Grazing and border-collision in piecewise-smooth systems: A unified analytical framework

A comprehensive derivation is presented of normal form maps for grazing bifurcations in piecewise smooth models of physical processes. This links gratings with border-collisions in nonsmooth maps. Contrary to previous literature, piecewise linear maps correspond only to nonsmooth discontinuity boundaries. All other maps have either square-root or (3/2)-type singularities.