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.
Cite
@article{arxiv.2607.25229,
title = {Grazing bifurcations of linear impact oscillators in the zero damping limit},
author = {Olivia J. Goodman and David J. W. Simpson},
journal= {arXiv preprint arXiv:2607.25229},
year = {2026}
}