Path integration method based on a decoupling probability mapping for fast solving the stochastic response of dynamical systems

An efficient path integration method is proposed for obtaining the stochastic response of dynamical systems. This method benefits from a new short-time transition probability density function (TPDF) matrix. First, a decoupling probability mapping is revealed, which achieves the decoupling of the short-time TPDF from stochastic processes that the system suffers. Then, the short-time transition paths carrying known probabilities can be determined by solving the probability mapping. Thus, the new TPDF matrix can be obtained simply and efficiently. The transient and steady responses of two 4-dimensional systems for different noise parameters and system parameters are explored by using the efficient path integration method. The results confirm that our method requires a cheap time cost to achieve good results that fit well with the Monte Carlo simulations.