Global stability for the three-dimensional logistic map

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