-Central question: Can we reconstruct the underlying topological space (chaotic attractor) out of our data? Apart of topological features, there may be useful geometrical properties like fractal dimension that characterize the strange attractor.
-Using differential mapping instead of time delay mappings to represent the data may reveal topological properties not captured by the time delay mappings. Specially, we want to make sure, that the topological structure (e.g. the spiral arms) does not depend on the embedding (representation of the dynamics) chosen. (c.f. Appendix B on Embeddings from the book 'The Topology Of Chaos')
-The flow dynamics of our attractor is unknown.
-How do we know that we are dealing with Torus attractors? If so, the system would be quasi-periodic and not chaotic(?)
-The attractors are surprisingly symmetric. Analyzing more complex sounds by verifying that the symmetric structures do not show up anymore may be useful to discard that the symmetric shapes are just artefacts.
-The winding number and rotation rate of the spirals may be potentially interesting topological quanties, that may turn out to be invariant under certain conditions.
- Intuitively, by increasing the embedding dimension the spiral arms get stretched and disentangle (?).Â