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

Year 2021, Volume: 4 Issue: 1, 25 - 37, 30.04.2021

Batool Raddad Mohammad Saleh

https://doi.org/10.33187/jmsm.843626

Cited By: 1

Abstract

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.

Keywords

Fixed points, Stability, Bifurcation

References

• [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.