{"department":[{"_id":"UlWa"}],"tmp":{"short":"CC BY (4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)"},"file":[{"file_id":"18595","checksum":"73309a57cc798d696caa57b6aa1467d8","file_name":"2025_predecomposition.pdf","content_type":"application/pdf","relation":"main_file","access_level":"open_access","date_updated":"2024-11-28T06:52:38Z","creator":"mlipinsk","date_created":"2024-11-28T06:52:38Z","file_size":1483668,"success":1}],"date_created":"2024-11-24T23:01:47Z","oa":1,"acknowledgement":"M.L. acknowledge support by the Dioscuri program initiated by the Max Planck Society, jointly managed with the National Science Centre (Poland), and mutually funded by the Polish Ministry of Science and Higher Education and the German Federal Ministry of Education and Research. M.L. also acknowledges that this project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 101034413. Research of M.M. is partially supported by the Polish National Science Center under Opus Grant No. 2019/35/B/ST1/00874. The work of K.M. was partially supported by the National Science Foundation under awards DMS-1839294 and HDR TRIPODS award CCF-1934924, DARPA contract HR0011-16-2-0033, National Institutes of Health award R01 GM126555, Air Force Office of Scientific Research under award numbers FA9550-23-1-0011, AWD00010853-MOD002 and MURI FA9550-23-1-0400. K.M. was also supported by a grant from the Simons Foundation. Open access funding provided by Institute of Science and Technology (IST Austria). ","OA_type":"hybrid","external_id":{"arxiv":["2312.08013"]},"article_type":"original","abstract":[{"text":"Motivated by the study of recurrent orbits and dynamics within a Morse set of a Morse decomposition we introduce the concept of Morse predecomposition of an isolated invariant set within the setting of both combinatorial and classical dynamical systems. While Morse decomposition summarizes solely the gradient part of a dynamical system, the developed generalization extends to the recurrent component as well. In particular, a chain recurrent set, which is indecomposable in terms of Morse decomposition, can be represented more finely in the Morse predecomposition framework. This generalization is achieved by forgoing the poset structure inherent to Morse decomposition and relaxing the notion of connection between Morse sets (elements of Morse decomposition) in favor of what we term ’links’. We prove that a Morse decomposition is a special case of Morse predecomposition indexed by a poset. Additionally, we show how a Morse predecomposition may be condensed back to retrieve a Morse decomposition.","lang":"eng"}],"day":"15","month":"11","oa_version":"Published Version","doi":"10.1007/s12346-024-01144-3","_id":"18580","date_published":"2024-11-15T00:00:00Z","publisher":"Springer Nature","status":"public","ddc":["514","510"],"title":"Morse predecomposition of an invariant set","publication_identifier":{"issn":["1575-5460"],"eissn":["1662-3592"]},"year":"2024","date_updated":"2024-12-03T11:24:14Z","OA_place":"publisher","ec_funded":1,"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","volume":24,"corr_author":"1","language":[{"iso":"eng"}],"scopus_import":"1","publication":"Qualitative Theory of Dynamical Systems","has_accepted_license":"1","type":"journal_article","intvolume":"        24","quality_controlled":"1","arxiv":1,"project":[{"_id":"fc2ed2f7-9c52-11eb-aca3-c01059dda49c","name":"IST-BRIDGE: International postdoctoral program","grant_number":"101034413","call_identifier":"H2020"}],"publication_status":"epub_ahead","citation":{"short":"M. Lipiński, K. Mischaikow, M. Mrozek, Qualitative Theory of Dynamical Systems 24 (2024).","ieee":"M. Lipiński, K. Mischaikow, and M. Mrozek, “Morse predecomposition of an invariant set,” <i>Qualitative Theory of Dynamical Systems</i>, vol. 24. Springer Nature, 2024.","apa":"Lipiński, M., Mischaikow, K., &#38; Mrozek, M. (2024). Morse predecomposition of an invariant set. <i>Qualitative Theory of Dynamical Systems</i>. Springer Nature. <a href=\"https://doi.org/10.1007/s12346-024-01144-3\">https://doi.org/10.1007/s12346-024-01144-3</a>","mla":"Lipiński, Michał, et al. “Morse Predecomposition of an Invariant Set.” <i>Qualitative Theory of Dynamical Systems</i>, vol. 24, 5, Springer Nature, 2024, doi:<a href=\"https://doi.org/10.1007/s12346-024-01144-3\">10.1007/s12346-024-01144-3</a>.","ama":"Lipiński M, Mischaikow K, Mrozek M. Morse predecomposition of an invariant set. <i>Qualitative Theory of Dynamical Systems</i>. 2024;24. doi:<a href=\"https://doi.org/10.1007/s12346-024-01144-3\">10.1007/s12346-024-01144-3</a>","ista":"Lipiński M, Mischaikow K, Mrozek M. 2024. Morse predecomposition of an invariant set. Qualitative Theory of Dynamical Systems. 24, 5.","chicago":"Lipiński, Michał, Konstantin Mischaikow, and Marian Mrozek. “Morse Predecomposition of an Invariant Set.” <i>Qualitative Theory of Dynamical Systems</i>. Springer Nature, 2024. <a href=\"https://doi.org/10.1007/s12346-024-01144-3\">https://doi.org/10.1007/s12346-024-01144-3</a>."},"author":[{"first_name":"Michał","orcid":"0000-0001-9789-9750","id":"dfffb474-4317-11ee-8f5c-fe3fc95a425e","full_name":"Lipiński, Michał","last_name":"Lipiński"},{"last_name":"Mischaikow","full_name":"Mischaikow, Konstantin","first_name":"Konstantin"},{"first_name":"Marian","last_name":"Mrozek","full_name":"Mrozek, Marian"}],"article_number":"5","file_date_updated":"2024-11-28T06:52:38Z","article_processing_charge":"Yes (via OA deal)"}