Bifurcations in impact oscillations

Models of impact oscillators using an instantaneous impact law are by their very nature discontinuous. These discontinuities geve rise to bifurcations which cannot be classified using the usual tools of bifurcation analysis. However, we present numerical evidence which suggests that these discontinuous bifurcations are just the limits (in some sense) of standard bifurcations of smooth dynamical systems as the impact is hardened. Finally we show how one dimensional maps of the interval with essentially similar characteristics can exhibit the same kinds of bifurcational behaviour, and how these bifurcations are related to standard bifurcations.