The Promotion Of The Path Integral Method And Its Application In The Study Of Nonlinear Stochastic Dynamical Systems

The main purpose of this paper is to extend the numerical path integration based on Gauss-Legendre formula and study its applications in complex nonlinear stochastic dynamical systems. The present study is organized as follows:Firstly, we provide a general summary about the study of all numerical methods to solve FPK equation. Especially, the applications and importance of path integration for solving FPK equation in nonlinear stochastic dynamical systems are interpreted in detail. The theory of path integration and the numerical method based on Gauss-Legendre formula are also listed out in brief. Then we educe the numerical algorithm for calculating the probability density averaged in time.Secondly, the numerical path integration based on Gauss-Legendre formula is extended to a nonlinear dynamical system under stochastic parametric and external excitations. For purposes of comparison between the numerical solutions and the analytic solution or Monte-Carlo simulation, we discuss the system under parametric and external Gaussian white noise excitations. The numerical method is shown to give accurate results and P-bifurcation of the stochastic system is successfully captured.Thirdly, the Duffing-Rayleigh oscillator subject to harmonic and stochastic excitations is concerned. We carry out the probability density averaged in time and contrast with the result by Monte Carlo simulation. The method is efficient to obtain double crater like probability density of the system. Then we analyze the relationship between stochastic jump and peaks of the probability density in three cases.Finally, we discuss the application of the path integration based on Gauss-Legendre formula in analyzing the chaotic response of the Mathieu-Duffing oscillator. The probability density of stochastic system is multi-peaked when the corresponding determinate system is chaotic and the intensity of white noise excitation is small. Meanwhile, the existence of attractors can be efficiently and clearly depicted by the evolution of a probability density.