Random P-bifurcation in a Duffing–van der Pol vibro-impact system with a Bingham model

The present work investigates stochastic P-bifurcation phenomena in a Duffing–van der Pol vibro-impact oscillator containing a Bingham model under Gaussian white noise excitation. By employing non-smooth transformations and stochastic averaging techniques, an approximate analytical method is proposed to analyze the stochastic response and bifurcation behavior of nonlinear systems with friction and vibro-impact effects. Using a non-smooth transformation, the stochastically excited vibro-impact oscillator is converted into an approximately equivalent system without velocity discontinuities. Subsequently, the friction term is handled, and stochastic averaging is applied to derive the averaged stochastic Itô equation. The corresponding Fokker–Planck–Kolmogorov equation is then solved to obtain the probability density function of the system's steady-state response. Numerical simulations are conducted to verify the reliability of the proposed method. Based on these results, the critical parameter conditions for stochastic P-bifurcation are derived using singularity theory, considering both the amplitude probability density and the joint probability density of system displacement and velocity. Bifurcation diagrams, extreme value plots, amplitude probability density plots, velocity probability density plots, and joint probability density plots of system displacement and velocity are constructed for different parameter spaces. The findings demonstrate that changes in the viscous damping coefficient of the magnetorheological damper, Coulomb damping force, noise intensity, vibro-impact coefficient, and nonlinear damping coefficient can all induce stochastic P-bifurcations.