Unstable manifolds of a limit cycle near grazing

We study the local dynamics of an impacting system near a grazing bifurcation point. In particular, we investigate local invariant manifolds of grazing periodic orbits. At a grazing bifurcation point the local return map does not have a Jacobian nor is of Lipschitz continuous, so that the classical theory does not apply. Nevertheless, we are able to use the Graph Transform technique and show that under certain conditions a local Lipschitz unstable manifold of the periodic orbit exists at the grazing bifurcation point.