Global stability for the 2-dimensional logistic map

Abstract

For the delayed logistic equation $x_{n+1} = a x_n (a-x_{n-1})$ it is well known that the nontrivial fixed point is locally stable for $1<a\leq 2$, and unstable for $a>2$. We prove that for $1<a\leq 2$ the fixed point is globally stable, in the sense that it is locally stable and attracts all points of $S$, where $S$ contains those $(x_0,x_1)\in \mathbb{R}_+^2$, for which the sequence $\left\lbrace x_n\right\rbrace \subset \mathbb{R}_+$. The proof is a combination of analytical and reliable numerical methods.