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.