One-dimensional Piecewise Smooth Rational Degree Maps

In this paper, we consider a class of continuous maps characterized by a singularity of order xq/p (with p,q ∈ N, $p>q$, and $(p,q)=1$) on one side of the discontinuity boundary Σ 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 Σ 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.