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 <tex-math id="FE1"> {document}$ \{{array}{ll}xₙ = max\{A_n , {zₙ₋₁}{yₙ₋₂}\}, \ yₙ = max \{B_n, {wₙ₋₁}{xₙ₋₂}\}, \ zₙ = max\{C_n , {xₙ₋₁}{wₙ₋₂}\}, \ wₙ = max \{D_n, {yₙ₋₁}{zₙ₋₂}\}, \ {array}. \ \ n∈ \{0, 1, 2, ⋯\}, document </tex-math> where A_n, B_n, C_n, D_n∈ (0, +∞) $ are periodic sequences with period 2 and the initial values $ x₋ᵢ, y₋ᵢ, z₋ᵢ, w₋ᵢ∈ (0, +∞)\ (1≤ i≤ 2) $. We show that if $ min\{A_0C_1, B_0D_1, A_1C_0, B_1D_0\} < 1 $, then this system has unbounded solutions. Also, if $ min\{A_0C_1, B_0D_1, A_1C_0, B_1D_0\}≥ 1 $, then every solution of this system is eventually periodic with period $ 4 $.