Analysis and Representation of Solutions of a Nonlinear Difference Equation System

Year 2025, Volume: 13 Issue: 2, 180 - 189, 31.10.2025

Mehmet Gümüş , Raafat Abo-zeid , Anwar Zeb , Aleyna Sezgin

Abstract

Difference equations and systems of difference equations have an important place in many branches of science. These equations (systems) play a critical role in modeling physical systems, engineering problems and many biological problems. In addition, it is widespread to use difference equations to obtain numerical solutions of differential equations. In this paper, we derive a representation of the well-defined solutions of the two-dimensional non-linear system of difference equations \begin{equation*}\label{1} x_{n+1}=\frac{y_{n-1}x_{n-2}}{ax_{n-2}+by_{n}},\quad y_{n+1}=\frac{x_{n-1}y_{n-2}}{cy_{n-2}+dx_{n}},\text{ }n=0,1,..., \end{equation*}% where $a,b,c,d,$ and the initial values $x_{-2},x_{-1},x_{0},y_{-2},y_{-1},y_{0}$ are real numbers. We determine the well-defined solutions to the above-mentioned system when $a=b=c=d=1$ as well as when $a=b=c=-d=1$. In addition, the theoretical results obtained here are also supported by numerical examples and simulations.

Keywords

Difference Equations , Well-defined solutions , Fibonacci Sequences , Periodicity , Explicit Formula

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