Article Contents
Article Contents

# Global dynamics of some system of second-order difference equations

• * Corresponding author

The first author is supported by UTEHY grand number UTEHY.L.2020.11

• 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.

Mathematics Subject Classification: Primary: 39A10, 39A30; Secondary: 40A05.

 Citation:

• Figure 1.  Plot of $x_n$ for the system 32

Figure 2.  Plot of $y_n$ for the system 32

Figure 3.  An attractor of the system 32

Figure 4.  Plot of $x_n$ for the system 33

Figure 5.  Plot of $y_n$ for the system 33

Figure 6.  Phase portrait of system 33

Figure 7.  Plot of $x_n$ for the system 34

Figure 8.  Plot of $y_n$ for the system 34

Figure 9.  An attractor of the system 34

Figure 10.  Plot of $x_n$ for the system 35

Figure 11.  Plot of $y_n$ for the system 35

Figure 12.  Phase portrait of system 35

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