Period Increment Cascades in a Discontinuous Map with Square-Root Singularity*

In this paper, we consider a discontinuous map with square-root singularity, which is relevant to many physical systems. Such maps occur especially in switching dynamical systems 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 for first time. Finally, we show that from a piecewise ordinary differential equation the same type of map can also be obtained.