# Using Zigzag Persistent Homology to Detect Hopf Bifurcations in Dynamical Systems

arXiv: https://arxiv.org/abs/2009.08972 · HTML: https://mlsanity.com/paper/2009.08972/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Sep 2020.

## Built on by (most cited first, 7 of 7)

- [Persistent Topological Features in Large Language Models](https://mlsanity.com/paper/2410.11042/) (2024)
- [Persistent Homology of Coarse Grained State Space Networks](https://mlsanity.com/paper/2206.02530/) (2022)
- [Let's do the time-warp-attend: Learning topological invariants of dynamical systems](https://mlsanity.com/paper/2312.09234/) (2023)
- [Neural Persistence Dynamics](https://mlsanity.com/paper/2405.15732/) (2024)
- [Topological Detection of Hopf Bifurcations via Persistent Homology: A Functional Criterion from Time Series](https://mlsanity.com/paper/2603.27395/) (2026)
- [Synergistic Fusion of Topological Structure and Temporal Semantics of Mobility for Urban Region Embedding](https://mlsanity.com/paper/2609.08268/) (2026)
- [A differentiability framework for zigzag persistent homology via linear interpolation](https://mlsanity.com/paper/2609.39242/) (2026)

## Abstract (arXiv)

Bifurcations in dynamical systems characterize qualitative changes in the system behavior. Therefore, their detection is important because they can signal the transition from normal system operation to imminent failure. While standard persistent homology has been used in this setting, it usually requires analyzing a collection of persistence diagrams, which in turn drives up the computational cost considerably. Using zigzag persistence, we can capture topological changes in the state space of the dynamical system in only one persistence diagram. Here we present Bifurcations using ZigZag (BuZZ), a one-step method to study and detect bifurcations using zigzag persistence. The BuZZ method is successfully able to detect this type of behavior in two synthetic examples as well as an example dynamical system.