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On Dynamics and Solutions Expressions of Higher-Order Rational Difference Equations

Year 2023, Volume: 5 Issue: 1, 39 - 61, 22.08.2023

Elsayed Elsayed , Faiza Al-rakhami

https://doi.org/10.54286/ikjm.1131769

Abstract

The principle goal of this paper is to look at some of the qualitative behavior of the critical point of the rational difference equation

Ψ_{n+1}=αΨ_{n-2}+((βΨ_{n-2}Ψ_{n-3})/(γΨ_{n-3}+δΨ_{n-6})), n=0,1,2,...,

where α,β,γ and δ are arbitrary positive real numbers. We also used the proposed equation to get the general solution for particular cases and provided numerical examples to demonstrate our results.

Keywords

Solutions expressions of difference equation local and global stability boundedness.

References

  • 1.R. Abo-Zeid and H. Kamal, Global behavior of two rational third order difference equations , Universal J. Math. Appl., (2019), 212-217.
  • 2.R. P. Agarwal and E. M. Elsayed, Periodicity and stability of solutions of higher order rational difference equation, Advanced Studies in Contemp. Math., 2008, 181-201.
  • 3.H. S. Alayachi, M. S. M. Noorani, A. Q. Khan and M. B. Almatrafi, Analytic solutions and stability of sixth order difference equations, Math. Prob. Eng., 2020 (2020), 1-12.
  • 4.H. S. Alayachi, A. Q. Khan, M. S. M. Noorani, and A. Khaliq, Displaying the structure of the solutions for some fifth-order systems of recursive equations, Math. Prob. Eng., 2021 (2021), 1-14.
  • 5.H. S. Alayachi, A. Q. Khan, and M. S. M. Noorani, On the solutions of three-dimensional rational difference equation systems, J. Math., 2021 (2021), 1- 15.
  • 6.M. B. Almatrafi , E. M. Elsayed and F. Alzahrani, Qualitative behavior of a quadratic second-order rational difference equation, Int. J. Adv. Math, 2019 (1), 1-14.
  • 7.A. Alshareef, F. Alzahrani, and A. Q. Khan, Dynamics and solutions' expressions of a higher-order nonlinear fractional recursive sequence, Math. Prob. Eng., 2021 (2021), 1 - 12.
  • 8.M. M. Alzubaidi and E. M. Elsayed, Analysis of qualitative behavior of fifth order difference equations, MathLAB J., 2019 (2) (1), 1-18.
  • 9.A. Asiri and E. M. Elsayed, Dynamics and solutions of some recursive sequences of a higher-order, J. Comp. Anal. Appl. 27 (4) (2019), 656-670.
  • 10.F. Belhannache, On the stability of a system of difference equations, Elec. J. Math. Anal. and Appl., 8(2) (2020), 109-114.
  • 11.E. Camouzis and G. Ladas. Dynamics of third-order rational difference equations with open problems and conjectures, 5. CRC Press, (2007).
  • 12.C. Çinar, On the positive solutions of the difference equation x_{n+1}=((ax_{n-1})/(1+bx_{n}x_{n-1})), Appl. Math. Comput., 156 (2) (2004), 587-590.
  • 13.Q. Din, Global stability and Neimark-Sacker bifurcation of a host parasitoid model, Int. J. Syst. Sci., 48 (6) (2017), 1194-1202.
  • 14.E. M. Elabbasy, H. El-Metwally, and E. M. Elsayed, On the difference equation x_{n+1}=ax_{n-}((bx_{n})/(cx_{n}-dx_{n-1})), Adv. Differ. Equations, 2006 ( June) (2006), 1-10.
  • 15.E. M. Elabbasy, and A. El-Biaty, Asymptotic behavior of some rational difference equations, Int. J. Comp. Appl., 136 (8) (2016), 18-24.
  • 16.S. Elaydi, An introduction to difference equations, Springer New York, NY,USA, (2005).
  • 17.H. A. El-Metwally, On the qualitative study of some difference equations, IOSR J. Math., 16 (2020), 48-54.
  • 18.E. M. Elsayed, Solution and attractivity for a rational recursive sequence, Dis. Dyn. Nat. Soc., 2011(2011), 1- 17.
  • 19.E. M. Elsayed, Solution and dynamics of a fourth rational difference equation, Int. J. Phy. Sci., 7(48)(2012), 6191-6202.
  • 20.E. M. Elsayed, Expression and behavior of the solutions of some rational recursive sequences, Math. Meth. Appl. Sci.(39) (2016), 5682-5694.
  • 21.E. M. Elsayed and T. F. Ibrahim, Solutions and periodicity of a rational recursive sequences of order five, Bull. Malays. Math. Sci. Soc. 38(1)(2015), 95-112.
  • 22.E. M. Elsayed, and A. M. Ahmed, Dynamics of a three-dimensional systems of rational difference equations, Math. Meth. Appl. Sci., 39 (5) (2016), 1026-1038.
  • 23.M. Folly-Gbetoula, N. Mnguni and A. H. Kara, A group theory approach towards some rational difference equations, J. Math., 2019 (2019), 1-9.
  • 24.M. Gümüş and R. Abo-Zeid, Qualitative study of a third order rational system of difference equations, Math. Moravica, 25(1) (2021), 81-97.
  • 25.T. F. Ibrahim, On the third order rational difference equation x_{n+1}=((x_{n}x_{n-2})/(x_{n-1}(a+bx_{n}x_{n-2}))), Int. J. Contemp. Math. Sci., 4 (27) (2009), 1321-1334.
  • 26.T. F. Ibrahim, A. Q. Khan and A. Ibrahim, Qualitative behavior of a nonlinear generalized recursive sequence with delay, Math. Prob. Eng., 2021 (2021),1-8.
  • 27.M. Kara, Y. Yazliky. Halim, On a solvable system of non-linear difference equations with variable coefficients, J. Scie. Arts, 1 (54) (2021), 145-162.
  • 28.R. Karatas, C. Cinar, D. Simsek, On positive solutions of the difference equationx_{n+1}=((x_{n-5})/(1+bx_{n-2}x_{n-5})), Int. J. Contemp. Math. Sci., 1 (10) (2006), 494-500.
  • 29.A. Khaliq and E. M. Elsayed, Global behavior and periodicities of some fractional recursive sequences, Proceedings of the Jangjeon Math. Soci.,20 (3) (2017), 421 - 441.
  • 30.A. Khaliq, H. S. Alayachi, M. S. M. Noorani, and A. Q. Khan, On stability analysis of higher-order rational difference equation, Discret. Dyn. Nat. Soc., 2020 (2020).
  • 31.A. Q. Khan, M. S. M. Noorani and H. S. Alayachi, Global dynamics of higher-order exponential systems of difference equations, Discret. Dyn. Nat. Soc., 2019 (2019), 1-19.
  • 32.M. R. S. Kulenovic and G. Ladas, Dynamics of second order rational difference equations: with open problems and conjectures. Chapman and Hall/CRC, (2001).
  • 33.A. S. Kurbanli, C. Çinar, and I. Yalçinkaya, On the behavior of positive solutions of the system of rational difference equations x_{n+1}=((x_{n-1})/(x_{n-1}y_{n}+1)), y_{n+1}=((y_{n-1})/(y_{n-1}x_{n}+1)), Math. Comput. Model., 53, (5--6) (2011), 1261-1267.
  • 34.R. E. Mickens, Difference equations: theory, applications and advanced topics, CRC Press, 2015.
  • 35.A. A. Muna and S. Mohammad, Dynamics of a higher order rational difference equation x_{n+1}=(((α+βx_{n}))/((A+Bx_{n}+Cx_{n-k}))), J. Nonlinear Anal. Appl, 8 (2) (2017), 363-379.
  • 36.B. Oğul, D. Şimşek, H. Öğünmez and A. S. Kurbanlı, Dynamical behavior of rational difference equation x_{n+1}=((x_{n-17})/(±1±x_{n-2}x_{n-5}x_{n-8}x_{n-11}x_{n-14}x_{n-17})), Bol. Soc. Mat. Mex., 27(49)(2021).
  • 37.D. Simşek, B. Oğul, and T. F. Ibrahim, Solution of the rational difference equation , Dyn. Cont., Discrete and Imp. Sys., 33 (5) (2021), 125-141.
  • 38.D. T. Tollu, I. Yalcinkaya, H. Ahmad, and S. W. Yao, A detailed study on a solvable system related to the linear fractional difference equation, Math. Biosci. Eng.,18 (5) (2021), 5392-5408.
  • 39.I. Yalçinkaya and C. Cinar, Global asymptotic stability of a system of two nonlinear difference equations, Fasciculi Math., 43 (2010), 171-180.
  • 40.E. M. E. Zayed, The dynamics of a new nonlinear rational difference equations, Math. Anal., 27 (2020), 153-165.

Year 2023, Volume: 5 Issue: 1, 39 - 61, 22.08.2023

Elsayed Elsayed , Faiza Al-rakhami

https://doi.org/10.54286/ikjm.1131769

Abstract

References

  • 1.R. Abo-Zeid and H. Kamal, Global behavior of two rational third order difference equations , Universal J. Math. Appl., (2019), 212-217.
  • 2.R. P. Agarwal and E. M. Elsayed, Periodicity and stability of solutions of higher order rational difference equation, Advanced Studies in Contemp. Math., 2008, 181-201.
  • 3.H. S. Alayachi, M. S. M. Noorani, A. Q. Khan and M. B. Almatrafi, Analytic solutions and stability of sixth order difference equations, Math. Prob. Eng., 2020 (2020), 1-12.
  • 4.H. S. Alayachi, A. Q. Khan, M. S. M. Noorani, and A. Khaliq, Displaying the structure of the solutions for some fifth-order systems of recursive equations, Math. Prob. Eng., 2021 (2021), 1-14.
  • 5.H. S. Alayachi, A. Q. Khan, and M. S. M. Noorani, On the solutions of three-dimensional rational difference equation systems, J. Math., 2021 (2021), 1- 15.
  • 6.M. B. Almatrafi , E. M. Elsayed and F. Alzahrani, Qualitative behavior of a quadratic second-order rational difference equation, Int. J. Adv. Math, 2019 (1), 1-14.
  • 7.A. Alshareef, F. Alzahrani, and A. Q. Khan, Dynamics and solutions' expressions of a higher-order nonlinear fractional recursive sequence, Math. Prob. Eng., 2021 (2021), 1 - 12.
  • 8.M. M. Alzubaidi and E. M. Elsayed, Analysis of qualitative behavior of fifth order difference equations, MathLAB J., 2019 (2) (1), 1-18.
  • 9.A. Asiri and E. M. Elsayed, Dynamics and solutions of some recursive sequences of a higher-order, J. Comp. Anal. Appl. 27 (4) (2019), 656-670.
  • 10.F. Belhannache, On the stability of a system of difference equations, Elec. J. Math. Anal. and Appl., 8(2) (2020), 109-114.
  • 11.E. Camouzis and G. Ladas. Dynamics of third-order rational difference equations with open problems and conjectures, 5. CRC Press, (2007).
  • 12.C. Çinar, On the positive solutions of the difference equation x_{n+1}=((ax_{n-1})/(1+bx_{n}x_{n-1})), Appl. Math. Comput., 156 (2) (2004), 587-590.
  • 13.Q. Din, Global stability and Neimark-Sacker bifurcation of a host parasitoid model, Int. J. Syst. Sci., 48 (6) (2017), 1194-1202.
  • 14.E. M. Elabbasy, H. El-Metwally, and E. M. Elsayed, On the difference equation x_{n+1}=ax_{n-}((bx_{n})/(cx_{n}-dx_{n-1})), Adv. Differ. Equations, 2006 ( June) (2006), 1-10.
  • 15.E. M. Elabbasy, and A. El-Biaty, Asymptotic behavior of some rational difference equations, Int. J. Comp. Appl., 136 (8) (2016), 18-24.
  • 16.S. Elaydi, An introduction to difference equations, Springer New York, NY,USA, (2005).
  • 17.H. A. El-Metwally, On the qualitative study of some difference equations, IOSR J. Math., 16 (2020), 48-54.
  • 18.E. M. Elsayed, Solution and attractivity for a rational recursive sequence, Dis. Dyn. Nat. Soc., 2011(2011), 1- 17.
  • 19.E. M. Elsayed, Solution and dynamics of a fourth rational difference equation, Int. J. Phy. Sci., 7(48)(2012), 6191-6202.
  • 20.E. M. Elsayed, Expression and behavior of the solutions of some rational recursive sequences, Math. Meth. Appl. Sci.(39) (2016), 5682-5694.
  • 21.E. M. Elsayed and T. F. Ibrahim, Solutions and periodicity of a rational recursive sequences of order five, Bull. Malays. Math. Sci. Soc. 38(1)(2015), 95-112.
  • 22.E. M. Elsayed, and A. M. Ahmed, Dynamics of a three-dimensional systems of rational difference equations, Math. Meth. Appl. Sci., 39 (5) (2016), 1026-1038.
  • 23.M. Folly-Gbetoula, N. Mnguni and A. H. Kara, A group theory approach towards some rational difference equations, J. Math., 2019 (2019), 1-9.
  • 24.M. Gümüş and R. Abo-Zeid, Qualitative study of a third order rational system of difference equations, Math. Moravica, 25(1) (2021), 81-97.
  • 25.T. F. Ibrahim, On the third order rational difference equation x_{n+1}=((x_{n}x_{n-2})/(x_{n-1}(a+bx_{n}x_{n-2}))), Int. J. Contemp. Math. Sci., 4 (27) (2009), 1321-1334.
  • 26.T. F. Ibrahim, A. Q. Khan and A. Ibrahim, Qualitative behavior of a nonlinear generalized recursive sequence with delay, Math. Prob. Eng., 2021 (2021),1-8.
  • 27.M. Kara, Y. Yazliky. Halim, On a solvable system of non-linear difference equations with variable coefficients, J. Scie. Arts, 1 (54) (2021), 145-162.
  • 28.R. Karatas, C. Cinar, D. Simsek, On positive solutions of the difference equationx_{n+1}=((x_{n-5})/(1+bx_{n-2}x_{n-5})), Int. J. Contemp. Math. Sci., 1 (10) (2006), 494-500.
  • 29.A. Khaliq and E. M. Elsayed, Global behavior and periodicities of some fractional recursive sequences, Proceedings of the Jangjeon Math. Soci.,20 (3) (2017), 421 - 441.
  • 30.A. Khaliq, H. S. Alayachi, M. S. M. Noorani, and A. Q. Khan, On stability analysis of higher-order rational difference equation, Discret. Dyn. Nat. Soc., 2020 (2020).
  • 31.A. Q. Khan, M. S. M. Noorani and H. S. Alayachi, Global dynamics of higher-order exponential systems of difference equations, Discret. Dyn. Nat. Soc., 2019 (2019), 1-19.
  • 32.M. R. S. Kulenovic and G. Ladas, Dynamics of second order rational difference equations: with open problems and conjectures. Chapman and Hall/CRC, (2001).
  • 33.A. S. Kurbanli, C. Çinar, and I. Yalçinkaya, On the behavior of positive solutions of the system of rational difference equations x_{n+1}=((x_{n-1})/(x_{n-1}y_{n}+1)), y_{n+1}=((y_{n-1})/(y_{n-1}x_{n}+1)), Math. Comput. Model., 53, (5--6) (2011), 1261-1267.
  • 34.R. E. Mickens, Difference equations: theory, applications and advanced topics, CRC Press, 2015.
  • 35.A. A. Muna and S. Mohammad, Dynamics of a higher order rational difference equation x_{n+1}=(((α+βx_{n}))/((A+Bx_{n}+Cx_{n-k}))), J. Nonlinear Anal. Appl, 8 (2) (2017), 363-379.
  • 36.B. Oğul, D. Şimşek, H. Öğünmez and A. S. Kurbanlı, Dynamical behavior of rational difference equation x_{n+1}=((x_{n-17})/(±1±x_{n-2}x_{n-5}x_{n-8}x_{n-11}x_{n-14}x_{n-17})), Bol. Soc. Mat. Mex., 27(49)(2021).
  • 37.D. Simşek, B. Oğul, and T. F. Ibrahim, Solution of the rational difference equation , Dyn. Cont., Discrete and Imp. Sys., 33 (5) (2021), 125-141.
  • 38.D. T. Tollu, I. Yalcinkaya, H. Ahmad, and S. W. Yao, A detailed study on a solvable system related to the linear fractional difference equation, Math. Biosci. Eng.,18 (5) (2021), 5392-5408.
  • 39.I. Yalçinkaya and C. Cinar, Global asymptotic stability of a system of two nonlinear difference equations, Fasciculi Math., 43 (2010), 171-180.
  • 40.E. M. E. Zayed, The dynamics of a new nonlinear rational difference equations, Math. Anal., 27 (2020), 153-165.
There are 40 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Elsayed Elsayed 0000-0003-0894-8472

Faiza Al-rakhami

Early Pub Date February 27, 2023
Publication Date August 22, 2023
Acceptance Date December 23, 2022
Published in Issue Year 2023 Volume: 5 Issue: 1

Cite

APA Elsayed, E., & Al-rakhami, F. (2023). On Dynamics and Solutions Expressions of Higher-Order Rational Difference Equations. Ikonion Journal of Mathematics, 5(1), 39-61. https://doi.org/10.54286/ikjm.1131769
AMA Elsayed E, Al-rakhami F. On Dynamics and Solutions Expressions of Higher-Order Rational Difference Equations. ikjm. August 2023;5(1):39-61. doi:10.54286/ikjm.1131769
Chicago Elsayed, Elsayed, and Faiza Al-rakhami. “On Dynamics and Solutions Expressions of Higher-Order Rational Difference Equations”. Ikonion Journal of Mathematics 5, no. 1 (August 2023): 39-61. https://doi.org/10.54286/ikjm.1131769.
EndNote Elsayed E, Al-rakhami F (August 1, 2023) On Dynamics and Solutions Expressions of Higher-Order Rational Difference Equations. Ikonion Journal of Mathematics 5 1 39–61.
IEEE E. Elsayed and F. Al-rakhami, “On Dynamics and Solutions Expressions of Higher-Order Rational Difference Equations”, ikjm, vol. 5, no. 1, pp. 39–61, 2023, doi: 10.54286/ikjm.1131769.
ISNAD Elsayed, Elsayed - Al-rakhami, Faiza. “On Dynamics and Solutions Expressions of Higher-Order Rational Difference Equations”. Ikonion Journal of Mathematics 5/1 (August 2023), 39-61. https://doi.org/10.54286/ikjm.1131769.
JAMA Elsayed E, Al-rakhami F. On Dynamics and Solutions Expressions of Higher-Order Rational Difference Equations. ikjm. 2023;5:39–61.
MLA Elsayed, Elsayed and Faiza Al-rakhami. “On Dynamics and Solutions Expressions of Higher-Order Rational Difference Equations”. Ikonion Journal of Mathematics, vol. 5, no. 1, 2023, pp. 39-61, doi:10.54286/ikjm.1131769.
Vancouver Elsayed E, Al-rakhami F. On Dynamics and Solutions Expressions of Higher-Order Rational Difference Equations. ikjm. 2023;5(1):39-61.
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