Bifurcation analysis of a birhythmic vibroimpact oscillator with two kinds of fractional derivative elements subjected to Gaussian white noise excitation

Abstract

This paper explores the stochastic bifurcations of a birhythmic vibroimpact oscillator incorporating two types of fractional derivative elements under Gaussian white noise excitation. The analysis begins with a nonsmooth coordinate transformation of the state variables, followed by the application of the stochastic averaging method to derive analytical solutions. To validate the proposed approach, numerical simulations based on the Monte Carlo method are conducted, demonstrating excellent agreement with the analytical predictions. A key finding of this study is that the fractional orders, fractional coefficients, and restitution coefficient play a significant role in triggering stochastic P bifurcations within the system.