Universal limit mapping in grazing bifurcations

Grazing bifurcations are nonsmooth bifurcations that occur in impacting mechanical oscillators. A sign of the lack of smoothness is a square root expression that arises in mappings describing the dynamics. Certain types of grazing bifurcations involve chaotic bands, or period adding cascades with or without chaotic bands. There is also a characteristic scaling behavior present. Here this type of bifurcation is investigated, and the self-similarity under scaling is used to derive a renormalized limit mapping. A study of the dynamics of the limit mapping identifying all attractors for all parameter values is also presented.