Exploring the topology of dynamical reconstructions

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 compute the homology efficiently and effectively without a full (diffeomorphic) reconstruction. Using trajectories from the classic Lorenz system, we reconstruct the dynamics using the method of delays, then build a simplicial complex whose vertices are a small subset of the data: the “witness complex”. 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 for assuring topological conjugacy between the true and reconstructed dynamics that are specified in the embedding theorems. We conjecture that this is because the requirements for reconstructing homology are less stringent: 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 exist at a lower dimension than that required to achieve an embedding.