Stochastic bifurcations of nonlinear vibroimpact system with time delay and fractional derivative excited by Gaussian white noise

Abstract

This paper investigates the stochastic dynamics in the nonlinear vibroimpact system with time delay and fractional derivative under Gaussian white noise excitation. Firstly, based on the definition of Caputo-type fractional derivative and the method of non-smooth transformation, the original system is transformed into an equivalent delayed stochastic vibroimpact system without fractional derivative. Then, by using the stochastic averaging method, the stationary density function of the stochastic Itoˆ equation is obtained. The effectiveness of the proposed method was validated by comparing the consistency between the original system and the optimized system without fractional derivative or the term of time delay. At last, we also explore the stochastic P-bifurcation induced by the power spectral density of two uncorrelated noises, time delay, fractional order, and restitution coefficient of the system.