Global Asymptotic Stability of a System of Difference Equations with Quadratic Terms

Global Asymptotic Stability of a System of Difference Equations with Quadratic Terms

In this article, we discuss the global asymptotic stability of following system of difference equations with quadratic terms: $x_{i+1}=\alpha+\beta \frac{y_{i-1}}{y_{i}^{2}},\quad y_{i+1}=\alpha+\beta \frac{x_{i-1}}{x_{i}^{2} }$ where $\alpha$, $\beta$ are positive numbers and the initial values are positive numbers. We also study the rate of convergence and oscillation behaviour of the solutions of related system. We will give also, some numerical examples to illustrate our results.

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  • [1] R. P. Agarwal, P. J. Wong, Advanced Topics in Difference Equations, volume 404, Springer Science & Business Media, 2013.
  • [2] M. B. Almatrafi, E. M. Elsayed, Solutions and formulae for some systems of difference equations, MathLAB J. , 1(3) (2018), 356-369.
  • [3] S. Abualrub, M. Aloqeili, Dynamics of positive solutions of a system of difference equations, J. Comput. Appl. Math., 392 (2021), 113489.
  • [4] E. Besˇo. Kalabusˇic ́, N. Mujic ́, E. Pilav, Boundedness of solutions and stability of certain second-order difference equation with quadratic term, Adv. Differ. Equ., 2020(1) (2020), 1-22.
  • [5] A. Bilgin, M. Kulenovic ́, Global asymptotic stability for discrete single species population model, Discrete Dynamics in Nature and Society, 2017.
  • [6] J. B. Bacani, J. F. T. Rabago, On two nonlinear difference equations, Dyn. Contin. Discrete Impuls Syst. Ser. A Math. Anal., 24 (2017), 375-394.
  • [7] F. Belhannache, N. Touafek, R. Abo-zeid, On a higher-order rational difference equation, J. Appl. Math. & Informatics, 34 (2016), 5-6, 369-382.
  • [8] D. Burgic, M. Kulenovic, M. Nurkanovic, Global dynamics of a rational system of difference equations in the plane, Commun. Appl. Nonlinear Anal., 15(1) (2008), 71-84.
  • [9] Q. Din, E. M. Elsayed, Stability analysis of a discrete ecological model, Comput. Ecol. Softw., 4(2) (2014), 89-103.
  • [10] Q. Din, Asymptotic behavior of an anti-competitive system of second order difference equations, J. Egypt. Math. Soc., 24(1) (2016), 37-43.
  • [11] E. Elabbasy, S. Eleissawy, Asymptotic behavior of two dimensional rational system of difference equations, Dyn. Contin. Discrete Impuls. Syst. Ser. B. Appl. Algorithms, 20 (2013), 221-235.
  • [12] S. N. Elaydi, An Introduction to Difference Equations, New York, 1996.
  • [13] E. Tasdemir, Dynamics of a system of higher order difference equations with quadratic terms, Preprints, 2021, 2021040082, doi: 10.20944/preprints202104.0082.v1.
  • [14] M. El-Dessoky, On a solvable for some systems of rational difference equations, J. Nonlinear Sci. Appl., 9(6) (2016), 3744-3759.
  • [15] M. A. El-Moneam, On the dynamics of the higher order nonlinear rational difference equation, Math. Sci. Lett., 3(2) (2014), 121-129.
  • [16] M. A. El-Moneam, On the dynamics of the solutions of the rational recursive sequences, British Journal of Mathematics, Computer Science, 5(5) (2015), 654-665.
  • [17] M. A. El-Moneam, S.O. Alamoudy, On study of the asymptotic behavior of some rational difference equations, DCDIS Series A: Math. Anal., 21 (2014), 89-109.
  • [18] M. A. El-Moneam, E. M. E. Zayed, Dynamics of the rational difference equation, Inf. Sci. Lett., 3 (2) (2014), 1-9.
  • [19] M. A. El-Moneam, E. M. E. Zayed, On the dynamics of the nonlinear rational difference equation xn+1 = Axn + Bxn−k + Cxn−l +((bxn−k)/(dxn−k −exn−l)), J. Egypt. Math. Soc., 23 (2015), 494-499.
  • [20] M. Garic-Demirovic, S. Hrustic, S. Morankic, Global dynamics of certain non-symmetric second order difference equation with quadratic terms, Sarajevo J. Math., 15(2) (2019), 155-167.
  • [21] M. Gocen, A. Cebeci, On the periodic solutions of some systems of higher order difference equations, Rocky Mountain J. Math., 48(3) (2018), 845-858.
  • [22] N. Haddad, N. Touafek, J. Rabago, Solution form of a higher-order system of difference equations and dynamical behavior of its special case, Math. Meth. App. Sci., 40(10) (2016), 3599-3607.
  • [23] V. Hadzˇiabdic ́ , M. R. S. Kulenovic , E. Pilav, Dynamics of a two-dimensional competitive system of rational difference equations with quadratic terms, Adv. Differ. Equ., 301 (2014), 1-32.
  • [24] A. Khan, M. Qureshi, Qualitative behavior of two systems of higher order difference equations, Math. Meth. Appl. Sci., 39(11) (2016), 3058-3074.
  • [25] A. Q. Khan, K. Sharif, Global dynamics, forbidden set, and trans critical bifurcation of a one-dimensional discrete-time laser model, Math. Meth. Appl. Sci., 43(7) (2020), 4409-4421.
  • [26] V. L. Kocic, G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, volume 256, Springer Science & Business Media, 1993.
  • [27] M. R. Kulenovic, G. Ladas, Dynamics of Second Order Rational Difference Equations: With Open Problems and Conjectures, Chapman and Hall/CRC, 2001.
  • [28] J. D. Murray, Mathematical Biology: I. An Introduction, 3rd Ed., Springer-Verlag, New York, 2001.
  • [29] I. Okumus ̧, Y. Soykan, Dynamical behavior of a system of three dimensional nonlinear difference equations, Adv. Differ. Equ, 2018(1) (2018), 1-15.
  • [30] M. Pituk, More on Poincare and Perron theorems for difference equations, J. Differ. Equ. Appl., 8 (2002), 201-216.
  • [31] L. Yang, J. Yang, Dynamics of a system of two nonlinear difference equations, Int. J. Contemp. Math. Sci., 6(5) (2011), 209-214.