{"tmp":{"name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","short":"CC BY (4.0)","image":"/images/cc_by.png"},"date_created":"2026-08-05T06:11:30Z","day":"04","main_file_link":[{"open_access":"1","url":"https://doi.org/10.1007/s10208-026-09766-6"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","keyword":["Multivector field","Conley index","Morse decomposition","Bifurcation","Continuation","Zigzag persistence","Persistence barcode","Gentle algebra"],"publication_status":"epub_ahead","OA_place":"publisher","status":"public","publication":"Foundations of Computational Mathematics","department":[{"_id":"HeEd"}],"publisher":"Springer","oa":1,"_id":"22648","OA_type":"hybrid","article_processing_charge":"Yes (via OA deal)","acknowledgement":"M.L. acknowledges support from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 101034413. T.D. acknowledges the support of NSF funds CCF-2437030 and DMS-2301360. The authors would like to thank the anonymous reviewers for their careful reading of the paper. Their feedback significantly improved the quality of the article. T.D. and M.L. would like to acknowledge many thought-provoking discussions with Marian Mrozek on combinatorial dynamical systems and their continuations. M.S.T. would like to thank Álvaro Sánchez for insightful discussions about representation theory. Open access funding provided by Institute of Science and Technology (IST Austria).","das_tickbox":"0","external_id":{"arxiv":["2504.17105"]},"quality_controlled":"1","ddc":["500"],"citation":{"chicago":"Dey, Tamal K., Michał Lipiński, and Manuel Soriano Trigueros. “Conley-Morse Persistence Barcode: A Homological Signature of Combinatorial Bifurcations.” <i>Foundations of Computational Mathematics</i>. Springer, 2026. <a href=\"https://doi.org/10.1007/s10208-026-09766-6\">https://doi.org/10.1007/s10208-026-09766-6</a>.","ista":"Dey TK, Lipiński M, Soriano Trigueros M. 2026. Conley-Morse persistence barcode: A homological signature of combinatorial bifurcations. Foundations of Computational Mathematics.","mla":"Dey, Tamal K., et al. “Conley-Morse Persistence Barcode: A Homological Signature of Combinatorial Bifurcations.” <i>Foundations of Computational Mathematics</i>, Springer, 2026, doi:<a href=\"https://doi.org/10.1007/s10208-026-09766-6\">10.1007/s10208-026-09766-6</a>.","ieee":"T. K. Dey, M. Lipiński, and M. Soriano Trigueros, “Conley-Morse persistence barcode: A homological signature of combinatorial bifurcations,” <i>Foundations of Computational Mathematics</i>. Springer, 2026.","short":"T.K. Dey, M. Lipiński, M. Soriano Trigueros, Foundations of Computational Mathematics (2026).","apa":"Dey, T. K., Lipiński, M., &#38; Soriano Trigueros, M. (2026). Conley-Morse persistence barcode: A homological signature of combinatorial bifurcations. <i>Foundations of Computational Mathematics</i>. Springer. <a href=\"https://doi.org/10.1007/s10208-026-09766-6\">https://doi.org/10.1007/s10208-026-09766-6</a>","ama":"Dey TK, Lipiński M, Soriano Trigueros M. Conley-Morse persistence barcode: A homological signature of combinatorial bifurcations. <i>Foundations of Computational Mathematics</i>. 2026. doi:<a href=\"https://doi.org/10.1007/s10208-026-09766-6\">10.1007/s10208-026-09766-6</a>"},"abstract":[{"lang":"eng","text":"Bifurcation characterizes the qualitative changes in parameterized dynamical systems and is one of the major topics in the field. In this work, we study combinatorial bifurcations within the framework of combinatorial dynamical systems—a young but already well-established theory. We introduce the Conley–Morse persistence barcode, a compact algebraic descriptor of combinatorial bifurcations. This barcode captures structural changes in a dynamical system at the level of Morse decompositions and provides a characterization of the nature of observed transitions in terms of the Conley index. The construction of the Conley–Morse persistence barcode builds upon ideas from topological persistence. Specifically, we consider a persistence module obtained from the Conley index of invariant sets indexed over a poset. Using gentle algebras, we prove that this module decomposes into simple intervals (bars) and compute them by adapting the zigzag persistence algorithm to our purpose."}],"supplementarymaterial":"yes","author":[{"last_name":"Dey","first_name":"Tamal K.","full_name":"Dey, Tamal K."},{"orcid":"0000-0001-9789-9750","first_name":"Michał","last_name":"Lipiński","full_name":"Lipiński, Michał","id":"dfffb474-4317-11ee-8f5c-fe3fc95a425e"},{"full_name":"Soriano Trigueros, Manuel","id":"15ebd7cf-15bf-11ee-aebd-bb4bb5121ea8","first_name":"Manuel","orcid":"0000-0003-2449-1433","last_name":"Soriano Trigueros"}],"oa_version":"Published Version","year":"2026","title":"Conley-Morse persistence barcode: A homological signature of combinatorial bifurcations","type":"journal_article","arxiv":1,"publication_identifier":{"eissn":["1615-3383"],"issn":["1615-3375"]},"date_updated":"2026-08-11T06:13:33Z","article_type":"original","researchdata_availability":"no","language":[{"iso":"eng"}],"corr_author":"1","has_accepted_license":"1","project":[{"call_identifier":"H2020","grant_number":"101034413","_id":"fc2ed2f7-9c52-11eb-aca3-c01059dda49c","name":"IST-BRIDGE: International postdoctoral program"}],"PlanS_conform":"1","ec_funded":1,"date_published":"2026-08-04T00:00:00Z","doi":"10.1007/s10208-026-09766-6","month":"08","scopus_import":"1"}