Global behavior of a max-type system of difference equations with four variables

Abstract

In this paper, we investigate global behavior of the following max-type system of difference equations with four variables

$$\begin{aligned} \left\{ \begin{array}{ll}x_{n} = \max \Big \{A,\frac{z_{n-1}}{y_{n-2}}\Big \},\\ y_{n} = \max \Big \{B ,\frac{w_{n-1}}{x_{n-2}}\Big \},\\ z_{n} = \max \Big \{C ,\frac{x_{n-1}}{w_{n-2}}\Big \},\\ w_{n} = \max \Big \{D,\frac{y_{n-1}}{z_{n-2}}\Big \},\\ \end{array}\right. \quad n\in \{0,1,2, \ldots \}, \end{aligned}$$

where $A, B,C,D\in (0,+\infty )$ with $A\le B$ and $C\le D$, and the initial conditions $x_{-2},y_{-2},z_{-2},w_{-2},x_{-1},y_{-1},z_{-1},w_{-1}\in (0,+\infty )$. We show that: (1) If $AC< 1$, then there exists a solution $\{(x_n,y_n,z_n,w_n)\}^{+\infty }_{n= -2}$ of this system such that $x_n=A$ and $z_n=C$ for any $n\ge -2$ and $\lim _{n\longrightarrow \infty }y_n=\lim _{n\longrightarrow \infty }w_n=\infty$. (2) If $AC=1$, then every solution of this system is eventually periodic with period 4. (3) If $AC>1$, then every solution of this system is eventually periodic with period 1.