Romanazzi Nicola, Lefranc Marc, Gilmore Robert
Physics Department, Drexel University, Philadelphia, Pennsylvania 19104, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2007 Jun;75(6 Pt 2):066214. doi: 10.1103/PhysRevE.75.066214. Epub 2007 Jun 25.
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.
当一个低维混沌吸引子嵌入三维空间时，其拓扑性质取决于嵌入方式。我们证明，仅存在三种依赖于嵌入的拓扑性质：奇偶性、整体挠率和纽结类型。我们讨论了它们如何随嵌入方式而变化。最后，我们表明导致混沌行为的机制是所有嵌入方式的一个不变量。这些结果仅适用于亏格为一的混沌吸引子，这涵盖了大多数对实验数据进行拓扑分析的情况。这意味着从先前分析得出的结论，例如产生混沌行为的机制是斯梅尔马蹄机制、反向马蹄、卷蛋糕、S -模板分支流形等，并非所选分析嵌入方式的人为产物。
