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

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