Period increment cascades in a discontinuous map with square-root singularity

We consider a discontinuous map with square-root singularity, which is relevant to many physical systems. Such maps occur in modeling grazing-sliding bifurcations in switching dynamical systems, or if the Poincaré plane coincides with the switching plane. It is shown that there are notable differences in the bifurcation scenarios between this type of discontinuous map and a continuous map with square-root singularity. We determine the bifurcation structures and the scaling constant analytically. A different kind of period increment is observed, and the possibility of breakdown of period increment cascade is detected. Finally, we show that a system of piecewise smooth ordinary differential equations can exhibit the same type of bifurcation behavior.