Global Topology from an Embedding

TL;DR

This paper investigates the topological differences between original chaotic attractors and their reconstructions via embeddings, extending previous results to attractors of any genus and revealing symmetries in Lorenz attractors.

Contribution

It generalizes earlier findings by demonstrating topological discrepancies for 3D attractors of any genus, enhancing understanding of attractor symmetries and embeddings.

Findings

Reconstructed attractors may not be topologically equivalent to original attractors.

The results apply to 3D attractors of arbitrary genus.

Symmetries in Lorenz attractors are characterized through this topological analysis.

Abstract

An embedding of chaotic data into a suitable phase space creates a diffeomorphism of the original attractor with the reconstructed attractor. Although diffeomorphic, the original and reconstructed attractors may not be topologically equivalent. In a previous work we showed how the original and reconstructed attractors can differ when the original is three-dimensional and of genus-one type. In the present work we extend this result to three-dimensional attractors of arbitrary genus. This result describes symmetries exhibited by the Lorenz attractor and its reconstructions.