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

Taixiang Sun; Guangwang Su; Bin Qin; Caihong Han; Taixiang Sun; Guangwang Su; Bin Qin; Caihong Han

doi:10.3934/math.20231220

AIMS Mathematics

2023, Volume 8, Issue 10: 23941-23952. doi: 10.3934/math.20231220

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

1.
Guangxi Key Laboratory of Big Data in Finance and Economics, Nanning, 530003, China
2.
Guangxi (ASEAN) Research Center of Finance and Economics, Nanning, 530003, China
3.
College of Mathematics and Quantitative Economy, Guangxi University of Finance and Economics, Nanning, 530003, China

Received: 26 June 2023 Revised: 20 July 2023 Accepted: 24 July 2023 Published: 07 August 2023
MSC : 39A10, 39A11

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

$ \left\{\begin{array}{ll}x_{n} = \max\Big\{A_n , \frac{z_{n-1}}{y_{n-2}}\Big\}, \ y_{n} = \max \Big\{B_n, \frac{w_{n-1}}{x_{n-2}}\Big\}, \ z_{n} = \max\Big\{C_n , \frac{x_{n-1}}{w_{n-2}}\Big\}, \ w_{n} = \max \Big\{D_n, \frac{y_{n-1}}{z_{n-2}}\Big\}, \ \end{array}\right. \ \ n\in \{0, 1, 2, \cdots\}, $

where $ A_n, B_n, C_n, D_n\in (0, +\infty) $ are periodic sequences with period 2 and the initial values $ x_{-i}, y_{-i}, z_{-i}, w_{-i}\in (0, +\infty)\ (1\leq i\leq 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\}\geq 1 $, then every solution of this system is eventually periodic with period $ 4 $.
- max-type system of difference equations,
- solution,
- eventual periodicity
Citation: Taixiang Sun, Guangwang Su, Bin Qin, Caihong Han. Global behavior of a max-type system of difference equations of the second order with four variables and period-two parameters[J]. AIMS Mathematics, 2023, 8(10): 23941-23952. doi: 10.3934/math.20231220

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Abstract

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

$ \left\{\begin{array}{ll}x_{n} = \max\Big\{A_n , \frac{z_{n-1}}{y_{n-2}}\Big\}, \ y_{n} = \max \Big\{B_n, \frac{w_{n-1}}{x_{n-2}}\Big\}, \ z_{n} = \max\Big\{C_n , \frac{x_{n-1}}{w_{n-2}}\Big\}, \ w_{n} = \max \Big\{D_n, \frac{y_{n-1}}{z_{n-2}}\Big\}, \ \end{array}\right. \ \ n\in \{0, 1, 2, \cdots\}, $

where $ A_n, B_n, C_n, D_n\in (0, +\infty) $ are periodic sequences with period 2 and the initial values $ x_{-i}, y_{-i}, z_{-i}, w_{-i}\in (0, +\infty)\ (1\leq i\leq 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\}\geq 1 $, then every solution of this system is eventually periodic with period $ 4 $.

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