Stability and convergence of a higher order rational difference Equation

TL;DR

This paper investigates the stability and convergence properties of a higher order rational difference equation, analyzing equilibrium and periodic points, and employing order reduction to understand its asymptotic behavior.

Contribution

It introduces a method to analyze the stability of a complex higher order rational difference equation using order reduction techniques.

Findings

Identified the forbidden set of the difference equation.

Established conditions for asymptotic stability of equilibria.

Analyzed the periodic points and their stability.

Abstract

In this paper the asymptotic stability of equilibria and periodic points of the following higher order rational difference Equation x_{n+1} =(alpha x_{n-k})/(1+x_{n}...x_{n-k}), k>=1, n=0,1,... is studied where the parameters ?alpha, betta, and gamma are positive real numbers, and the initial conditions x_{-k}, ..., x_{0} are given arbitrary real numbers. The forbidden set of this equation is found and then, the order reduction method is used to facilitate the analysis of its asymptotic dynamics