Embeddings of low-dimensional strange attractors: Topological invariants and degrees of freedom

TL;DR

This paper investigates how the topological properties of low-dimensional chaotic attractors vary with embedding choices, identifying only three embedding-dependent properties and establishing an invariant mechanism for chaos creation.

Contribution

It demonstrates that only three topological properties depend on embedding and that the chaos-generating mechanism remains invariant across embeddings for genus one attractors.

Findings

Only three topological properties depend on embedding: parity, global torsion, and knot type.

The chaos-generating mechanism is invariant across all embeddings.

Previous topological analyses are not artifacts of embedding choices.

Abstract

When a low dimensional chaotic attractor is embedded in a three dimensional space its topological properties are embedding-dependent. We show that there are just three topological properties that depend on the embedding: parity, global torsion, and knot type. We discuss how they can change with the embedding. Finally, we show that the mechanism that is responsible for creating chaotic behavior is an invariant of all embeddings. These results apply only to chaotic attractors of genus one, which covers the majority of cases in which experimental data have been subjected to topological analysis. This means that the conclusions drawn from previous analyses, for example that the mechanism generating chaotic behavior is a Smale horseshoe mechanism, a reverse horseshoe, a gateau roule, an S-template branched manifold, ..., are not artifacts of the embedding chosen for the analysis.