Numerical Solution Of Probability Density Function Of Stochastic Delay Dynamical Systems Based On Path Integral Method

In the real world,stochastic phenomena and time-delay phenomena are ubiquitous,so stochastic delayed dynamical systems are widely used in various research fields,such as physics,biology,mechanical engineering and so on.In these complex dynamic systems,by analyzing the probability density function of the system,we can reveal all the probability characteristics and statistical laws of the system,and simplify the complex system.The formal probability density function of stochastic differential equations with time delay has been given,but the calculation is very difficult.In practical engineering,it is necessary to construct an appropriate numerical method to solve the probability density function of stochastic delay differential equation.In this paper,the path integral method is used to construct a numerical solution to the probability density function of a stochastic dynamical system with time delay excited by Gaussian white noise.For each solution of a stochastic delay differential equation,we can uniquely construct a solution of an associated stochastic differential equation with continuous conditions.Therefore,the conditional probability density function of associated stochastic differential equation is included in the formal probability density function of stochastic delay differential equation.Theoretically,any numerical method,such as finite element method and Galerkin type method,can be used to solve the corresponding FokkerPlanck equation to calculate the conditional probability density function of the associated stochastic differential equation,and then calculate the probability density function of the stochastic delay differential equation.However,in the actual numerical solution process,due to the huge amount of calculation,the general numerical method can not be implemented.The path integration method is mainly a step-by-step iterative solution technology based on conditional probability density function,which saves computing time and memory.Therefore,in this paper,the path integral method is used to solve the conditional probability density function of the solution of the associated stochastic differential equation.Then,the probability density function in the form of stochastic delay differential equation is solved numerically.In order to verify the effectiveness of the path integration method,an example of stochastic differential equation with time delay under Gaussian white noise excitation is selected for numerical calculation.The results show that the approximate probability density function provided by the path integration method is basically consistent with the accurate probability density function,and the error is small,especially in the tail region with very low probability level.