One-dimensional Piecewise Smooth Rational Degree Maps

TL;DR

This paper classifies bifurcation scenarios, including period doubling and chaos, in a specific class of one-dimensional piecewise smooth rational maps with singularities, relevant to grazing bifurcations in hybrid systems.

Contribution

It provides a comprehensive classification of bifurcation behaviors in a novel class of piecewise smooth rational maps with singularities.

Findings

Classification of all bifurcation scenarios including period doubling.

Identification of robust chaos in the studied maps.

Analysis of grazing bifurcations in hybrid and piecewise flows.

Abstract

In this paper, we consider a class of continuous maps characterized by a singularity of order $x^{q/p}$ (with $p,q \in \mathbb{N}$, $p>q$, and $(p,q)=1$) on one side of the discontinuity boundary $\Sigma$ 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 $\Sigma$ 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.