2007/05/23 by Nicola Romanazzi, Marc Lefranc, Robert Gilmore
Mathematics · Physics and Astronomy · #Chaos control and synchronization #Mathematical Dynamics and Fractals #Quantum chaos and dynamical systems #nlin.CD
paper · pdf · doi:10.1103/physreve.75.066214
published as Phys. Rev. E 75, 066214 (2007)
arxiv created 2007/05/23 · openalex publication_date 2007/06/25 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 roulé, an S -template branched manifold, etc., are not artifacts of the embedding chosen for the analysis.