Qualitative Behavior of Sixth Order of Rational Difference Equation

Article Sidebar

PDF
DOI: https://doi.org/10.28924/2291-8639-22-2024-106
Published: Jun 24, 2024
Cite as:
J. G. AL-Juaid, Qualitative Behavior of Sixth Order of Rational Difference Equation, Int. J. Anal. Appl., 22 (2024), 106.

Main Article Content

J. G. AL-Juaid

Abstract

The field of difference equations (DEs)has gained considerable prominence in applied analysis. The primary aim of this research is to conduct a comprehensive analysis on the periodicity of solutions, local asymptotic stability, and global behavior of DEs
Un+1=sUn−2+tUn−5+hUn−2+rUn−5/cUn−5−e, n=0,1,2, . . . .

Article Details

References

  1. R.P. Agarwal, E.M. Elsayed, Periodicity and Stability of Solutions of Higher Order Rational Difference Equation, Adv. Stud. Contemp. Math. 17 (2008), 181–201.
  2. T.D. Alharbi, E.M. Elsayed, The Solution Expressions and the Periodicity Solutions of Some Nonlinear Discrete Systems, Pan-Amer. J. Math. 2 (2023), 3. https://doi.org/10.28919/cpr-pajm/2-3.
  3. J.G. AL-Juaid, E.M. Elsayed, H. Malaikah, Behavior and Formula of the Solutions of Rational Difference Equations of Order Six, Ann. Commun. Math. 6 (2023), 72–85.
  4. R.J. Beverton, S.J. Holt, On the Dynamics of Exploited Fish Populations, Vol. 19, MAFF Fish. Invest., London, 1957.
  5. D.S. Dilip, S.M. Mathew, Dynamics of a Second-Order Nonlinear Difference System With Exponents, J. Egypt. Math. Soc. 29 (2021), 10. https://doi.org/10.1186/s42787-021-00119-6.
  6. Q. Din, E.M. Elabbasy, A.A. Elsadany, S. Ibrahim, Bifurcation Analysis and Chaos Control of a Second-Order Exponential Difference Equation, Filomat 33 (2019), 5003–5022. https://doi.org/10.2298/fil1915003d.
  7. E.M. Elabbasy, S.M. Eleissawy, Asymptotic Behavior of Two Dimensional Rational System of Difference Equations, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms. 20 (2013), 221–235.
  8. E.M. Elabbasy, H.A. El-Metwally, E.M. Elsayed, Global Behavior of the Solutions of Some Difference Equations, Adv. Differ. Equ. 2011 (2011), 28. https://doi.org/10.1186/1687-1847-2011-28.
  9. M.M. El-Dessoky, On a Solvable for Some Systems of Rational Difference Equations, J. Nonlinear Sci. Appl. 9 (2016), 3744–3759.
  10. M.F. Elettreby, H. El-Metwally, On a System of Difference Equations of an Economic Model, Discrete Dyn. Nat. Soc. 2013 (2013), 405628. https://doi.org/10.1155/2013/405628.
  11. H. El-Metwally, Solutions Form for Some Rational Systems of Difference Equations, Discrete Dyn. Nat. Soc. 2013 (2013), 903593. https://doi.org/10.1155/2013/903593.
  12. H. El-Metwally, E.A. Grove, G. Ladas, et al. On the Difference Equation Xn+1 = α + βxn−1e −xn , Nonlinear Anal.: Theory Meth. Appl. 47 (2001), 4623–4634. https://doi.org/10.1016/s0362-546x(01)00575-2.
  13. M.A. El-Moneam, E.M.E. Zayed, Dynamics of the Rational Difference Equation, Inf. Sci. Lett. 3 (2014), 45–53. https://doi.org/10.12785/isl/030202.
  14. E.M. Elsayed, Behavior of a Rational Recursive Sequences, Stud. Univ. Babes-Bolyai Math. LVI (2011), 27–42.
  15. E.M. Elsayed, Solution of a Recursive Sequence of Order Ten, Gen. Math. 19 (2011), 145–162.
  16. E.M. Elsayed, Solution and Attractivity for a Rational Recursive Sequence, Discrete Dyn. Nat. Soc. 2011 (2011), 982309. https://doi.org/10.1155/2011/982309.
  17. E. M. Elsayed and A. Alghamdi, Dynamics and Global Stability of Higher Order Nonlinear Difference Equation, J. Comput. Anal. Appl. 21 (2016), 493–503.
  18. E. Elsayed, J. Al-Juaid, The Form of Solutions and Periodic Nature for Some System of Difference Equations, Fundam. J. Math. Appl. 6 (2023), 24–34. https://doi.org/10.33401/fujma.1166022.
  19. E.M. Elsayed, J.G. AL-Juaid, H. Malaikah, On the Solutions of Systems of Rational Difference Equations, J. Prog. Res. Math. 19 (2022), 49–59.
  20. E.M. Elsayed, J.G. AL-Juaid, H. Malaikah, On the Dynamical Behaviors of a Quadratic Difference Equation of Order Three, Eur. J. Math. Appl. 3 (2023), 1. https://doi.org/10.28919/ejma.2023.3.1.
  21. E.M. Elsayed, B.S. Alofi, The Periodic Nature and Expression on Solutions of Some Rational Systems of Difference Equations, Alexandria Eng. J. 74 (2023), 269–283. https://doi.org/10.1016/j.aej.2023.05.026.
  22. E. Elsayed, F. Al-Rakhami, On Dynamics and Solutions Expressions of Higher-Order Rational Difference Equations, Ikonion J. Math. 5 (2023), 39–61. https://doi.org/10.54286/ikjm.1131769.
  23. E.M. Elsayed, M.M. Alzubaidi, On a Higher-Order Systems of Difference Equations, Pure Appl. Anal. 2023 (2023), 2.
  24. A. Gelisken, On A System of Rational Difference Equations, J. Comput. Anal. Appl. 23 (2017), 593–606.
  25. A. Khaliq, E.M. Elsayed, Qualitative Properties of Difference Equation of Order Six, Mathematics 4 (2016), 24. https://doi.org/10.3390/math4020024.
  26. A. Khaliq, T.F. Ibrahim, A.M. Alotaibi, M. Shoaib, M.A. El-Moneam, Dynamical Analysis of Discrete-Time TwoPredators One-Prey Lotka-Volterra Model, Mathematics. 10 (2022), 4015. https://doi.org/10.3390/math10214015.
  27. A. Qadeer Khan, M. Tasneem, B.A.I. Younis, T.F. Ibrahim, Dynamical Analysis of a Discrete-Time COVID-19 Epidemic Model, Math. Meth. Appl. Sci. 46 (2022), 4789–4814. https://doi.org/10.1002/mma.8806.
  28. A.Q. Khan, H.S. Alayachi, Bifurcation and Chaos in a Phytoplankton-Zooplankton Model with Holling Type-II Response and Toxicity, Int. J. Bifurcation Chaos. 32 (2022), 2250176. https://doi.org/10.1142/s0218127422501760.
  29. A.Q. Khan, F. Nazir, M.B. Almatrafi, Bifurcation Analysis of a Discrete Phytoplankton–zooplankton Model With Linear Predational Response Function and Toxic Substance Distribution, Int. J. Biomath. 16 (2022), 22500954. https://doi.org/10.1142/s1793524522500954.
  30. V.L. Kocic, G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications, Springer, Dordrecht, 1993. https://doi.org/10.1007/978-94-017-1703-8.
  31. Y. Kostrov, Z. Kudlak, On a Second-Order Rational Difference Equation with a Quadratic Term, Int. J. Diff. Equ. 11 (2016), 179–202.
  32. A.S. Kurbanlı, C. Çinar, I. Yalçinkaya, On the behavior of positive solutions of the system of rational difference equations xn+1 = xn−1 / ynxn−1+1 , yn+1 = yn−1 / xnyn−1+1 , Math. Comput. Model. 53 (2011), 1261–1267. https://doi.org/10.1016/j.mcm.2010.12.009.
  33. C.Y. Ma, B. Shiri, G.C. Wu, D. Baleanu, New Fractional Signal Smoothing Equations With Short Memory and Variable Order, Optik 218 (2020), 164507. https://doi.org/10.1016/j.ijleo.2020.164507.
  34. B. Ogul, D. Simsek, On the Recursive Sequence, Dynamics of Continuous, Discrete Impuls. Syst. Ser. B Appl. Algorithms. 29 (2022), 423–435.
  35. J. Tariboon, S.K. Ntouyas, P. Agarwal, New Concepts of Fractional Quantum Calculus and Applications to Impulsive Fractional q-Difference Equations, Adv. Differ. Equ. 2015 (2015), 18. https://doi.org/10.1186/s13662-014-0348-8.
  36. E. Ta¸sdemir, On the Global Asymptotic Stability of a System of Difference Equations With Quadratic Terms, J. Appl. Math. Comput. 66 (2020), 423–437. https://doi.org/10.1007/s12190-020-01442-4.
  37. N.L. Wang, P. Agarwal, S. Kanemitsu, Limiting Values and Functional and Difference Equations, Mathematics. 8 (2020), 407. https://doi.org/10.3390/math8030407.
  38. G.C. Wu, Z.G. Deng, D. Baleanu, D.Q. Zeng, New Variable-Order Fractional Chaotic Systems for Fast Image Encryption, Chaos. 29 (2019), 083103. https://doi.org/10.1063/1.5096645.
  39. E.M.E. Zayed, M. El-Moneam, On The Rational Recursive Sequence xn+1 = Axn + Bxn−k + βxn+γxn−k/Cxn+Dxn−k, Acta Appl. Math. 111 (2010), 287–301.