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

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