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

* Corresponding author

Received  June 2021 Early access October 2021

Fund Project: 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.
Citation: Tran Hong Thai, Nguyen Anh Dai, Pham Tuan Anh. Global dynamics of some system of second-order difference equations. Electronic Research Archive, doi: 10.3934/era.2021077
References:
[1]

R. P. Agarwal, Difference Equations and Inequalities, 2$^{nd}$ edition, Dekker, New York, 2000.  Google Scholar

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Q. Din, Global stability of a population models, Chaos, Solitons & Fractals, 59 (2014), 119-128.  doi: 10.1016/j.chaos.2013.12.008.  Google Scholar

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Q. Din and E. M. Elsayed, Stability analysis of a discrete ecological model, Computational Ecology and Software, 4 (2014), 89-103.   Google Scholar

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H. El-MetwallyE. A. GroveG. LadasR. Levins and M. Radin, On the difference equation $x_{n+1} = \alpha + \beta x_{n-1}e^{-x_n}$, Nonlinear Anal., 47 (2001), 4623-4634.  doi: 10.1016/S0362-546X(01)00575-2.  Google Scholar

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N. Fotiades and G. Papaschinopoulos, Existence, uniqueness and attractivity of prime period two solution for a difference equation of exponential form, Appl. Math. Comput., 218 (2012), 11648-11653.  doi: 10.1016/j.amc.2012.05.047.  Google Scholar

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E. A. Grove and G. Ladas, Periodicities in Nonlinear Difference Equations, Chapman and Hall/CRC, Boca Raton, Fla, 2005.  Google Scholar

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E. A. GroveG. LadasN. R. Prokup and R. Levins, On the global behavior of solutions of a biological model, Comm. Appl. Nonlinear Anal., 7 (2000), 33-46.   Google Scholar

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T. Hong Thai, Asymptotic behavior of the solution of a system of difference equations, Int. J. Difference Equ., 13 (2018), 157-171.   Google Scholar

[9]

T. Hong Thai and V. Van Khuong, Stability analysis of a system of second-order difference equations, Math. Methods Appl. Sci., 39 (2016), 3691-3700.  doi: 10.1002/mma.3816.  Google Scholar

[10]

V. L. Kocić and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1993. doi: 10.1007/978-94-017-1703-8.  Google Scholar

[11]

M. R. S. Kulenović and G. Ladas, Dynamics of Second Order Rational Difference Equations, Chapman and Hall/CRC, Boca Raton, Fla, 2002.  Google Scholar

[12]

I. OzturkF. Bozkurt and S. Ozen, On the difference equation $y_{n+1} = \dfrac{\alpha+\beta e^{-y_n}}{\gamma + y_{n-1}}$, Appl. Math. Comput., 181 (2006), 1387-1393.  doi: 10.1016/j.amc.2006.03.007.  Google Scholar

[13]

G. PapaschinopoluosG. Ellina and K. B. Papadopoulos, Asymptotic behavior of the positive solutions of an exponential type system of difference equations, Appl. Math. Comput., 245 (2014), 181-190.  doi: 10.1016/j.amc.2014.07.074.  Google Scholar

[14]

G. PapaschinopoluosN. Fotiades and C. J. Schinas, On a system of difference equations including negative exponential terms, J. Difference Equ. Appl., 20 (2014), 717-732.  doi: 10.1080/10236198.2013.814647.  Google Scholar

[15]

G. PapaschinopoluosM. A. Radin and C. J. Schinas, On the system of two difference equations of exponential form: $x_{n+1} = a+b x_{n-1}e^{-y_n}, y_{n+1} = c+d y_{n-1}e^{-x_n}$, Math. Comput. Modelling, 54 (2011), 2969-2977.  doi: 10.1016/j.mcm.2011.07.019.  Google Scholar

[16]

G. PapaschinopoluosM. Radin and C. J. Schinas, Study of the asymptotic behavior of the solutions of three systems of difference equations of exponential form, Appl. Math. Comput., 218 (2012), 5310-5318.  doi: 10.1016/j.amc.2011.11.014.  Google Scholar

[17]

G. Papaschinopoluos and C. J. Schinas, On the dynamics of two exponential type systems of difference equations, Comput. Math. Appl., 64 (2012), 2326-2334.  doi: 10.1016/j.camwa.2012.04.002.  Google Scholar

[18]

M. Pituk, More on Poincaré's and Peron's theorems for difference equations, J. Difference Equ. Appl., 8 (2002), 201-216.  doi: 10.1080/10236190211954.  Google Scholar

[19]

H. Sedaghat, Nonlinear Difference Equations: Theory with Applications to Social Science Models, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2003. doi: 10.1007/978-94-017-0417-5.  Google Scholar

[20]

W. Wang and H. Feng, On the dynamics of positive solutions for the difference equation in a new population model, J. Nonlinear Sci. Appl., 9 (2016), 1748-1754.  doi: 10.22436/jnsa.009.04.30.  Google Scholar

show all references

References:
[1]

R. P. Agarwal, Difference Equations and Inequalities, 2$^{nd}$ edition, Dekker, New York, 2000.  Google Scholar

[2]

Q. Din, Global stability of a population models, Chaos, Solitons & Fractals, 59 (2014), 119-128.  doi: 10.1016/j.chaos.2013.12.008.  Google Scholar

[3]

Q. Din and E. M. Elsayed, Stability analysis of a discrete ecological model, Computational Ecology and Software, 4 (2014), 89-103.   Google Scholar

[4]

H. El-MetwallyE. A. GroveG. LadasR. Levins and M. Radin, On the difference equation $x_{n+1} = \alpha + \beta x_{n-1}e^{-x_n}$, Nonlinear Anal., 47 (2001), 4623-4634.  doi: 10.1016/S0362-546X(01)00575-2.  Google Scholar

[5]

N. Fotiades and G. Papaschinopoulos, Existence, uniqueness and attractivity of prime period two solution for a difference equation of exponential form, Appl. Math. Comput., 218 (2012), 11648-11653.  doi: 10.1016/j.amc.2012.05.047.  Google Scholar

[6]

E. A. Grove and G. Ladas, Periodicities in Nonlinear Difference Equations, Chapman and Hall/CRC, Boca Raton, Fla, 2005.  Google Scholar

[7]

E. A. GroveG. LadasN. R. Prokup and R. Levins, On the global behavior of solutions of a biological model, Comm. Appl. Nonlinear Anal., 7 (2000), 33-46.   Google Scholar

[8]

T. Hong Thai, Asymptotic behavior of the solution of a system of difference equations, Int. J. Difference Equ., 13 (2018), 157-171.   Google Scholar

[9]

T. Hong Thai and V. Van Khuong, Stability analysis of a system of second-order difference equations, Math. Methods Appl. Sci., 39 (2016), 3691-3700.  doi: 10.1002/mma.3816.  Google Scholar

[10]

V. L. Kocić and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1993. doi: 10.1007/978-94-017-1703-8.  Google Scholar

[11]

M. R. S. Kulenović and G. Ladas, Dynamics of Second Order Rational Difference Equations, Chapman and Hall/CRC, Boca Raton, Fla, 2002.  Google Scholar

[12]

I. OzturkF. Bozkurt and S. Ozen, On the difference equation $y_{n+1} = \dfrac{\alpha+\beta e^{-y_n}}{\gamma + y_{n-1}}$, Appl. Math. Comput., 181 (2006), 1387-1393.  doi: 10.1016/j.amc.2006.03.007.  Google Scholar

[13]

G. PapaschinopoluosG. Ellina and K. B. Papadopoulos, Asymptotic behavior of the positive solutions of an exponential type system of difference equations, Appl. Math. Comput., 245 (2014), 181-190.  doi: 10.1016/j.amc.2014.07.074.  Google Scholar

[14]

G. PapaschinopoluosN. Fotiades and C. J. Schinas, On a system of difference equations including negative exponential terms, J. Difference Equ. Appl., 20 (2014), 717-732.  doi: 10.1080/10236198.2013.814647.  Google Scholar

[15]

G. PapaschinopoluosM. A. Radin and C. J. Schinas, On the system of two difference equations of exponential form: $x_{n+1} = a+b x_{n-1}e^{-y_n}, y_{n+1} = c+d y_{n-1}e^{-x_n}$, Math. Comput. Modelling, 54 (2011), 2969-2977.  doi: 10.1016/j.mcm.2011.07.019.  Google Scholar

[16]

G. PapaschinopoluosM. Radin and C. J. Schinas, Study of the asymptotic behavior of the solutions of three systems of difference equations of exponential form, Appl. Math. Comput., 218 (2012), 5310-5318.  doi: 10.1016/j.amc.2011.11.014.  Google Scholar

[17]

G. Papaschinopoluos and C. J. Schinas, On the dynamics of two exponential type systems of difference equations, Comput. Math. Appl., 64 (2012), 2326-2334.  doi: 10.1016/j.camwa.2012.04.002.  Google Scholar

[18]

M. Pituk, More on Poincaré's and Peron's theorems for difference equations, J. Difference Equ. Appl., 8 (2002), 201-216.  doi: 10.1080/10236190211954.  Google Scholar

[19]

H. Sedaghat, Nonlinear Difference Equations: Theory with Applications to Social Science Models, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2003. doi: 10.1007/978-94-017-0417-5.  Google Scholar

[20]

W. Wang and H. Feng, On the dynamics of positive solutions for the difference equation in a new population model, J. Nonlinear Sci. Appl., 9 (2016), 1748-1754.  doi: 10.22436/jnsa.009.04.30.  Google Scholar

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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