A path integration algorithm for stochastic dynamical systems with multiple non-smooth events

An extended path integration method is designed for obtaining the stochastic response of systems with multiple non-smooth events. The concept of the ordered non-smooth events set is proposed, which makes complex non-smooth events orderly. Then we cleverly establish a regular mapping on this set according to the continuity in time and space, it allows multiple non-smooth events to be handled uniformly. Based on this, an extended path integration method is proposed to capture the non-smooth behavior of the system in a direct manner. The Duffing vibro-impact systems with various bilateral constraints are used as examples to show the novel results on constraints obtained by the extended path integration algorithm. Finally, Monte Carlo simulations prove that our algorithm has wide applicability and good accuracy.