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

TL;DR

This paper analyzes the stable and unstable invariant manifolds of a specific second-order nonlinear difference equation, providing asymptotic approximations and numerical validation to understand the global dynamics around fixed points and periodic solutions.

Contribution

It offers new asymptotic methods for approximating invariant manifolds in a nonlinear difference equation, enhancing understanding of the equation's global behavior.

Findings

Asymptotic approximations of invariant manifolds are derived.

Numerical examples support the theoretical results.

The manifolds fully determine the global dynamics of the system.

Abstract

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.