Representation of Solutions to a Two-Dimensional System of Difference Equations

Year 2025, Volume: 13 Issue: 1, 108 - 116, 30.04.2025

Mehmet Gümüş , Raafat Abo-zeid , Ana Catarina Carapito

Abstract

In this paper, we conduct a comprehensive investigation to introduce a representation to the well-defined solutions of the following system of higher-order difference equations $$ x_{n+1}=\frac{y_{n-1}(a_1y_{n-1}+b_1x_{n-3})}{c_1x_{n-3}+d_1y_{n-1}},\quad y_{n+1}=\frac{x_{n-1}(a_2x_{n-1}+b_2y_{n-3})}{c_2y_{n-3}+d_2x_{n-1}},\quad n=0,1,\ldots, $$ where $a_i,b_i,c_i,d_i$, $i=1,2$, and the initial values $x_{-3},...,x_{0},y_{-3},...,y_{0}$ are real numbers such that $|b_1|+|b_2|+|c_1|+|c_2|\neq0$. Finally, the theoretical findings of the study are supported by some numerical examples.

Keywords

Well-defined solutions, forbidden sets, system of difference equations, numerical simulations

Supporting Institution

This work was partially supported by the Portuguese Foundation for Science and Technology (FCT- Fundação para a Ciência e a Tecnologia), through the Center of Mathematics and Applications of University of Beira Interior (CMA-UBI), within Project UIDB/00212/2020.

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