Global Behavior of a Nonlinear System of Difference Equations

Abstract

In this paper, we study the admissible solutions of the nonlinear system of difference equations $$x_{n+1}=\dfrac{y_{n}}{x_{n}},\quad y_{n+1}=\dfrac{y_{n} }{\check{a}x_{n}+\check{b}y_{n}},\quad n=0,1,\ldots,$$ where $\check{a}$, $\check{b}$ are real numbers and the initial values $x_{0},y_{0}$ are nonzero real numbers. In case $\check{b}<0$ and ${\check{a}}^2<-4\check{b}$, we show that there are eventually periodic solutions when either $tan^{-1}\frac{\sqrt{-4\check{b}-{\check{a}}^2}}{\check{a}} \in]\dfrac{\pi}{2},\pi[$ (with $\check{a}<0$) is a rational multiple of $\pi$ or $tan^{-1}\frac{\sqrt{-4\check{b}-{\check{a}}^2}}{\check{a}} \in]0,\dfrac{\pi}{2}[$ (with $\check{a}>0$) as well.

Keywords

• System of difference equations
• Forbidden set
• Periodic solution
• Convergence

References

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