A discontinuity-geometry view of the relationship between saddle–node and grazing bifurcations

In this study we will introduce a topological approach termed as discontinuity-geometry for both analysis and visualization of the dynamics of periodically-forced impact oscillators. After a brief outline of the methodology and the general representation of trajectories with impacts the framework will be used to analyse grazing bifurcations. In particular it will be used to geometrically explain why there is sometimes a saddle–node bifurcation present in the vicinity of grazing bifurcations and why it is sometimes absent.