Araştırma Makalesi
BibTex RIS Kaynak Göster
Yıl 2021, Cilt: 4 Sayı: 1, 25 - 37, 30.04.2021
https://doi.org/10.33187/jmsm.843626

Öz

Kaynakça

  • [1] G.-M. Tang, L.-X. Hu, G. Ma, Global Stability of a Rational Difference Equation, Discrete Dynamics in Nature and Society, Volume 2010.
  • [2] E. Camouzis, G. Ladas, Dynamics of Third-Order Rational Difference Equations With Open Problems And Conjectures, Chapman. Hall/CRC, Boca Raton, 2008.
  • [3] M. Saleh, N. Alkoumi, A. Farhat, On the dynamics of a rational difference equation $ x_{n+1}=\frac{ \alpha +\beta x_{n}+\gamma x_{n-k}}{Bx_{n}+Cx_{n-k}}$, Chaos Soliton, (2017), 76-84.
  • [4] M. Saleh, A. Farhat, Global asymptotic stability of the higher order equation $x_{n+1} = \frac{ ax_{n}+bx_{n-k}}{A+Bx_{n-k}}$, J. Appl. Math. Comput, (2017), 135-148, doi: 10.1007/s12190-016-1029-4.
  • [5] M. Saleh, A.Asad, Dynamics of kth order rational difference equation, J. Appl. Nonlinear Dyn., (2021), 125-149, doi: 10.5890/JAND.2021.03.008.
  • [6] M. Saleh, S.Hirzallah, Dynamics and bifurcation of a second order rational difference equation with quadratic terms, J. Appl. Nonlinear Dyn., (to appear).
  • [7] M. Saleh, S.Hirzallah, Dynamics and bifurcation of a second order quadratic rational difference equation, J. Math. Sci. Model., 3(3) (2020), 102-119.
  • [8] S. Elaydi, An Introduction to Difference Equations, 3rd Edition, Springer, 2000.
  • [9] A. Y. Kuznetsov, Elements of Applied Bifurcation Theory, 2nd Ed., Springer-Verlag, 1998.

Dynamics and Bifurcation of $x_{n+1}=\frac{\alpha+\beta x_{n-2}}{A+Bx_{n}+C x_{n-2}}$

Yıl 2021, Cilt: 4 Sayı: 1, 25 - 37, 30.04.2021
https://doi.org/10.33187/jmsm.843626

Öz

In this paper, we study dynamics and bifurcation of the third order rational difference equation \begin{eqnarray*} x_{n+1}=\frac{\alpha+\beta x_{n-2}}{A+Bx_{n}+Cx_{n-2}}, ~~n=0, 1, 2, \ldots \end{eqnarray*} with positive parameters $\alpha, \beta, A, B, C$ and non-negative initial conditions $\{x_{-k}, x_{-k+1}, \ldots, x_{0}\}$. We study the dynamic behavior, the sufficient conditions for the existence of the Neimark-Sacker bifurcation, and the direction of the Neimark-Sacker bifurcation. Then, we give numerical examples with figures to support our results.

Kaynakça

  • [1] G.-M. Tang, L.-X. Hu, G. Ma, Global Stability of a Rational Difference Equation, Discrete Dynamics in Nature and Society, Volume 2010.
  • [2] E. Camouzis, G. Ladas, Dynamics of Third-Order Rational Difference Equations With Open Problems And Conjectures, Chapman. Hall/CRC, Boca Raton, 2008.
  • [3] M. Saleh, N. Alkoumi, A. Farhat, On the dynamics of a rational difference equation $ x_{n+1}=\frac{ \alpha +\beta x_{n}+\gamma x_{n-k}}{Bx_{n}+Cx_{n-k}}$, Chaos Soliton, (2017), 76-84.
  • [4] M. Saleh, A. Farhat, Global asymptotic stability of the higher order equation $x_{n+1} = \frac{ ax_{n}+bx_{n-k}}{A+Bx_{n-k}}$, J. Appl. Math. Comput, (2017), 135-148, doi: 10.1007/s12190-016-1029-4.
  • [5] M. Saleh, A.Asad, Dynamics of kth order rational difference equation, J. Appl. Nonlinear Dyn., (2021), 125-149, doi: 10.5890/JAND.2021.03.008.
  • [6] M. Saleh, S.Hirzallah, Dynamics and bifurcation of a second order rational difference equation with quadratic terms, J. Appl. Nonlinear Dyn., (to appear).
  • [7] M. Saleh, S.Hirzallah, Dynamics and bifurcation of a second order quadratic rational difference equation, J. Math. Sci. Model., 3(3) (2020), 102-119.
  • [8] S. Elaydi, An Introduction to Difference Equations, 3rd Edition, Springer, 2000.
  • [9] A. Y. Kuznetsov, Elements of Applied Bifurcation Theory, 2nd Ed., Springer-Verlag, 1998.
Toplam 9 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Matematik
Bölüm Makaleler
Yazarlar

Batool Raddad Bu kişi benim

Mohammad Saleh

Yayımlanma Tarihi 30 Nisan 2021
Gönderilme Tarihi 19 Aralık 2020
Kabul Tarihi 19 Nisan 2021
Yayımlandığı Sayı Yıl 2021 Cilt: 4 Sayı: 1

Kaynak Göster

APA Raddad, B., & Saleh, M. (2021). Dynamics and Bifurcation of $x_{n+1}=\frac{\alpha+\beta x_{n-2}}{A+Bx_{n}+C x_{n-2}}$. Journal of Mathematical Sciences and Modelling, 4(1), 25-37. https://doi.org/10.33187/jmsm.843626
AMA Raddad B, Saleh M. Dynamics and Bifurcation of $x_{n+1}=\frac{\alpha+\beta x_{n-2}}{A+Bx_{n}+C x_{n-2}}$. Journal of Mathematical Sciences and Modelling. Nisan 2021;4(1):25-37. doi:10.33187/jmsm.843626
Chicago Raddad, Batool, ve Mohammad Saleh. “Dynamics and Bifurcation of $x_{n+1}=\frac{\alpha+\beta x_{n-2}}{A+Bx_{n}+C x_{n-2}}$”. Journal of Mathematical Sciences and Modelling 4, sy. 1 (Nisan 2021): 25-37. https://doi.org/10.33187/jmsm.843626.
EndNote Raddad B, Saleh M (01 Nisan 2021) Dynamics and Bifurcation of $x_{n+1}=\frac{\alpha+\beta x_{n-2}}{A+Bx_{n}+C x_{n-2}}$. Journal of Mathematical Sciences and Modelling 4 1 25–37.
IEEE B. Raddad ve M. Saleh, “Dynamics and Bifurcation of $x_{n+1}=\frac{\alpha+\beta x_{n-2}}{A+Bx_{n}+C x_{n-2}}$”, Journal of Mathematical Sciences and Modelling, c. 4, sy. 1, ss. 25–37, 2021, doi: 10.33187/jmsm.843626.
ISNAD Raddad, Batool - Saleh, Mohammad. “Dynamics and Bifurcation of $x_{n+1}=\frac{\alpha+\beta x_{n-2}}{A+Bx_{n}+C x_{n-2}}$”. Journal of Mathematical Sciences and Modelling 4/1 (Nisan 2021), 25-37. https://doi.org/10.33187/jmsm.843626.
JAMA Raddad B, Saleh M. Dynamics and Bifurcation of $x_{n+1}=\frac{\alpha+\beta x_{n-2}}{A+Bx_{n}+C x_{n-2}}$. Journal of Mathematical Sciences and Modelling. 2021;4:25–37.
MLA Raddad, Batool ve Mohammad Saleh. “Dynamics and Bifurcation of $x_{n+1}=\frac{\alpha+\beta x_{n-2}}{A+Bx_{n}+C x_{n-2}}$”. Journal of Mathematical Sciences and Modelling, c. 4, sy. 1, 2021, ss. 25-37, doi:10.33187/jmsm.843626.
Vancouver Raddad B, Saleh M. Dynamics and Bifurcation of $x_{n+1}=\frac{\alpha+\beta x_{n-2}}{A+Bx_{n}+C x_{n-2}}$. Journal of Mathematical Sciences and Modelling. 2021;4(1):25-37.

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