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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 xn+ 1 = α + βxn− 1 + xn− 1/xn, where α > 0, $0\leqslant \beta <1$ and the initial conditions x− 1 and x0 are positive numbers. These manifolds determine completely global dynamics of this equation. The theoretical results are supported by some numerical examples.

On the Invariant Manifolds of the Fixed Point of a Second-Order Nonlinear Difference Equation