
- Ornstein-Uhlenbeck process /
- eigenvalue /
- eigenvector /
- the time-dependent solution
[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 51 977]
[10] Dong X J 2007 Acta Phys. Sin. 56 5618 (in Chinese) [董小娟 2007 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 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 60 120501]
[14] Wang C J 2012 Acta Phys. Sin. 61 010503 (in Chinese) [王参军 2012 61 010503]
[15] Zhang G T, Huang J J 2012 Acta Phys. Sin. 61 140205 (in Chinese) [张国亭, 黄俊杰 2012 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 51 977]
[10] Dong X J 2007 Acta Phys. Sin. 56 5618 (in Chinese) [董小娟 2007 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 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 60 120501]
[14] Wang C J 2012 Acta Phys. Sin. 61 010503 (in Chinese) [王参军 2012 61 010503]
[15] Zhang G T, Huang J J 2012 Acta Phys. Sin. 61 140205 (in Chinese) [张国亭, 黄俊杰 2012 61 140205]
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