Global behavior of solutions of a two-dimensional system of difference equations

Year 2024, Volume: 6 Issue: 2, 13 - 29

Mehmet Gümüş , Raafat Abo-zeid , Kemal Türk

https://doi.org/10.54286/ikjm.1457991

Abstract

In this paper, we mainly investigate the qualitative and quantitative behavior of the solutions of a discrete system of difference equations
$$x_{n+1}=\frac{x_{n-1}}{y_{n-1}},\quad y_{n+1}=\frac{x_{n-1} }{ax_{n-1}+by_{n-1}},\quad n=0,1,\ldots, $$
where $a$, $b$ and the initial values $x_{-1},x_{0},y_{-1},y_{0}$ are non-zero real numbers. For $a\in \mathbb{R}_+-\{1\}$, we show any admissible solution $\{(x_n,y_n)\}_{n=-1}^\infty$ is either entirely located in a certain quadrant of the plane or there exists a natural number $N>0$ (we calculate its value) such that $\{(x_n,y_n)\}_{n=N}^\infty$ is located. Besides, some numerical simulations with graphs are given to emphasize the efficiency of our theoretical results in the article.

Keywords

Discrete dynamical system, forbidden set, invariant set, convergence

References

• R. Abo-Zeid, Global behavior and oscillation of a third order difference equation, Quaest. Math., 44(9) (2021), 1261−1280.
• R. Abo-Zeid, Global behavior of a fourth order difference equation with quadratic term, Bol. Soc. Mat. Mexicana, 25 (2019), 187−194.
• R. Abo-Zeid Forbidden sets and stability in some rational difference equations, J. Difference Equ. Appl., 24(2) (2018), 220−239.
• R. Abo-Zeid, Global behavior of a higher order rational difference equation, Filomat 30(12) (2016), 3265−3276.
• R. Abo-Zeid, Global behavior of a third order rational difference equation,Math. Bohem., 139(1) (2014), 25−37.
• A.M. Amleh, E. Camouzis and G. Ladas On the dynamics of a rational difference equation, Part 2, Int. J. Difference Equ., 3(2) (2008), 195−225.
• A.M. Amleh, E. Camouzis and G. Ladas On the dynamics of a rational difference equation, Part 1, Int. J. Difference Equ., 3(1) (2008), 1−35.
• M. Bekker,M. Bohner and H. Voulovc, Asymptotic behavior of solutions of a rational system of difference equations, J. Nonlinear Sci. Appl. 7 (2014), 3479−382.
• E. Camouzis, C.M. Kent, G. Ladas, C. D. Lynd, On the global character of solutions of the system xn+1 = α1+yn xn and yn+1 = α2+β2xn+γ2 yn A2+B2xn+C2 yn , J. Difference Equ. Appl., 18(7) (2012), 1205−1252.
• E. Camouzis, G. Ladas and L. Wu, On the global character of the system xn+1 = α1+γ1 yn xn and yn+1 = β2xn+γ2 yn B2xn+C2 yn , Inter. J. Pure Appl.Math., 53(1) (2009), 21−36.
• E. Camouzis, M.R.S. Kulenovic´, G. Ladas and O. Merino, Rational systems in the plane, J. Difference Equ. Appl., 15(3), (2009), 303−323.
• E. Camouzis and G. Ladas, Dynamics of Third Order Rational Difference Equations: With Open Problems and Conjectures, Chapman & Hall/CRC, Boca Raton, 2008.
• Q. Din, T.F. Ibrahim and A.Q. Khan, Behavior of a competitive system of second-order difference equations, Sci.World J., Volume 2014, Article ID 283982, 9 pages.
• E.M. Elsayed, Solution for systems of difference equations of rational form of order two, Comput. Appl. Math., 33(3) (2014), 751−765.
• M. Folly-Gbetoula and D. Nyirenda, Lie Symmetry Analysis and Explicit Formulas for Solutions of some Third-order Difference Equations, Quaest.Math., 42 (2019), 907−917.
• M. Folly-Gbetoula and D. Nyirenda, On some sixth-order rational recursive sequences, J. Comput. Anal. Appl., 27 (2019), 1057−1069.
• M. Gümüs and R. Abo Zeid, Qualitative study of a third order rational system of difference equations, Math.Moravica, 25(1) (2021), 81−97.
• M. Gümüs and Ö. Öcalan, The qualitative analysis of a rational system of diffrence equations, J. Fract. Calc. Appl., 9(2) (2018), 113-126.
• Y.Halim, A. Khelifa and M. Berkal, Representation of solutions of a two dimensional system of difference equations,MiskolcMath. Notes, 21(1) (2020), 203−218. doi: 10.18514/MMN.2020.3204.
• Y. Halim, Global character of systems of rational difference equations, Electron. J.Math. Analysis. Appl., 3(1) (2018), 204−214.
• Y. Halim, A system of difference equations with solutions associated to Fibonacci numbers, Int. J. Difference. Equ., 11(1) (2016), 65−77.
• T.F. Ibrahim and N. Touafek, On a third order rational difference equation with variable coefficients, Dyn. Contin. Discrete Impuls. Syst., Ser. B, Appl. Algorithms, 20 (2013), 251−264.
• T.F. Ibrahim, Closed form solution of a symmetric competitive system of rational difference equations, Stud.Math. Sci., 5(1) (2012), 49-57.
• M. Kara and Y. Yazlik, Solvable Three-Dimensional System of Higher-Order Nonlinear Difference Equations, Filomat, 36(10) (2022), 3449−3469.
• M. Kara and Y. Yazlik, On the solutions of three dimensional systemof difference equations via recursive relations of order two and applications, J. Appl. Anal. Comput., 12(2) 2022, 736−753.
• R. Khalaf-Allah, Asymptotic behavior and periodic nature of two difference equations, Ukrainian Math. J., 61(6) (2009), 988−993.
• V.L. Kocic and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic, Dordrecht, 1993.
• Z. Kudlak and R. Vernon, Unbounded rational systems with nonconstant coefficients, Nonauton. Dyn. Syst., 9 2022, 307−316.
• M.R.S. Kulenovi´c, Senada Kalabuši´c and Esmir Pilav, Basins of Attraction of Certain Linear Fractional Systems of Difference Equations in the Plane, Inter. J. Difference Equ., 9(2) (2014), 207−222.
• M.R.S. Kulenovi´c and G. Ladas, Dynamics of Second Order Rational Difference Equations: With Open Problems and Conjectures, Chapman and Hall/HRC, Boca Raton, 2002.
• M.R.S. Kulenovi´c and O. Merino, Discrete Dynamical Systems and Difference Equations with Mathematica, Chapman & Hall/CRC, Boca Raton, 2002.
• H. Sedaghat, On third order rational equations with quadratic terms, J. Difference Equ. Appl., 14(8) (2008), 889−897.
• S. Stevi´c, Solvability and representations of the general solutions to some nonlinear difference equations of second order, AIMSMath., 8(7) (2023), 15148−15165.
• S. Stevi´c, Representations of solutions to linear and bilinear difference equations and systems of bilinear difference equations, Adv. Differ. Equ., 2018 (2018), 1−21. https://doi.org/10.1186/s13662-018-1930-2.
• D.T. Tollu, Y. Yazlik and N. Taskara, On fourteen solvable systems of difference equations, Appl. Math. Comput., 233 (2014), 310−319.
• N. Touafek, On Some Fractional Systems of Difference Equations, Iranian Journal ofMath. Sci. Inf., 9(2) (2014), 73−86.
• N. Touafek and E.M. Elsayed, On the solutions of systems of rational difference equations, Math. Comput. Model., 55 (2012), 1987−1997.
• I. Yalçinkaya and C. Cinar, Global asymptotic stability of a system of two nonlinear difference equations, Fasc.Math., 43 (2010), 171-180.