Analytical determination of bifurcations in an impact oscillator

It is well known that some solutions for a sinusoidally driven oscillator with linear stiffness and impacts at rigid stops modelled with a coefficient of restitution impact law can be located analytically. Recently, new co-dimension one bifurcations called grazing bifurcations have been found in such systems. Here we present analytical results which show how the type of grazing bifurcation changes with parameter, and that when the type of grazing bifurcation changes a codimension two bifurcation occurs. The simplest grazing bifurcations involve orbits of period-1, but we show that the same analytical methods can be used to locate some subharmonics and their bifurcations.