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

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

1. R. Abo-Zeid, Global behavior and oscillation of a third order difference equation, Quaest. Math., 44(9) (2021), 1261−1280.
2. R. Abo-Zeid, Global behavior of a fourth order difference equation with quadratic term, Bol. Soc. Mat. Mexicana, 25 (2019), 187−194.
3. R. Abo-Zeid Forbidden sets and stability in some rational difference equations, J. Difference Equ. Appl., 24(2) (2018), 220−239.
4. R. Abo-Zeid, Global behavior of a higher order rational difference equation, Filomat 30(12) (2016), 3265−3276.
5. R. Abo-Zeid, Global behavior of a third order rational difference equation,Math. Bohem., 139(1) (2014), 25−37.
6. 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.
7. 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.
8. 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.

9. 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.
10. 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.
11. 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.
12. E. Camouzis and G. Ladas, Dynamics of Third Order Rational Difference Equations: With Open Problems and Conjectures, Chapman & Hall/CRC, Boca Raton, 2008.
13. 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.
14. E.M. Elsayed, Solution for systems of difference equations of rational form of order two, Comput. Appl. Math., 33(3) (2014), 751−765.
15. 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.
16. M. Folly-Gbetoula and D. Nyirenda, On some sixth-order rational recursive sequences, J. Comput. Anal. Appl., 27 (2019), 1057−1069.
17. M. Gümüs and R. Abo Zeid, Qualitative study of a third order rational system of difference equations, Math.Moravica, 25(1) (2021), 81−97.
18. M. Gümüs and Ö. Öcalan, The qualitative analysis of a rational system of diffrence equations, J. Fract. Calc. Appl., 9(2) (2018), 113-126.
19. 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.
20. Y. Halim, Global character of systems of rational difference equations, Electron. J.Math. Analysis. Appl., 3(1) (2018), 204−214.
21. Y. Halim, A system of difference equations with solutions associated to Fibonacci numbers, Int. J. Difference. Equ., 11(1) (2016), 65−77.
22. 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.
23. T.F. Ibrahim, Closed form solution of a symmetric competitive system of rational difference equations, Stud.Math. Sci., 5(1) (2012), 49-57.
24. M. Kara and Y. Yazlik, Solvable Three-Dimensional System of Higher-Order Nonlinear Difference Equations, Filomat, 36(10) (2022), 3449−3469.
25. 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.
26. R. Khalaf-Allah, Asymptotic behavior and periodic nature of two difference equations, Ukrainian Math. J., 61(6) (2009), 988−993.
27. V.L. Kocic and G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic, Dordrecht, 1993.
28. Z. Kudlak and R. Vernon, Unbounded rational systems with nonconstant coefficients, Nonauton. Dyn. Syst., 9 2022, 307−316.
29. 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.
30. 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.
31. M.R.S. Kulenovi´c and O. Merino, Discrete Dynamical Systems and Difference Equations with Mathematica, Chapman & Hall/CRC, Boca Raton, 2002.
32. H. Sedaghat, On third order rational equations with quadratic terms, J. Difference Equ. Appl., 14(8) (2008), 889−897.
33. S. Stevi´c, Solvability and representations of the general solutions to some nonlinear difference equations of second order, AIMSMath., 8(7) (2023), 15148−15165.
34. 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.
35. D.T. Tollu, Y. Yazlik and N. Taskara, On fourteen solvable systems of difference equations, Appl. Math. Comput., 233 (2014), 310−319.
36. N. Touafek, On Some Fractional Systems of Difference Equations, Iranian Journal ofMath. Sci. Inf., 9(2) (2014), 73−86.
37. N. Touafek and E.M. Elsayed, On the solutions of systems of rational difference equations, Math. Comput. Model., 55 (2012), 1987−1997.
38. I. Yalçinkaya and C. Cinar, Global asymptotic stability of a system of two nonlinear difference equations, Fasc.Math., 43 (2010), 171-180.