A New Algorithm for Proving Global Asymptotic Stability of Rational Difference Equations

TL;DR

This paper introduces an algorithmic method based on contraction principles to prove the global asymptotic stability of rational difference equations, offering a systematic alternative to existing techniques.

Contribution

The paper presents a novel algorithmic approach employing contractions to establish stability, with implementation details using Maple.

Findings

Successfully proved stability for certain classes of rational difference equations.

Provided general theoretical results supporting the algorithm's effectiveness.

Implemented the algorithm in Maple for practical use.

Abstract

Global asymptotic stability of rational difference equations is an area of research that has been well studied. In contrast to the many current methods for proving global asymptotic stability, we propose an algorithmic approach. The algorithm we summarize here employs the idea of contractions. Given a particular rational difference equation, defined by a function $Q$ which maps the $k+1$ dimensional real numbers to itself, we attempt to find an integer, $K$, for which $Q^K$ shrinks distances to the difference equation's equilibrium point. We state some general results that our algorithm has been able to prove, and also mention the implementation of our algorithm using Maple.