Global stability for the three-dimensional logistic map

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 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 thoseThe 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 3-dimensional logistic map by using center manifold techniques and the Neimark-Sacker bifurcational normal form.

An investigation on the Lasota-Wazewska model with a piecewise constant argument

“…Besides, global attractivity and stability are also among the most popular subjects in the study of both difference equations and equations with piecewise constant arguments [15,17,18,21,24,27,35,41,[45][46][47][48][49]59,60]. Another current issue used in the study of difference equations is the semi-cycle analysis, which was initiated by [28] and then attracted great attention in different fields of the difference equations [1,20,[36][37][38][39][40]57].…”

Global stability for the delayed logistic map

“…where a > 0 and d ∈ N. The thesis is based on papers [4,7] of the author. Chapter 1 is the introduction where we describe the problem and also outline the main parts of the dissertation.…”

“…In Chapter 3, based on the article [7] we extend the method of [4,5,6] from twodimensional to higher-dimensional maps. As the delayed logistic dierence equation is interesting in its own right for d = 2, we demonstrate on the three-dimensional nonlinear map (1.2) with d = 2 how the extension goes to higher dimension.…”