Exploring the Topology of Dynamical Reconstructions

Abstract: Computing the state-space topology of a dynamical system from scalar data requires accurate reconstruction of those dynamics and construction of an appropriate simplicial complex from the results. The reconstruction process involves a number of free parameters and the computation of homology for a large number of simplices can be expensive. This paper is a study of how to avoid a full (diffeomorphic) reconstruction and how to decrease the computational burden. Using trajectories from the classic Lorenz system, we reconstruct the dynamics using the method of delays, then build a parsimonious simplicial complex---the "witness complex"---to compute its homology. Surprisingly, we find that the witness complex correctly resolves the homology of the underlying invariant set from noisy samples of that set even if the reconstruction dimension is well below the thresholds specified in the embedding theorems for assuring topological conjugacy between the true and reconstructed dynamics. We conjecture that this is because the requirements for reconstructing homology, are less stringent than those in the embedding theorems. In particular, to faithfully reconstruct the homology, a homeomorphism is sufficient---as opposed to a diffeomorphism, as is necessary for the full dynamics. We provide preliminary evidence that a homeomorphism, in the form of a delay-coordinate reconstruction map, may manifest at a lower dimension than that required to achieve an embedding.