| Abstract: |
| Distinguishing deterministic chaos from stochastic dynamics in discrete time series remains a fundamental and challenging problem, particularly in the presence of noise and limited data. Prior work has explored subjective metrics like entropy as a framework for capturing structural invariants between such signals---motivating the search for a robust, theoretically grounded method. In this work, we develop a nonparametric, theory-driven method based on excursion statistics of stochastic processes.
Building on classical results on the persistent homology for continuous semimartingales, we exploit a universal scaling law relating the number of barcodes of size $\eps$ to the quadratic variation of the signal. This yields a simple and interpretable diagnostic: stochastic diffusion processes exhibit an inverse-squared scaling in barcodes/excursion counts, while deterministic systems---periodic or chaotic---systematically violate this behavior. Using this principle, we construct a data-driven classification framework that operates directly on a single observed time series without embedding, symbolic transformations, or model training.
We demonstrate the effectiveness of the method across a wide range of systems, including canonical stochastic diffusions, deterministic chaotic maps and flows, noisy hybrid systems, and real-world datasets including financial time series and audio signals. |
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