Solution Form of a Higher Order System of Difference Equation and Dynamical Behavior of Its Special Case

TL;DR

This paper derives the explicit solution form for a class of nonlinear difference equations and investigates the dynamical behavior of a special case where p=1, supported by numerical examples.

Contribution

It provides the explicit solution form for a higher order nonlinear difference system and analyzes the dynamics when p=1, which is a novel extension.

Findings

Explicit solution form derived for the difference system.

Behavior of solutions characterized for the case p=1.

Numerical examples illustrate theoretical results.

Abstract

The solution form of the system of nonlinear difference equations \begin{equation*} x_{n+1} = \frac{x_{n-k+1}^{p}y_{n}}{a y_{n-k}^{p}+b y_{n}},\ y_{n+1} = \frac{y_{n-k+1}^{p}x_{n}}{\alpha x_{n-k}^{p}+\beta x_{n}}, \quad n, p \in \mathbb{N}_{0},\ k\in \mathbb{N}, \end{equation*} where the coefficients $a, b, \alpha, \beta$ and the initial values $x_{-i},y_{-i},i\in\{0,1,\ldots,k\}$ are real numbers, is obtained. Furthermore, the behavior of solutions of the above system when $p=1$ is examined. Numerical examples are presented to illustrate the results exhibited in the paper.