Stochastic dynamics analysis of nonlinear systems with time delay based on path integration method

In this paper, an approach based on path integration method is proposed to study the stochastic nonlinear systems with time delay. The delay terms are processed according to the idea of multi-scale representation, so that the response of obtained approximate system is Markovian process. Then the probability density function of the equivalent system is solved by the path integration method, including both transient and steady-state aspects. The method is applied to a system with nonlinear delay term and a coupled two-degree-of-freedom system with multiple delay terms, respectively. The technique of stretching and deformation is used to deal with the dilemma of heavy computational cost caused by transition probability density function when solving high-dimensional system. Meanwhile, the huge matrix is stored sparsely to reduce the running memory usage. Finally, the influence mechanism of different time delay and parameters on the system response is discussed, and the stochastic P-bifurcation phenomenon of the probability density function is observed. The results are compared with Monte Carlo simulation results to verify the effectiveness of the proposed method.