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
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
收藏
页码:5243 / 5269
页数:27
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