---
OA_place: publisher
OA_type: hybrid
_id: '18580'
abstract:
- lang: eng
  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.
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). '
article_number: '5'
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Michał
  full_name: Lipiński, Michał
  id: dfffb474-4317-11ee-8f5c-fe3fc95a425e
  last_name: Lipiński
  orcid: 0000-0001-9789-9750
- first_name: Konstantin
  full_name: Mischaikow, Konstantin
  last_name: Mischaikow
- first_name: Marian
  full_name: Mrozek, Marian
  last_name: Mrozek
citation:
  ama: Lipiński M, Mischaikow K, Mrozek M. Morse predecomposition of an invariant
    set. <i>Qualitative Theory of Dynamical Systems</i>. 2025;24(1). doi:<a href="https://doi.org/10.1007/s12346-024-01144-3">10.1007/s12346-024-01144-3</a>
  apa: Lipiński, M., Mischaikow, K., &#38; Mrozek, M. (2025). 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>
  chicago: Lipiński, Michał, Konstantin Mischaikow, and Marian Mrozek. “Morse Predecomposition
    of an Invariant Set.” <i>Qualitative Theory of Dynamical Systems</i>. Springer
    Nature, 2025. <a href="https://doi.org/10.1007/s12346-024-01144-3">https://doi.org/10.1007/s12346-024-01144-3</a>.
  ieee: M. Lipiński, K. Mischaikow, and M. Mrozek, “Morse predecomposition of an invariant
    set,” <i>Qualitative Theory of Dynamical Systems</i>, vol. 24, no. 1. Springer
    Nature, 2025.
  ista: Lipiński M, Mischaikow K, Mrozek M. 2025. Morse predecomposition of an invariant
    set. Qualitative Theory of Dynamical Systems. 24(1), 5.
  mla: Lipiński, Michał, et al. “Morse Predecomposition of an Invariant Set.” <i>Qualitative
    Theory of Dynamical Systems</i>, vol. 24, no. 1, 5, Springer Nature, 2025, doi:<a
    href="https://doi.org/10.1007/s12346-024-01144-3">10.1007/s12346-024-01144-3</a>.
  short: M. Lipiński, K. Mischaikow, M. Mrozek, Qualitative Theory of Dynamical Systems
    24 (2025).
corr_author: '1'
date_created: 2024-11-24T23:01:47Z
date_published: 2025-02-01T00:00:00Z
date_updated: 2025-04-14T07:54:56Z
day: '01'
ddc:
- '514'
- '510'
department:
- _id: UlWa
doi: 10.1007/s12346-024-01144-3
ec_funded: 1
external_id:
  arxiv:
  - '2312.08013'
  isi:
  - '001356000500005'
file:
- access_level: open_access
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  creator: mlipinsk
  date_created: 2024-11-28T06:52:38Z
  date_updated: 2024-11-28T06:52:38Z
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has_accepted_license: '1'
intvolume: '        24'
isi: 1
issue: '1'
language:
- iso: eng
license: https://creativecommons.org/licenses/by/4.0/
month: '02'
oa: 1
oa_version: Published Version
project:
- _id: fc2ed2f7-9c52-11eb-aca3-c01059dda49c
  call_identifier: H2020
  grant_number: '101034413'
  name: 'IST-BRIDGE: International postdoctoral program'
publication: Qualitative Theory of Dynamical Systems
publication_identifier:
  eissn:
  - 1662-3592
  issn:
  - 1575-5460
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Morse predecomposition of an invariant set
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 24
year: '2025'
...