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Topological Detection of Hopf Bifurcations via Persistent...
[Submitted on 28 Mar 2026 (v1), last revised 20 Aug 2026 (this v · 2026-03-29 · via stat updates on arXiv.org

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Abstract:We propose a topological framework for detecting Hopf-type dynamical transitions directly from scalar time series. The method combines delay-coordinate reconstruction with persistent homology and uses the maximum persistence of one-dimensional homology classes as a scalar descriptor of cyclic structure. For the supercritical Hopf setting, we derive finite-resolution persistence bounds that relate detectability of the reconstructed periodic orbit to its geometry, sampling quality, and finite-data perturbations. A derivative-based estimator is then introduced to localize the critical parameter from the sampled topological functional. The approach is evaluated on the Hopf normal form, the Lorenz system, and a reduced Belousov--Zhabotinsky model. The numerical experiments show accurate finite-resolution localization of the corresponding transitions and illustrate the effects of embedding parameters, temporal sampling, smoothing, observational noise, and transient removal. These results support persistent homology as an interpretable data-driven tool for detecting geometric reorganizations in nonlinear time series.

Submission history

From: Jhonathan Barrios [view email]
[v1] Sat, 28 Mar 2026 20:25:38 UTC (1,499 KB)
[v2] Thu, 20 Aug 2026 19:15:40 UTC (189 KB)

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