On the invariant manifolds of the fixed point of a second order nonlinear difference equation

This paper addresses the asymptotic approximations of the stable and unstable manifolds for the saddle fixed point and the 2-periodic solutions of the difference equation $x_{n+1} = \alpha + \beta x_{n-1}+x_{n-1}/x_{n},$ where $\alpha>0,$ $0\leqslant \beta <1$ and the initial conditions $x_{-1}$ and $x_0$ are positive numbers. These manifolds determine completely global dynamics of this equation. The theoretical results are supported by some numerical examples.