---
OA_place: publisher
OA_type: hybrid
PlanS_conform: '1'
_id: '22648'
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.
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).
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Tamal K.
  full_name: Dey, Tamal K.
  last_name: Dey
- first_name: Michał
  full_name: Lipiński, Michał
  id: dfffb474-4317-11ee-8f5c-fe3fc95a425e
  last_name: Lipiński
  orcid: 0000-0001-9789-9750
- first_name: Manuel
  full_name: Soriano Trigueros, Manuel
  id: 15ebd7cf-15bf-11ee-aebd-bb4bb5121ea8
  last_name: Soriano Trigueros
  orcid: 0000-0003-2449-1433
citation:
  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>'
  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>'
  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>.'
  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.'
  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>.'
  short: T.K. Dey, M. Lipiński, M. Soriano Trigueros, Foundations of Computational
    Mathematics (2026).
corr_author: '1'
das_tickbox: '0'
date_created: 2026-08-05T06:11:30Z
date_published: 2026-08-04T00:00:00Z
date_updated: 2026-08-11T06:13:33Z
day: '04'
ddc:
- '500'
department:
- _id: HeEd
doi: 10.1007/s10208-026-09766-6
ec_funded: 1
external_id:
  arxiv:
  - '2504.17105'
has_accepted_license: '1'
keyword:
- Multivector field
- Conley index
- Morse decomposition
- Bifurcation
- Continuation
- Zigzag persistence
- Persistence barcode
- Gentle algebra
language:
- iso: eng
license: https://creativecommons.org/licenses/by/4.0/
main_file_link:
- open_access: '1'
  url: https://doi.org/10.1007/s10208-026-09766-6
month: '08'
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: Foundations of Computational Mathematics
publication_identifier:
  eissn:
  - 1615-3383
  issn:
  - 1615-3375
publication_status: epub_ahead
publisher: Springer
quality_controlled: '1'
researchdata_availability: no
scopus_import: '1'
status: public
supplementarymaterial: yes
title: 'Conley-Morse persistence barcode: A homological signature of combinatorial
  bifurcations'
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
year: '2026'
...