Grazing-Induced Evolutions of Coexisting Orbits in a Vibro-Impact Smooth–Discontinuous Oscillator with a Unilateral Adjustable Rigid Wall

In this paper, a new vibro-impact bistable oscillator by installing an adjustable rigid wall to one side of the smooth–discontinuous (SD) oscillator, is constructed to analytically and numerically reveal its unique global dynamic characteristics. In the unperturbed case, there exist complex unilateral homoclinic structures transversal intersecting, grazing and nonintersecting with the impact boundary due to the adjustability of the rigid constraint. When weak vicious damping, weak periodic excitation and impact energy dissipation are considered, the influences of the adjustable rigid constraint and the evolutions from coexisting periodic orbits to chaos are analyzed through bifurcation diagrams, phase diagrams and Poincaré map and basins of attraction. As the position of the impact boundary moves, the orbits graze multiple times with the boundary, causing period-doubling bifurcations and presenting some new impacting periodic orbits and disconnected chaotic attractors. The vibro-impact SD oscillator also gradually loses stability due to the rapid migration of attractors among coexistence orbits.