# Topological Detection of Hopf Bifurcations via Persistent Homology: A Functional Criterion from Time Series

arXiv: https://arxiv.org/abs/2603.27395 · HTML: https://mlsanity.com/paper/2603.27395/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 38 references that are in the map.
- First posted on arXiv 28 Mar 2026.

## Builds on (most cited first, 6 of 6)

- [An introduction to Topological Data Analysis: fundamental and practical aspects for data scientists](https://mlsanity.com/paper/1710.04019/) (2017)
- [Ripser: efficient computation of Vietoris-Rips persistence barcodes](https://mlsanity.com/paper/1908.02518/) (2019)
- [Topological Time Series Analysis](https://mlsanity.com/paper/1812.05143/) (2018)
- [Using Zigzag Persistent Homology to Detect Hopf Bifurcations in Dynamical Systems](https://mlsanity.com/paper/2009.08972/) (2020)
- [Critical Transitions In a Model of a Genetic Regulatory System](https://mlsanity.com/paper/1309.7919/) (2013)
- [Topological descriptors of foot clearance gait dynamics improve differential diagnosis of Parkinsonism](https://mlsanity.com/paper/2603.06212/) (2026)

## Abstract (arXiv)

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.