Global stability for the three-dimensional logistic map

Abstract

Abstract For the delayed logistic equation x n+1 = ax n (1 − x n−2) it is well known that the nontrivial fixed point is locally stable for 1 < a ⩽ ( 5 + 1 ) / 2 , and unstable for }\left(\sqrt{5}+1\right)/2$?> a > ( 5 + 1 ) / 2 . We prove that for 1 < a ⩽ ( 5 + 1 ) / 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 , x 2 ) ∈ R + 3 for which the sequence ( x n ) n = 0 ∞ remains in R + . The proof is a combination of analytical and reliable numerical methods. The novelty of this article is an explicit construction of a relatively large attracting neighborhood of the nontrivial fixed point of the three-dimensional logistic map by using centre manifold techniques and the Neimark–Sacker bifurcational normal form.