Dynamics of a piecewise smooth map with singularity

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 Letter 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.