On Periodic and Chaotic Orbits in a Rational Planar System

TL;DR

This paper demonstrates that a rational planar system can exhibit coexisting periodic orbits of all periods and stable aperiodic orbits by transforming it into a scalar difference equation, revealing complex dynamical behaviors.

Contribution

It introduces a method to analyze rational planar systems by folding them into scalar difference equations, uncovering rich dynamical phenomena including all possible periodic orbits.

Findings

Existence of coexisting periodic orbits of all periods

Presence of stable aperiodic orbits for certain parameters

Method of folding rational systems into scalar difference equations

Abstract

By folding an autonomous system of rational equations in the plane to a scalar difference equation, we show that the rational system has coexisting periodic orbits of all possible periods as well as stable aperiodic orbits for certain parameter ranges.