Global Dynamical Behaviours and Periodicity of a Certain Quadratic-Rational Difference Equation with Delay

TL;DR

This paper investigates the complex dynamics, periodicity, and stability of a higher order quadratic-rational difference equation with delay, providing insights into its long-term behavior and convergence properties.

Contribution

It introduces a detailed analysis of the dynamics, periodicity, and stability of a specific quadratic-rational difference equation with delay, including convergence rates.

Findings

Identification of conditions for boundedness and stability

Characterization of periodic solutions and semi-cycles

Analysis of the rate of convergence of solutions

Abstract

Our aim in this paper is to deal with the dynamics of following higher order difference equation x_{n+1}=A+B((x_{n-m})/(x_{n}^2)) where A,B>0, and initial values are positive, and m={1,2,...}. Furthermore, we discuss the periodicity, boundedness, semi-cycles, global asymptotic stability of solutions of these equations. We also handle the rate of convergence of solutions of these difference equations.