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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
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.

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
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.