On the Dynamics of a Higher Order Nonlinear System of Difference Equations

TL;DR

This paper explores the complex dynamics of a higher order nonlinear rational difference system, focusing on boundedness, oscillations, unbounded solutions, and global behavior of positive solutions.

Contribution

It provides new insights into the global behavior and stability conditions of a specific higher order nonlinear difference system.

Findings

Conditions for boundedness of solutions

Existence of oscillatory solutions

Characterization of unbounded solutions

Abstract

The aim of this paper is to investigate the dynamics of a higher order system of rational difference equations. Our concentration is on boundedness character, the oscillatory, the existence of unbounded solutions and the global behavior of positive solutions for the following system of difference equations \begin{equation*} x_{n+1}=A+\frac{x_{n-m}}{z_{n}},\ y_{n+1}=A+\frac{y_{n-m}}{z_{n}},\ z_{n+1}=A+\frac{z_{n-m}}{y_{n}},\ n=0,1,...,\end{equation*} where $A$ and the initial values $x_{-i}$, $y_{-i}$, $z_{-i}$, for $i=0,1,...,m$, are positive real numbers.