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 = α + βx_n− 1 + x_n− 1/x_n, 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.

中文翻译：

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二阶非线性差分方程不动点的不变流形

本文地址鞍固定点和稳定的渐近近似和不稳定流形的差分方程的2周期解X _{Ñ + 1} = α + β X _{ñ - 1} + X _{ñ - 1} / X _Ñ，其中α > 0，\（0 \ leqslant \ beta <1 \）且初始条件x _{− 1}和x ₀为正数。这些流形完全确定了该方程的整体动力学。理论结果得到一些数值例子的支持。