Global dynamics of some system of second-order difference equations

Tran Hong Thai; Nguyen Anh Dai; Pham Tuan Anh; Tran Hong Thai; Nguyen Anh Dai; Pham Tuan Anh

doi:10.3934/era.2021077

Electronic Research Archive

2021, Volume 29, Issue 6: 4159-4175. doi: 10.3934/era.2021077

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Global dynamics of some system of second-order difference equations

Department of Mathematics, Hung Yen University of Technology and Education, Hung Yen 160000, Vietnam

Received: 01 June 2021 Published: 08 October 2021
Primary: 39A10, 39A30; Secondary: 40A05

In this paper, we study the boundedness and persistence of positive solution, existence of invariant rectangle, local and global behavior, and rate of convergence of positive solutions of the following systems of exponential difference equations
$ \begin{align*} x_{n+1} = \dfrac{\alpha_1+\beta_1e^{-x_{n-1}}}{\gamma_1+y_n},\ y_{n+1} = \dfrac{\alpha_2+\beta_2e^{-y_{n-1}}}{\gamma_2+x_n},\\ x_{n+1} = \dfrac{\alpha_1+\beta_1e^{-y_{n-1}}}{\gamma_1+x_n},\ y_{n+1} = \dfrac{\alpha_2+\beta_2e^{-x_{n-1}}}{\gamma_2+y_n}, \end{align*} $
where the parameters $ \alpha_i,\ \beta_i,\ \gamma_i $ for $ i \in \{1,2\} $ and the initial conditions $ x_{-1}, x_0, y_{-1}, y_0 $ are positive real numbers. Some numerical example are given to illustrate our theoretical results.
- System of difference equations,
- boundedness,
- persistence,
- asymptotic behavior,
- rate of convergence
Citation: Tran Hong Thai, Nguyen Anh Dai, Pham Tuan Anh. Global dynamics of some system of second-order difference equations[J]. Electronic Research Archive, 2021, 29(6): 4159-4175. doi: 10.3934/era.2021077

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Abstract

In this paper, we study the boundedness and persistence of positive solution, existence of invariant rectangle, local and global behavior, and rate of convergence of positive solutions of the following systems of exponential difference equations

$ \begin{align*} x_{n+1} = \dfrac{\alpha_1+\beta_1e^{-x_{n-1}}}{\gamma_1+y_n},\ y_{n+1} = \dfrac{\alpha_2+\beta_2e^{-y_{n-1}}}{\gamma_2+x_n},\\ x_{n+1} = \dfrac{\alpha_1+\beta_1e^{-y_{n-1}}}{\gamma_1+x_n},\ y_{n+1} = \dfrac{\alpha_2+\beta_2e^{-x_{n-1}}}{\gamma_2+y_n}, \end{align*} $

where the parameters $ \alpha_i,\ \beta_i,\ \gamma_i $ for $ i \in \{1,2\} $ and the initial conditions $ x_{-1}, x_0, y_{-1}, y_0 $ are positive real numbers. Some numerical example are given to illustrate our theoretical results.

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