Global Analysis of a $(1,2)$-Type System of Non-Linear Difference Equations

Year 2021, Volume: 4 Issue: 1, 26 - 38, 29.03.2021

Emine Yener , Mehmet Gümüş

https://doi.org/10.33434/cams.837197

Abstract

This paper deals with the study of global analysis of following $(1,2)-$type system of non-linear difference equations:\[ u_{n+1}=\frac{\alpha v_{n-1}}{\beta +\gamma v_{n}^{p}v_{n-2}^{q}},\ \ \ \ \ \ v_{n+1}=\frac{\alpha _{1}u_{n-1}}{\beta _{1}+\gamma _{1}u_{n}^{p}u_{n-2}^{q}},\ \ \ \ n=0,1,... \] where the parameters $\alpha ,\beta ,\gamma ,\alpha _{1},\beta _{1},\gamma _{1,}p,q$ and the initial conditions $u_{i},v_{i},$ $i=-2,-1,0$ are non negative real numbers.

Keywords

(1,2)-type system, difference equations, equilibrium, global stability, rate of convergence

References

• [1] R. Agarwal, Difference equations and inequalities, theory, methods and applications, Marcel Dekker Inc., New York 1992.
• [2] R. Abo-Zeid and H. Kamal, Global behavior of two rational third order difference equations, Univers. J. Math. Appl., 2(4) (2019), 212-217.
• [3] S. Abualrub and M. Aloqeili, Dynamics of the System of Difference Equations $x_{n+1}=A+y_{n-k}/y_{n},$\textit{\ }$% y_{n+1}=B+x_{n-k}/x_{n}$, Qual. Theory Dyn. Syst., 19(2) (2020), 1-19.
• [4] A. M. Ahmed, On the Dynamics of Higher-Order Rational Difference Equation, Discrete Dyn. Nat. Soc., 2011, Article ID: 419789, 8 pages.
• [5] A. M. Ahmed, The Dynamics of the Recursive Sequences $x_{n+1}=\frac{bx_{n-1}}{A+Bx_{n}^{p}x_{n-2}^{p}}$, J.Pure Appl. Math.: Adv. Applic., 1(2) (2009), 215-223.
• [6] M. M. Alzubaidi and E. M. Elsayed, Analytical and Solutions of Fourth Order Difference Equations, Commun. Adv. Math. Sci., 2(1) (2019), 9-21.
• [7] Q. Din, T. F. Ibrahim and A.Q. Khan, Behavior of a competitive system of second order difference equations, Sci. World J., Article ID: 283982.
• [8] Q. Din, Asymptotic behavior of an anti-competitive system of second-order difference equations, J. Egyptian Math. Soc., 24(1) (2016), 37-43.
• [9] H. M. El-Owaidy, A. M. Ahmed and A. M. Youssef, The Dynamics of the Recursive Sequence $x_{n+1}=\frac{\alpha x_{n-1}% }{\beta +\gamma x_{n-2}^{p}}$, Appl. Math. Lett., 18 (2005), 1013-1018.
• [10] M. M. El-Dessoky, E. M. Elsayed, E. M. Elabbasy and A. Asiri, Expressions of the solutions of some systems of difference equations, J. Comput. Anal. Appl., 27(7) (2019), 1161-1172.
• [11] S. Elaydi, An Introduction to Difference Equations, 3rd ed., Springer-Verlag, New York, 2005.
• [12] M. Gocen and M. Guneysu, The Global Attractivity of some ratinal difference equations, J. Comp. Anal. Appl., 25 (7) (2018), 1233-1243.
• [13] M. Gümüş and Y. Soykan, Global Character of a Six-Dimensional Nonlinear System of Difference Equations, Discrete Dyn. Nat. Soc., 2016, Article ID 6842521.
• [14] M. Gumus and O. Ocalan, The Qualitative Analysis of a Rational System of Difference Equations, J. Fract. Calc. Appl., 9(2) (2018), 113-126.
• [15] M. Gumus and Y. Soykan, Dynamics of Positive Solutions of a Higher Order Fractional Difference Equation with Arbitrary Powers , J. Appl. Math. Inf., 35(3-4) (2017), 267-276.
• [16] V. Kocic and G. Ladas, Global behavior of nonlinear difference equations of higher order with applications, Kluwer Academic Publishers, Dordrecht, (1993).
• [17] F. H. Gumus, Yuksek Mertebeden Fark Denklemlerinin Global Davranışları Uzerine, Master Thesis, Afyon Kocatepe Universitesi Fen Bilimleri Enstitu¨su¨ Afyon (2015).
• [18] A. Khelifa, Y. Halim and M. Berkal, Solutions of a system of two higher-order difference equations in terms of lucas sequence, Univers. J. Math. Appl., 2(4) (2019), 202-211.
• [19] M. A. Kerker, E. Hadidi, and A. Salmi, Qualitative behavior of a higher-order nonautonomous rational difference equation, J. Appl. Math. Comput., 64 (2020), 399–409.
• [20] M. R. S. Kulenovic and O. Merino, Discrete Dynamical Systems and Difference Equations with Mathematica. Chapman and Hall/CRC, Boca Raton, London. (2002).
• [21] M. R. S. Kulenovic and G. Ladas, Dynamics of second order rational difference equations, Chapman and Hall/CRC, (2001).
• [22] M. Kara, N. Touafek and Y. Yazlık, Well-Defined Solutions of a Three-Dimensional System of Difference Equations, Gazi Univ. J.Sci., 33(3), 767-778.
• [23] W. X. Ma, Global Behavior of an Arbitrary-Order Nonlinear Difference Equation with a Nonnegative Function, Mathematics, 8(5) (2020), 825.
• [24] M. Merdan and S. Sisman, Investigation of linear difference equations with random effects, Adv. Difference Equ., 2020 (1) (2020), 1-19.
• [25] M. Pituk, More on Poincares and Perrons Theorems for Difference Equations, J. Difference Equ. Applic., 8(3) (2002), 201-216.
• [26] A. Sanbo, E. M. Elsayed and F. Alzahrani, Dynamics of the nonlinear rational difference equation $x_{n+1}=(Ax_{n-\alpha }x_{n-\beta }+Bx_{n-\gamma })/(Cx_{n-\alpha }x_{n-\beta }+Dx_{n-\gamma }),$; Indian J. Pure Appl. Math., 50(2) (2019), 385-401.
• [27] M. N. Qureshi and A. Q. Khan, Global dynamics of $(1,2)$-type systems of difference equations, Malaya J. Matematik, 6(2), 408-416.
• [28] M. N. Qureshi, Q. Din and A. Q. Khan, Asymptotic behavior of an anti-competitive system of rational difference equations, Life Sci. J., 11(2014): 1-7.
• [29] M. N. Qureshi and Q. Din, Oualitative behavior of an anti-competitive system of third-order rational difference equations, Comput. Ecology Software, 4(2) (2014), 104-115.
• [30] H. Sedaghat, Nonlinear difference equations theory with applications to social science models, 15. Springer Science and Business Media, (2003).
• [31] D. T. Tollu and I. Yalc¸ınkaya, Global behavior of a three dimensional system of difference equations of order three, Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat., 68(1) (2019), 1-16.
• [32] I. Yalcınkaya, On the Global Asymptotic Stability of a Second-Order System of Difference Equations, Discrete Dyn. Nat. Soc., 2008, Article ID 860152.
• [33] D. Zhang, W. Ji, L. Wang and X. Li, On the symmetrical system of rational difference equation $x_{n+1}=A+y_{n-k}/y_{n}$% \textit{, }$y_{n+1}=A+x_{n-k}/x_{n}$, Appl. Math., 4 (2013), 834-837.