---
OA_place: repository
OA_type: green
_id: '20980'
abstract:
- lang: eng
  text: 'Morse decompositions partition the flows in a vector field into equivalent
    structures. Given such a decomposition, one can define a further summary of its
    flow structure by what is called a connection matrix. These matrices, a generalization
    of Morse boundary operators from classical Morse theory, capture the connections
    made by the flows among the critical structures—such as attractors, repellers,
    and orbits—in a vector field. Recently, in the context of combinatorial dynamics,
    an efficient persistence-like algorithm to compute connection matrices has been
    proposed in Dey, Lipiński, Mrozek, and Slechta [SIAM J. Appl. Dyn. Syst., 23 (2024),
    pp. 81–97]. We show that, actually, the classical persistence algorithm with exhaustive
    reduction retrieves connection matrices, both simplifying the algorithm of Dey
    et al. and bringing the theory of persistence closer to combinatorial dynamical
    systems. We supplement this main result with an observation: the concept of persistence
    as defined for scalar fields naturally adapts to Morse decompositions whose Morse
    sets are filtered with a Lyapunov function. We conclude by presenting preliminary
    experimental results.'
acknowledgement: "This research was supported by NSF grants DMS-2301360 and CCF-2437030
  as well as from the European Union's Horizon 2020 research and innovation programme
  under Marie Sk\\lodowska-Curie grant 101034413.\r\n"
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Tamal K.
  full_name: Dey, Tamal K.
  last_name: Dey
- first_name: Andrew
  full_name: Haas, Andrew
  last_name: Haas
- first_name: Michał
  full_name: Lipiński, Michał
  id: dfffb474-4317-11ee-8f5c-fe3fc95a425e
  last_name: Lipiński
  orcid: 0000-0001-9789-9750
citation:
  ama: Dey TK, Haas A, Lipiński M. Computing a connection matrix and persistence efficiently
    from a morse decomposition. <i>SIAM Journal on Applied Dynamical Systems</i>.
    2026;25(1):108-130. doi:<a href="https://doi.org/10.1137/25m1739406">10.1137/25m1739406</a>
  apa: Dey, T. K., Haas, A., &#38; Lipiński, M. (2026). Computing a connection matrix
    and persistence efficiently from a morse decomposition. <i>SIAM Journal on Applied
    Dynamical Systems</i>. Society for Industrial &#38; Applied Mathematics. <a href="https://doi.org/10.1137/25m1739406">https://doi.org/10.1137/25m1739406</a>
  chicago: Dey, Tamal K., Andrew Haas, and Michał Lipiński. “Computing a Connection
    Matrix and Persistence Efficiently from a Morse Decomposition.” <i>SIAM Journal
    on Applied Dynamical Systems</i>. Society for Industrial &#38; Applied Mathematics,
    2026. <a href="https://doi.org/10.1137/25m1739406">https://doi.org/10.1137/25m1739406</a>.
  ieee: T. K. Dey, A. Haas, and M. Lipiński, “Computing a connection matrix and persistence
    efficiently from a morse decomposition,” <i>SIAM Journal on Applied Dynamical
    Systems</i>, vol. 25, no. 1. Society for Industrial &#38; Applied Mathematics,
    pp. 108–130, 2026.
  ista: Dey TK, Haas A, Lipiński M. 2026. Computing a connection matrix and persistence
    efficiently from a morse decomposition. SIAM Journal on Applied Dynamical Systems.
    25(1), 108–130.
  mla: Dey, Tamal K., et al. “Computing a Connection Matrix and Persistence Efficiently
    from a Morse Decomposition.” <i>SIAM Journal on Applied Dynamical Systems</i>,
    vol. 25, no. 1, Society for Industrial &#38; Applied Mathematics, 2026, pp. 108–30,
    doi:<a href="https://doi.org/10.1137/25m1739406">10.1137/25m1739406</a>.
  short: T.K. Dey, A. Haas, M. Lipiński, SIAM Journal on Applied Dynamical Systems
    25 (2026) 108–130.
date_created: 2026-01-12T11:17:06Z
date_published: 2026-01-01T00:00:00Z
date_updated: 2026-01-20T07:40:39Z
day: '01'
ddc:
- '510'
department:
- _id: HeEd
doi: 10.1137/25m1739406
ec_funded: 1
external_id:
  arxiv:
  - '2502.19369'
intvolume: '        25'
issue: '1'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.2502.19369
month: '01'
oa: 1
oa_version: Preprint
page: 108-130
project:
- _id: fc2ed2f7-9c52-11eb-aca3-c01059dda49c
  call_identifier: H2020
  grant_number: '101034413'
  name: 'IST-BRIDGE: International postdoctoral program'
publication: SIAM Journal on Applied Dynamical Systems
publication_identifier:
  issn:
  - 1536-0040
publication_status: published
publisher: Society for Industrial & Applied Mathematics
quality_controlled: '1'
scopus_import: '1'
status: public
title: Computing a connection matrix and persistence efficiently from a morse decomposition
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 25
year: '2026'
...