THE GRAPH OF THE LOGISTIC MAP IS A TOWER

被引：4

作者：

De Leo, Roberto ^{[1
]}

Yorke, James A. ^{[2
,3
,4
]}

机构：

[1] Howard Univ, Dept Math, Washington, DC 20059 USA

[2] Univ Maryland, Inst Phys Sci & Technol, College Pk, MD 20742 USA

[3] Univ Maryland, Dept Math, College Pk, MD 20742 USA

[4] Univ Maryland, Dept Phys, College Pk, MD 20742 USA

来源：

DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS | 2021年 / 41卷 / 11期

基金：

美国国家科学基金会;

关键词：

chain-recurrent sets; graph of a dynamical system; towers; spectral theorem; Logistic map; DYNAMICS;

D O I：

10.3934/dcds.2021075

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

The qualitative behavior of a dynamical system can be encoded in a graph. Each node of the graph is an equivalence class of chain-recurrent points and there is an edge from node A to node B if, using arbitrary small perturbations, a trajectory starting from any point of A can be steered to any point of B. In this article we describe the graph of the logistic map. Our main result is that the graph is always a tower, namely there is an edge connecting each pair of distinct nodes. Notice that these graphs never contain cycles. If there is an edge from node A to node B, the unstable manifold of some periodic orbit in A contains points that eventually map onto B. For special parameter values, this tower has infinitely many nodes.

引用

页码：5243 / 5269

页数：27

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