非线性漂移的Fokker-Planck方程的近似非定态解

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.关键词：奥恩斯坦-乌伦贝克过程本征值本征矢非定态解

Approximate time-dependent solution of Fokker-Planck equation with non-linear drift force

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 processeigenvalueeigenvectorthe time-dependent solution