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Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain


Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

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The Solution and Dynamic Behaviour of some Difference Equations of Seventh Order

Journal of Applied Nonlinear Dynamics 10(4) (2021) 709--719 | DOI:10.5890/JAND.2021.12.010

M. B. Almatrafi$^1$ , Marwa M. Alzubaidi$^2$

$^1$ Department of Mathematics, Faculty of Science, Taibah University, Al-Madinah Al-Munawarah, Saudi Arabia

$^2$ Department of Mathematics, College of Duba, University of Tabuk, Saudia Arabia

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Nonlinear difference equations are mostly utilized to describe some natural phenomena. The exact solutions of some models cannot be sometimes extracted. Therefore, investigating the long behaviour of the model may give the future pattern of the respective problem. This article manifests the long behaviour of a seventh order rational difference equation with positive real coefficients. Through this paper, the local stability, global stability, boundedness are obtained analytically and some figures are illustrated to test the accuracy of the solutions. The used technique can be extended to be utilized in solving some higher order difference equations.


  1. [1]  Elaydi, S.An Introduction to Difference Equations An Introduction to Difference Equations, 3rd Ed., Springer, USA.
  2. [2]  Murray, J.D.Mathematical Biology: I. An Introduction Mathematical Biology: I. An Introduction, 3rd Ed., Springer-Verlag, New York.
  3. [3]  Almatrafi, M.B. and Alzubaidi, M.M.Open Journal of Discrete Applied Mathematics Analysis of the qualitative behaviour of an eighth-order fractional difference equation, Open Journal of Discrete Applied Mathematics, 2(1), 41-47.
  4. [4]  Almatrafi, M.B.Open Journal of Mathematical Analysis Solutions structures for some systems of fractional difference equations, Open Journal of Mathematical Analysis, 3(1), 51-61.
  5. [5]  Cinar, C.Applied Mathematics and Computation On the positive solutions of the difference equation $x_{n+1}=ax_{n-1}/(1+bx_{n}x_{n-1}),$\ Applied Mathematics and Computation, 156, 587-590.
  6. [6]  Elabbasy, E.M., El-Metwally, H., and Elsayed, E.M.Advances in difference Equations On the difference equation $x_{n+1}=ax_{n}-(bx_{n})/(cx_{n}-dx_{n-1}),$ Advances in difference Equations, 2006, Article ID 82579, 1-10.
  7. [7]  Gari\{c}-Demirovi\{c}, M., Nurkanovi\{c}, M., and Nurkanovi\{c}, Z.International Journal of Difference Equations Stability, periodicity and neimark-sacker bifurcation of certain homogeneous fractional difference equations, International Journal of Difference Equations, 12(1), 27-53.
  8. [8]  Ghazel, M., Elsayed, E.M., Matouk, A.E., and Mousallam, A.M.J. Nonlinear Sci. Appl. Investigating dynamical behaviors of the difference equation $x_{n+1}=Cx_{n-5}/(A+Bx_{n-2}x_{n-5}),$ J. Nonlinear Sci. Appl., 10, 4662-4679.
  9. [9]  Saleh, M., Alkoumi, N., and Farhat, A.Chaos, Solitons and Fractals On the dynamic of a rational difference equation $x_{n+1}=\alpha+\beta x_{n}+\gamma x_{n-k}/B x_{n}+C x_{n-k}, $ Chaos, Solitons and Fractals, 96, 76-84.
  10. [10]  Almatrafi, M.B. and Elsayed, E.M.MathLAB Journal Solutions and formulae for some systems of difference equations, MathLAB Journal, 1(3), 356-369.
  11. [11]  Almatrafi, M.B., Elsayed, E.M., and Alzahrani, F.International Journal of Advances in Mathematics Qualitative behavior of a quadratic second-order rational difference equation, International Journal of Advances in Mathematics, 2019(1), 1-14.
  12. [12]  Belhannache, F., Touafek, N., and Abo-zeid, R.J. Appl. Math. $\&$ Informatics On a higher-order rational difference equation, J. Appl. Math. $\&$ Informatics, 34(5-6), 369-382.
  13. [13]  Khaliq, A. and Hassan, Sk.S.International Journal of Advances in Mathematics Dynamics of a rational difference equation $x_{n+1}=ax_{n}+(\alpha+\beta x_{n-k})/(A+Bx_{n-k}),$ International Journal of Advances in Mathematics, 2018(1), 159-179.
  14. [14]  Khyat, T. and Kulenovi\{c}, M.R.S.International Journal of Difference Equations The invariant curve caused by Neimark-Sacker bifurcation of a perturbed Beverton-Holt difference equation, International Journal of Difference Equations, 12(2), 267-280.
  15. [15]  Almatrafi, M.B., Elsayed, E.M., and Alzahrani, F.Fundamental Journal of Mathematics and Applications Qualitative behavior of two rational difference equations, Fundamental Journal of Mathematics and Applications, 1(2), 194-204.
  16. [16]  Kostrov, Y. and Kudlak, Z.International Journal of Difference Equations On a second-order rational difference equation with a quadratic term, International Journal of Difference Equations, 11(2), 179-202.
  17. [17]  Liu, K., Li, P., Han, F., and Zhong, W.Journal of Computational Analysis and Applications Global dynamics of nonlinear difference equation $x_{n+1}=A+x_{n}/x_{n-1}x_{n-2}$, Journal of Computational Analysis and Applications, 24(6), 1125-1132.
  18. [18]  Moranjki\{c}, S. and Nurkanovi\{c}, Z.Advances in Dynamical Systems and Applications Local and global dynamics of certain second-order rational difference equations containing quadratic terms, Advances in Dynamical Systems and Applications, 12(2), 123-157.
  19. [19]  Saleh, M. and Aloqeili, M.Appl. Math. Comput. On the rational difference equation $y_{n+1}=A+\frac{y_{n-k}}{y_{n}}$, Appl. Math. Comput., 171(1), 862-869.
  20. [20]  Simsek, D., Cinar, C., and Yalcinkaya, I.Int. J. Contemp. Math. Sci. On the recursive sequence $x_{n+1}=\frac{x_{n-3}}{1+x_{n-1}}$, Int. J. Contemp. Math. Sci., 1(10), 475-480.
  21. [21]  Kulenovic, M.R.S. and Ladas, G.Chapman and Hall / CRC Press Dynamics of second order rational difference equations with open problems and conjectures, Chapman and Hall / CRC Press.
  22. [22]  Elsayed, E.M.Kyungpook Math. J. Dynamics of recursive sequence of order two, Kyungpook Math. J., 50, 483-497.
  23. [23]  Kocic, V.L. and Ladas, G.Kluwer Academic Publishers Global behavior of nonlinear difference equations of higher order with applications, Kluwer Academic Publishers, Dordrecht.