三类随机系统广义概率密度演化方程的解析解

近年来逐步发展的概率密度演化方法理论为随机动力系统的分析与控制研究提供了新的途径.过去若干年来,已经发展了一系列数值方法如有限差分法、无网格法用于求解广义概率密度演化方程.但是,针对典型随机系统,关于这一方程解析解尚比较缺乏.本文以李群方法为工具,研究给出了Van der Pol振子、Riccati方程和Helmholtz振子3类典型随机非线性系统的广义概率密度演化方程解析解.这些结果,不仅可以作为检验求解广义概率密度演化方程的数值方法结果正确性的判别依据,也为概率密度演化理论的进一步深入研究提供了若干分析实例. 

ANALYTICAL SOLUTIONS OF THE GENERALIZED PROBABILITY DENSITY EVOLUTION EQUATION OF THREE CLASSES STOCHASTIC SYSTEMS

As a gradually improving and developed method, generalized probability density evolution equation(GDEE) provides a new methodology for the analysis and control of stochastic dynamic system.A variety of numerical methods such as finite difference method and meshfree method were introduced to solve the generalized probability density evolution equation.However, the analytical solution of the GDEE corresponding to typical stochastic systems is scarce relatively.In this paper, the explicit solutions of the GDEE corresponding to three classes of nonlinear stochastic systems including Van der Pol oscillator, Riccati system, and Helmholtz oscillator with random parameters are studied by using Lie group method.The results not only can be the benchmark of numerical methods, but also can provide more information for the further research.