Global behavior of a max-type system of difference equations of the second order with four variables and period-two parameters

In this paper, we study global behavior of the following max-type system of difference equations of the second order with four variables and period-two parameters

$$\{\begin{array}{l}{x}_n=max\{A_n,\frac{z_{n-1}}{y_{n-2}}\},y_n=max\{B_n,\frac{w_{n-1}}{x_{n-2}}\},z_n=max\{C_n,\frac{x_{n-1}}{w_{n-2}}\},w_n=max\{D_n,\frac{y_{n-1}}{z_{n-2}}\},\end{array}n\in \{0,1,2,\cdots \},$$

where $A_n,B_n,C_n,D_n\in (0,+\mathrm{\infty })$ are periodic sequences with period 2 and the initial values $x_{-i},y_{-i},z_{-i},w_{-i}\in (0,+\mathrm{\infty })(1\le i\le 2)$. We show that if $min\{A_0{C}_1,B_0{D}_1,A_1{C}_0,B_1{D}_0\}<1$, then this system has unbounded solutions. Also, if $min\{A_0{C}_1,B_0{D}_1,A_1{C}_0,B_1{D}_0\}\ge 1$, then every solution of this system is eventually periodic with period $4$.