Araştırma Makalesi
BibTex RIS Kaynak Göster
Yıl 2019, Cilt: 2 Sayı: 3, 176 - 182, 26.12.2019
https://doi.org/10.33187/jmsm.560049

Öz

Kaynakça

  • [1] M. B. Almatrafi, E. M. Elsayed, F. Alzahrani, Qualitative behavior of two rational difference equations, Fundam. J. Math. Appl., 1(2) (2018), 194-204.
  • [2] C. Cinar, On the positive solutions of the difference equation $x_{n+1}=ax_{n-1}/(1+bx_{n}x_{n-1})$, Appl. Math. Comput., 156 (2004), 587-590.
  • [3] E. M. Elabbasy, H. El-Metwally, E. M. Elsayed, On the difference equation $x_{n+1}=ax_{n}-(bx_{n})/(cx_{n}-dx_{n-1})$, Adv. Difference Equ., 2006 (2006), Article ID 82579, 1-10.
  • [4] M. Garic-Demirovic, M. Nurkanovic, Z. Nurkanovic, Stability, periodicity and Neimark-Sacker bifurcation of certain homogeneous fractional difference equations, Int. J. Difference Equ., 12(1) (2017), 27-53.
  • [5] M. Ghazel, E.M. Elsayed, A. E. Matouk, A. M. Mousallam, 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 (2017), 4662–4679.
  • [6] T. Khyat, M. R. S. Kulenovic, The invariant curve caused by Neimark-Sacker bifurcation of a perturbed Beverton-Holt difference equation, Int. J. Difference Equ., 12(2) (2017), 267-280.
  • [7] M. Saleh, N. Alkoumi, Aseel Farhat, 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 Fractals, 96(2017), 76–84.
  • [8] D. Simsek, C. Cinar, I. Yalcinkaya, On the recursive sequence $x_{n+1}=\frac{x_{n-3}}{1+x_{n-1}}$, Int. J. Contemp. Math. Sci., 1(10) (2006), 475-480.
  • [9] M. B. Almatrafi, E. M. Elsayed, Solutions and formulae for some systems of difference equations, MathLAB J., 1(3) (2018), 356-369.
  • [10] M. B. Almatrafi, E. M. Elsayed, Faris Alzahrani, Qualitative behavior of a quadratic second-order rational difference equation, Int. J. Adv. Math., 2019(1) (2019), 1-14.
  • [11] F. Belhannache, N. Touafek, R. Abo-zeid, On a higher-order rational difference equation, J. Appl. Math. & Informatics, 34(5-6) (2016), 369-382.
  • [12] E. M. Elabbasy, H. El-Metawally, E. M. Elsayed, On the difference equation $x_{n+1}=(ax_{n}^{2}+bx_{n-1}x_{n-k})/(cx_{n}^{2}+dx_{n-1}x_{n-k})$, Sarajevo J. Math., 4(17) (2008), 1-10.
  • [13] M.A. El-Moneam, E.M.E. Zayed, Dynamics of the rational difference equation, Inform. Sci. Letters, 3(2) (2014), 45-53.
  • [14] A. Khaliq, Sk.S. Hassan, Dynamics of a rational difference equation $x_{n+1}=ax_{n}+(\alpha+\beta x_{n-k})/(A+Bx_{n-k})$, Int. J. Adv. Math., 2018(1) (2018), 159-179.
  • [15] V. L. Kocic, G. Ladas, Global Behaviour of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic Publishers, Dordrecht, 1993.
  • [16] Y. Kostrov, Z. Kudlak, On a second-order rational difference equation with a quadratic term, Int. J. Difference Equ., 11(2) (2016), 179-202.
  • [17] K. Liu, P. Li, F. Han, W. Zhong, Global dynamics of nonlinear difference equation $x_{n+1}=A+x_{n}/x_{n-1}x_{n-2}$, J. Comput. Anal. Appl., 24(6) (2018), 1125-1132.
  • [18] S. Moranjkic, Z. Nurkanovic, Local and global dynamics of certain second-order rational difference equations containing quadratic terms, Adv. Dyn. Syst. Appl., 12(2) (2017), 123-157.
  • [19] M. Saleh, M. Aloqeili, On the rational difference equation $x_{n+1}=\frac{ax_{n}+bx_{n-k}}{A+Bx_{n-k}}$, Appl. Math. Comput. 171(1) (2005), 862-869.
  • [20] E. M. Elsayed, Dynamics of recursive sequence of order two, Kyungpook Math. J., 50 (2010), 483-497.

Analysis of the Convergence and Periodicity of a Rational Difference Equation

Yıl 2019, Cilt: 2 Sayı: 3, 176 - 182, 26.12.2019
https://doi.org/10.33187/jmsm.560049

Öz

The exact solutions of most difference equations cannot be obtained sometimes. This can be attributed to the fact that there is no a specific approach from which one can find the exact solution. Therefore, many researchers tend to study the qualitative behaviours of these equations.  In this paper, we will investigate some qualitative properties such as local stability, global stability, periodicity and solutions of the following eighth order recursive equation \begin{eqnarray*} x_{n+1}=c_{1}x_{n-3}-\frac{c_{2}x_{n-3}}{c_{3} x_{n-3}- c_{4} x_{n-7}},\;\;\;n=0,1,..., \end{eqnarray*} {\Large \noindent }where the coefficients $c_{i},\ \textit{for all} \ i=1,...,4,$ are assumed to be positive real numbers and the initial conditions $x_{i} \ \textit{ for all} \ i=-7,-6,...,0, $ are arbitrary non-zero real numbers.

Kaynakça

  • [1] M. B. Almatrafi, E. M. Elsayed, F. Alzahrani, Qualitative behavior of two rational difference equations, Fundam. J. Math. Appl., 1(2) (2018), 194-204.
  • [2] C. Cinar, On the positive solutions of the difference equation $x_{n+1}=ax_{n-1}/(1+bx_{n}x_{n-1})$, Appl. Math. Comput., 156 (2004), 587-590.
  • [3] E. M. Elabbasy, H. El-Metwally, E. M. Elsayed, On the difference equation $x_{n+1}=ax_{n}-(bx_{n})/(cx_{n}-dx_{n-1})$, Adv. Difference Equ., 2006 (2006), Article ID 82579, 1-10.
  • [4] M. Garic-Demirovic, M. Nurkanovic, Z. Nurkanovic, Stability, periodicity and Neimark-Sacker bifurcation of certain homogeneous fractional difference equations, Int. J. Difference Equ., 12(1) (2017), 27-53.
  • [5] M. Ghazel, E.M. Elsayed, A. E. Matouk, A. M. Mousallam, 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 (2017), 4662–4679.
  • [6] T. Khyat, M. R. S. Kulenovic, The invariant curve caused by Neimark-Sacker bifurcation of a perturbed Beverton-Holt difference equation, Int. J. Difference Equ., 12(2) (2017), 267-280.
  • [7] M. Saleh, N. Alkoumi, Aseel Farhat, 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 Fractals, 96(2017), 76–84.
  • [8] D. Simsek, C. Cinar, I. Yalcinkaya, On the recursive sequence $x_{n+1}=\frac{x_{n-3}}{1+x_{n-1}}$, Int. J. Contemp. Math. Sci., 1(10) (2006), 475-480.
  • [9] M. B. Almatrafi, E. M. Elsayed, Solutions and formulae for some systems of difference equations, MathLAB J., 1(3) (2018), 356-369.
  • [10] M. B. Almatrafi, E. M. Elsayed, Faris Alzahrani, Qualitative behavior of a quadratic second-order rational difference equation, Int. J. Adv. Math., 2019(1) (2019), 1-14.
  • [11] F. Belhannache, N. Touafek, R. Abo-zeid, On a higher-order rational difference equation, J. Appl. Math. & Informatics, 34(5-6) (2016), 369-382.
  • [12] E. M. Elabbasy, H. El-Metawally, E. M. Elsayed, On the difference equation $x_{n+1}=(ax_{n}^{2}+bx_{n-1}x_{n-k})/(cx_{n}^{2}+dx_{n-1}x_{n-k})$, Sarajevo J. Math., 4(17) (2008), 1-10.
  • [13] M.A. El-Moneam, E.M.E. Zayed, Dynamics of the rational difference equation, Inform. Sci. Letters, 3(2) (2014), 45-53.
  • [14] A. Khaliq, Sk.S. Hassan, Dynamics of a rational difference equation $x_{n+1}=ax_{n}+(\alpha+\beta x_{n-k})/(A+Bx_{n-k})$, Int. J. Adv. Math., 2018(1) (2018), 159-179.
  • [15] V. L. Kocic, G. Ladas, Global Behaviour of Nonlinear Difference Equations of Higher Order with Applications, Kluwer Academic Publishers, Dordrecht, 1993.
  • [16] Y. Kostrov, Z. Kudlak, On a second-order rational difference equation with a quadratic term, Int. J. Difference Equ., 11(2) (2016), 179-202.
  • [17] K. Liu, P. Li, F. Han, W. Zhong, Global dynamics of nonlinear difference equation $x_{n+1}=A+x_{n}/x_{n-1}x_{n-2}$, J. Comput. Anal. Appl., 24(6) (2018), 1125-1132.
  • [18] S. Moranjkic, Z. Nurkanovic, Local and global dynamics of certain second-order rational difference equations containing quadratic terms, Adv. Dyn. Syst. Appl., 12(2) (2017), 123-157.
  • [19] M. Saleh, M. Aloqeili, On the rational difference equation $x_{n+1}=\frac{ax_{n}+bx_{n-k}}{A+Bx_{n-k}}$, Appl. Math. Comput. 171(1) (2005), 862-869.
  • [20] E. M. Elsayed, Dynamics of recursive sequence of order two, Kyungpook Math. J., 50 (2010), 483-497.
Toplam 20 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Makaleler
Yazarlar

Mohammed Almatrafi 0000-0002-6859-2028

Marwa Alzubaidi 0000-0002-3314-5244

Yayımlanma Tarihi 26 Aralık 2019
Gönderilme Tarihi 2 Mayıs 2019
Kabul Tarihi 28 Temmuz 2019
Yayımlandığı Sayı Yıl 2019 Cilt: 2 Sayı: 3

Kaynak Göster

APA Almatrafi, M., & Alzubaidi, M. (2019). Analysis of the Convergence and Periodicity of a Rational Difference Equation. Journal of Mathematical Sciences and Modelling, 2(3), 176-182. https://doi.org/10.33187/jmsm.560049
AMA Almatrafi M, Alzubaidi M. Analysis of the Convergence and Periodicity of a Rational Difference Equation. Journal of Mathematical Sciences and Modelling. Aralık 2019;2(3):176-182. doi:10.33187/jmsm.560049
Chicago Almatrafi, Mohammed, ve Marwa Alzubaidi. “Analysis of the Convergence and Periodicity of a Rational Difference Equation”. Journal of Mathematical Sciences and Modelling 2, sy. 3 (Aralık 2019): 176-82. https://doi.org/10.33187/jmsm.560049.
EndNote Almatrafi M, Alzubaidi M (01 Aralık 2019) Analysis of the Convergence and Periodicity of a Rational Difference Equation. Journal of Mathematical Sciences and Modelling 2 3 176–182.
IEEE M. Almatrafi ve M. Alzubaidi, “Analysis of the Convergence and Periodicity of a Rational Difference Equation”, Journal of Mathematical Sciences and Modelling, c. 2, sy. 3, ss. 176–182, 2019, doi: 10.33187/jmsm.560049.
ISNAD Almatrafi, Mohammed - Alzubaidi, Marwa. “Analysis of the Convergence and Periodicity of a Rational Difference Equation”. Journal of Mathematical Sciences and Modelling 2/3 (Aralık 2019), 176-182. https://doi.org/10.33187/jmsm.560049.
JAMA Almatrafi M, Alzubaidi M. Analysis of the Convergence and Periodicity of a Rational Difference Equation. Journal of Mathematical Sciences and Modelling. 2019;2:176–182.
MLA Almatrafi, Mohammed ve Marwa Alzubaidi. “Analysis of the Convergence and Periodicity of a Rational Difference Equation”. Journal of Mathematical Sciences and Modelling, c. 2, sy. 3, 2019, ss. 176-82, doi:10.33187/jmsm.560049.
Vancouver Almatrafi M, Alzubaidi M. Analysis of the Convergence and Periodicity of a Rational Difference Equation. Journal of Mathematical Sciences and Modelling. 2019;2(3):176-82.

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