Homoclinic Bifurcations and Chaotic Dynamics in a Bistable Vibro-Impact SD Oscillator Subject to Gaussian White Noise

This paper studies the effect of Gaussian white noise on homoclinic bifurcations and chaotic dynamics of a bistable, vibro-impact Smooth-and-Discontinuous (SD) oscillator. First, the SD oscillator is reproduced and generalized by installing a slider on a fixed rod, so the slider is connected by a pair of linear springs initially pre-compressed in the vertical direction to achieve bistable vibration characteristics, and two screw nuts are installed on the rod as two adjustable bilateral rigid constraints to generate the vibro-impact. A discontinuous dynamical equation with a map defined on switching boundaries to represent velocity loss during each collision is derived to describe the vibration pattern of the bistable, vibro-impact SD oscillator through studying the persistence of the unique, unperturbed, nonsmooth, homoclinic structure. Second, the general framework of random Melnikov process for a class of bistable, vibro-impact systems contaminated with Gaussian white noise is derived and employed through the corresponding Melnikov function to obtain the necessary parameter thresholds for homoclinic tangency and possible chaos of the bistable, vibro-impact SD oscillator. Third, the effectiveness of a semi-analytical prediction by the Melnikov function is verified using the largest Lyapunov exponent, bifurcation series, and 0–1 test. Finally, the sensitivity to the initial values of chaos is verified by the fractal attractor basins, and the influence of the Gaussian white noise on periodic and chaotic structures is studied through Poincaré mapping to show the rich dynamical geometric structures.