  Experiments observing the liquid surface in a vertically oscillating
container have indicated that modeling the dynamics of such systems require
maps that admit states at infinity. In this paper we investigate the
bifurcations in such a map. We show that though such maps in general fall in
the category of piecewise smooth maps, the mechanisms of bifurcations are quite
different from those in other piecewise smooth maps. We obtain the conditions
of occurrence of infinite states, and show that periodic orbits containing such
states are superstable. We observe period-adding cascade in this system, and
obtain the scaling law of the successive periodic windows.
