Global behavior of solutions of a two-dimensional system of difference equations

Year 2024, , 13 - 29, 18.12.2024

Mehmet Gümüş , Raafat Abo-zeid , Kemal Türk

https://doi.org/10.54286/ikjm.1457991

Abstract

In this paper, we mainly investigate the qualitative and quantitative behavior of the solutions of a discrete system of difference equations
$$x_{n+1}=\frac{x_{n-1}}{y_{n-1}},\quad y_{n+1}=\frac{x_{n-1} }{ax_{n-1}+by_{n-1}},\quad n=0,1,\ldots, $$
where $a$, $b$ and the initial values $x_{-1},x_{0},y_{-1},y_{0}$ are non-zero real numbers. For $a\in \mathbb{R}_+-\{1\}$, we show any admissible solution $\{(x_n,y_n)\}_{n=-1}^\infty$ is either entirely located in a certain quadrant of the plane or there exists a natural number $N>0$ (we calculate its value) such that $\{(x_n,y_n)\}_{n=N}^\infty$ is located. Besides, some numerical simulations with graphs are given to emphasize the efficiency of our theoretical results in the article.

Keywords

Discrete dynamical system, forbidden set, invariant set, convergence

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