Naimark-Sacker bifurcation of a certain second order quadratic fractional difference equation

Abstract

We investigate the Naimark-Sacker Bifurcation of the equilibrium of some special cases of the difference equation $x_{n+1}=\frac{\beta x_n x_{n-1}+ \gamma x_{n-1}^2 +\delta x_n}{ B x_n x_{n-1}+C x_{n-1}^2 +D x_n}$ where the parameters $\beta, \gamma,\delta, B, C, D$ are nonnegative numbers which satisfy $B+C+D>0$ and the initial conditions $x_{-1}$ and $x_0$ are arbitrary nonnegative numbers such that $B x_n x_{n-1}+C x_{n-1}^2 +D x_n >0$ for all $n \geq 0$.