Grazing bifurcations of linear impact oscillators in the zero damping limit

Abstract

We consider a harmonically forced linear impact oscillator, where impact events are instantaneous with energy loss. We study the dynamics at the grazing bifurcation of the non-impacting periodic solution in the limit that the damping coefficient of the oscillator is zero. Through numerical computations we show that a recurring sequence of bifurcations exists between points of resonance. Specifically, resonance creates a stable periodic solution that subsequently loses stability in a secondary grazing bifurcation, then regains stability in a saddle-node bifurcation, then transitions to a chaotic attractor through a period-doubling cascade. The dynamics persist under mild parameter variation, so apply to weakly-damped impact oscillators near grazing.