一类随机微分动力系统的转移概率密度函数的研究

Study on the Transfer Probability Density Function of a Class of Stochastic Differential Dynamical Systems

摘要: 本文探究了一类随机动力系统转移概率密度函数的求解及其有效性问题。首先，提出了一种计算简便、精度较高的数值模拟方法，用于得到FPK (Fokker-Planck-Kolmogorov)方程的近似解。其次，通过比较该近似解与系统的精确解，我们发现此模拟方法的相对误差保持在2 × 10⁽⁻³⁾以内，误差基本在10⁽⁻³⁾以内。最后，通过比较由近似解得到的状态变量出现在某一稳态区域的概率和系统离散后的相图，我们得出此模拟方法可以有效地描述系统的性态。因此，它是求解系统FPK方程的一种有效算法。

Abstract: In this paper, the solution and validity of the transfer probability density function for a class of stochastic dynamical systems are investigated. Firstly, a simple and accurate numerical simulation method is proposed to obtain the approximate solution of FPK (Fokker-Planck-Kolmogorov) equation. Secondly, by comparing the approximate solution with the exact solution of the system, we found that the relative error of this simulation method is less than 2 × 10⁽⁻³⁾ and the error is basically within 10⁽⁻³⁾. Finally, by comparing the probability that the state variables obtained by the approximate solution appear in a steady-state region with the phase diagram of the discrete system, we conclude that the simulation method can effectively describe the system’s behavior. Therefore, it is an effective algorithm to solve the FPK equation of the system.