Topological Data Analysis of Phase Space Reconstructions in Chaotic Dynamical Systems

We study the topological structure of phase space reconstructions of chaotic dynamical systems through the lens of persistent homology. Starting from scalar observations, delay-coordinate embeddings produce finite point clouds whose shape is governed, by Takens-type theorems, by the topology of the underlying attractor. We give a self-contained account of persistent homology and barcodes, and we prove a Hausdorff-stability theorem for Vietoris–Rips persistence. The theorem yields explicit noise and subsampling thresholds for the reliability of topological features. We also establish a bi-Lipschitz invariance property in logarithmic scale, which explains the robustness of the signatures with respect to the choice of observable, and we prove elementary bounds for persistent entropy. We then describe a complete computational pipeline, comprising delay parameter selection, landmark subsampling, Rips filtrations and boundary matrix reduction. Finally, we formulate a numerical protocol for the Lorenz, Rössler and Hénon systems, with null models based on surrogate data and structural hypotheses to be tested. The resulting framework provides rigorous, noise-aware topological descriptors of chaotic attractors reconstructed from data.