Infinite towers in the graph of a dynamical system

TL;DR

This paper explores the complex structure of dynamical systems by analyzing graphs of chain-recurrent sets, highlighting systems like the logistic map that exhibit infinitely many coexisting states, and comparing their bifurcation behaviors.

Contribution

It introduces the concept of infinite towers of chain-recurrent sets in dynamical systems and compares different systems to illustrate this phenomenon.

Findings

Logistic map has infinitely many disjoint chain-recurrent nodes.

Lorenz system and logistic map show similar bifurcation diagrams.

Infinite collections of chain-recurrent sets occur at specific parameter values.

Abstract

Chaotic attractors, chaotic saddles and periodic orbits are examples of chain-recurrent sets. Using arbitrary small controls, a trajectory starting from any point in a chain-recurrent set can be steered to any other in that set. The qualitative behavior of a dynamical system can be encapsulated in a graph. Its nodes are chain-recurrent sets. There is an edge from node A to node B if, using arbitrary small controls, a trajectory starting from any point of A can be steered to any point of B. We discuss physical systems that have infinitely many disjoint coexisting nodes. Such infinite collections can occur for many carefully chosen parameter values. The logistic map is such a system, as we showed in arXiv:2008.08338. To illustrate these very common phenomena, we compare the Lorenz system and the logistic map and we show how extremely similar their bifurcation diagrams are in some parameter ranges.