Journal of Applied Mathematics and Computing ( IF 2.2 ) Pub Date : 2021-03-30 , DOI: 10.1007/s12190-021-01543-8 Taixiang Sun , Guangwang Su , Caihong Han , Lue Li , Weizhen Quan
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.
中文翻译：
带有四个变量的差分方程的最大型系统的整体性质
在本文中，我们研究了具有四个变量的以下最大型差分方程系统的整体行为
$$ \ begin {aligned} \ left \ {\ begin {array} {ll} x_ {n} = \ max \ Big \ {A，\ frac {z_ {n-1}} {y_ {n-2}} \ Big \}，\\ y_ {n} = \ max \ Big \ {B，\ frac {w_ {n-1}} {x_ {n-2}} \ Big \}，\\ z_ {n} = \ max \ Big \ {C，\ frac {x_ {n-1}} {w_ {n-2}} \ Big \}，\\ w_ {n} = \ max \ Big \ {D，\ frac {y_ {n-1}} {z_ {n-2}} \ Big \}，\\ \ end {array} \ right。\ quad n \ in \ {0,1,2，\ ldots \}，\ end {aligned} $$其中\（A，B，C，D \ in（0，+ \ infty）\）与\（A \ le B \）和\（C \ le D \）以及初始条件\（x _ {-2 }，y _ {-2}，z _ {-2}，w _ {-2}，x _ {-1}，y _ {-1}，z _ {-1}，w _ {-1} \ in（0，+ \ infty）\）。我们证明：（1）如果\（AC <1 \），则存在一个解决方案\（\ {（x_n，y_n，z_n，w_n）\} ^ {+ \ infty} _ {n = -2} \ ）该系统，使得\（x_n = A \）和\（z_n = C \）对于任何\（N \ GE -2 \）和\（\ LIM _ {N \ longrightarrow \ infty} y_n = \ LIM _ {n \ longrightarrow \ infty} w_n = \ infty \）。（2）如果\（AC = 1 \），则该系统的每个解最终都是周期为4的周期。（3）如果\（AC> 1 \），那么该系统的每个解决方案最终都是周期为1的周期。
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