{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:21:34Z","timestamp":1760239294074,"version":"build-2065373602"},"reference-count":48,"publisher":"MDPI AG","issue":"11","license":[{"start":{"date-parts":[[2020,10,31]],"date-time":"2020-10-31T00:00:00Z","timestamp":1604102400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["CCF-1907591, DMS-1800446, CMMI-1759823,DMS1759824"],"award-info":[{"award-number":["CCF-1907591, DMS-1800446, CMMI-1759823,DMS1759824"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>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.<\/jats:p>","DOI":"10.3390\/a13110278","type":"journal-article","created":{"date-parts":[[2020,10,31]],"date-time":"2020-10-31T21:39:56Z","timestamp":1604180396000},"page":"278","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":14,"title":["Using Zigzag Persistent Homology to Detect Hopf Bifurcations in Dynamical Systems"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4932-6373","authenticated-orcid":false,"given":"Sarah","family":"Tymochko","sequence":"first","affiliation":[{"name":"Department of Computational Mathematics, Science and Engineering, Michigan State University, East Lansing, MI 48824, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9459-9493","authenticated-orcid":false,"given":"Elizabeth","family":"Munch","sequence":"additional","affiliation":[{"name":"Department of Computational Mathematics, Science and Engineering, Michigan State University, East Lansing, MI 48824, USA"},{"name":"Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7817-7445","authenticated-orcid":false,"given":"Firas A.","family":"Khasawneh","sequence":"additional","affiliation":[{"name":"Department of Mechanical Engineering, Michigan State University, East Lansing, MI 48824, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,10,31]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"799","DOI":"10.1007\/s10208-014-9206-z","article-title":"Sliding Windows and Persistence: An Application of Topological Methods to Signal Analysis","volume":"15","author":"Perea","year":"2015","journal-title":"Found. 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