Stochastic responses of a vibro-impact system with additive and multiplicative colored noise excitations

By the aid of nonsmooth transformation, the stochastic averaging method is extended to solve the stationary response of a vibro-impact system with additive and multiplicative colored noise excitations. The stationary responses are obtained by using the concepts of total phase and residual phase. A Duffing–Van der Pol vibro-impact system is given to illustrate the application of the method. And stochastic P-bifurcation is also investigated.