The theory of difference equations got a significant position in the applicable analysis. Therefore, studying the qualitative behavior of the difference equations is a fruitful area of research that has increasingly attracted many researchers. In this paper, we demonstrate the stability and the existence of periodic solutions of the nonlinear difference equation. Moreover, we provide some numerical simulations to confirm our results.

The principle purpose of this paper is to investigate the long-term behaviors of some nonlinear systems of difference equations and obtain these solution expressions and periodicity character. Furthermore, we use MATLAB programming to simulate the dynamics and confirm our results.