One-dimensional Piecewise Smooth Rational Degree Maps

In this paper, we consider a class of continuous maps characterized by a singularity of order $x^{q\mathrm{/}p}$ (with $p,q\in \mathbb{N}$, $p>q$, and $(p,q)=1$) on one side of the discontinuity boundary $\mathrm{\Sigma }$ and a linear behaviour on the other side. Such maps arise naturally in the study of grazing bifurcations of hybrid and piecewise flows. In this context the boundary collision of a fixed point of the map with $\mathrm{\Sigma }$ then corresponds to a grazing bifurcation of the flow. We will start by studying one-dimensional maps, and the main result of this paper is a classification of all bifurcation scenarios, including: period doubling and robust chaos.