Topological Signatures of Classical Chaos

Classical chaos, characterized by sensitive dependence on initial conditions and unpredictable long-term behavior, is a ubiquitous phenomenon across various scientific disciplines. While traditional metrics like Lyapunov exponents and fractal dimensions offer valuable insights, they often fall short in capturing the intrinsic geometric and structural properties of chaotic attractors. This paper explores the emerging field of topological data analysis (TDA) as a powerful framework to uncover and characterize topological signatures embedded within classical chaotic systems. We delve into the theoretical underpinnings of persistent homology, specifically focusing on its application to phase space reconstructions from time series data. By constructing simplicial complexes from point cloud representations of chaotic attractors, persistent homology allows for the multi-scale quantification of topological features such as connected components, loops, and voids. We discuss how these features, represented by persistence diagrams and barcodes, can serve as robust invariants to distinguish between different chaotic regimes, identify bifurcations, and provide a deeper understanding of the "shape" of chaos. The paper reviews relevant literature, outlines a methodological approach for applying TDA to common chaotic systems, and discusses potential implications for classifying and understanding complex dynamics.