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Open AccessArticle

Using Zigzag Persistent Homology to Detect Hopf Bifurcations in Dynamical Systems

1
Department of Computational Mathematics, Science and Engineering, Michigan State University, East Lansing, MI 48824, USA
2
Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA
3
Department of Mechanical Engineering, Michigan State University, East Lansing, MI 48824, USA
*
Author to whom correspondence should be addressed.
Algorithms 2020, 13(11), 278; https://doi.org/10.3390/a13110278
Received: 18 September 2020 / Revised: 23 October 2020 / Accepted: 29 October 2020 / Published: 31 October 2020
(This article belongs to the Special Issue Topological Data Analysis)
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. In an experimental setting, this transition could lead to incorrect data or damage to the entire experiment. 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. View Full-Text
Keywords: topological data analysis; time-series analysis; zigzag persistent homology topological data analysis; time-series analysis; zigzag persistent homology
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MDPI and ACS Style

Tymochko, S.; Munch, E.; Khasawneh, F.A. Using Zigzag Persistent Homology to Detect Hopf Bifurcations in Dynamical Systems. Algorithms 2020, 13, 278. https://doi.org/10.3390/a13110278

AMA Style

Tymochko S, Munch E, Khasawneh FA. Using Zigzag Persistent Homology to Detect Hopf Bifurcations in Dynamical Systems. Algorithms. 2020; 13(11):278. https://doi.org/10.3390/a13110278

Chicago/Turabian Style

Tymochko, Sarah; Munch, Elizabeth; Khasawneh, Firas A. 2020. "Using Zigzag Persistent Homology to Detect Hopf Bifurcations in Dynamical Systems" Algorithms 13, no. 11: 278. https://doi.org/10.3390/a13110278

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