---
_id: '64979'
abstract:
- lang: eng
  text: We investigate homogeneous coupled cell systems with high-dimensional internal
    dynamics. In many studies on network dynamics, the analysis is restricted to networks
    with one-dimensional internal dynamics. Here, we show how symmetry explains the
    relation between dynamical behavior of systems with one-dimensional internal dynamics
    and with higher dimensional internal dynamics, when the underlying network topology
    is the same. Fundamental networks of homogeneous coupled cell systems (B. Rink,
    J. Sanders. Coupled Cell Networks and Their Hidden Symmetries. SIAM J. Math. Anal.
    46.2 (2014)) can be expressed in terms of monoid representations, which uniquely
    decompose into indecomposable subrepresentations. In the high-dimensional internal
    dynamics case, these subrepresentations are isomorphic to multiple copies of those
    one computes in the one-dimensional internal dynamics case. This has interesting
    implications for possible center subspaces in bifurcation analysis. We describe
    the effect on steady state and Hopf bifurcations in l-parameter families of network
    vector fields. The main results in that regard are that (1) generic one-parameter
    steady state bifurcations are qualitatively independent of the dimension of the
    internal dynamics and that, (2) in order to observe all generic l-parameter bifurcations
    that may occur for internal dynamics of any dimension, the internal dynamics has
    to be at least l-dimensional for steady state bifurcations and 2l-dimensional
    for Hopf bifurcations. Furthermore, we illustrate how additional structure in
    the network can be exploited to obtain even greater understanding of bifurcation
    scenarios in the high-dimensional case beyond qualitative statements about the
    collective dynamics. One-parameter steady state bifurcations in feedforward networks
    exhibit an unusual amplification in the asymptotic growth rates of individual
    cells, when these are one-dimensional (S. von der Gracht, E. Nijholt, B. Rink.
    Amplified steady state bifurcations in feedforward networks. Nonlinearity 35.4
    (2022)). As another main result, we prove that (3) the same cells exhibit this
    amplifying effect with the same growth rates when the internal dynamics is high-dimensional.
article_number: '118196'
article_type: original
author:
- first_name: Sören
  full_name: von der Gracht, Sören
  id: '97359'
  last_name: von der Gracht
  orcid: 0000-0002-8054-2058
- first_name: Eddie
  full_name: Nijholt, Eddie
  last_name: Nijholt
- first_name: Bob
  full_name: Rink, Bob
  last_name: Rink
citation:
  ama: von der Gracht S, Nijholt E, Rink B. Homogeneous coupled cell systems with
    high-dimensional internal dynamics. <i>Chaos, Solitons &#38; Fractals</i>. 2026;208.
    doi:<a href="https://doi.org/10.1016/j.chaos.2026.118196">10.1016/j.chaos.2026.118196</a>
  apa: von der Gracht, S., Nijholt, E., &#38; Rink, B. (2026). Homogeneous coupled
    cell systems with high-dimensional internal dynamics. <i>Chaos, Solitons &#38;
    Fractals</i>, <i>208</i>, Article 118196. <a href="https://doi.org/10.1016/j.chaos.2026.118196">https://doi.org/10.1016/j.chaos.2026.118196</a>
  bibtex: '@article{von der Gracht_Nijholt_Rink_2026, title={Homogeneous coupled cell
    systems with high-dimensional internal dynamics}, volume={208}, DOI={<a href="https://doi.org/10.1016/j.chaos.2026.118196">10.1016/j.chaos.2026.118196</a>},
    number={118196}, journal={Chaos, Solitons &#38; Fractals}, publisher={Elsevier
    BV}, author={von der Gracht, Sören and Nijholt, Eddie and Rink, Bob}, year={2026}
    }'
  chicago: Gracht, Sören von der, Eddie Nijholt, and Bob Rink. “Homogeneous Coupled
    Cell Systems with High-Dimensional Internal Dynamics.” <i>Chaos, Solitons &#38;
    Fractals</i> 208 (2026). <a href="https://doi.org/10.1016/j.chaos.2026.118196">https://doi.org/10.1016/j.chaos.2026.118196</a>.
  ieee: 'S. von der Gracht, E. Nijholt, and B. Rink, “Homogeneous coupled cell systems
    with high-dimensional internal dynamics,” <i>Chaos, Solitons &#38; Fractals</i>,
    vol. 208, Art. no. 118196, 2026, doi: <a href="https://doi.org/10.1016/j.chaos.2026.118196">10.1016/j.chaos.2026.118196</a>.'
  mla: von der Gracht, Sören, et al. “Homogeneous Coupled Cell Systems with High-Dimensional
    Internal Dynamics.” <i>Chaos, Solitons &#38; Fractals</i>, vol. 208, 118196, Elsevier
    BV, 2026, doi:<a href="https://doi.org/10.1016/j.chaos.2026.118196">10.1016/j.chaos.2026.118196</a>.
  short: S. von der Gracht, E. Nijholt, B. Rink, Chaos, Solitons &#38; Fractals 208
    (2026).
date_created: 2026-03-16T08:39:07Z
date_updated: 2026-03-16T08:42:56Z
ddc:
- '510'
department:
- _id: '101'
- _id: '841'
doi: 10.1016/j.chaos.2026.118196
external_id:
  arxiv:
  - '2510.06740'
file:
- access_level: closed
  content_type: application/pdf
  creator: svdg
  date_created: 2026-03-16T08:40:04Z
  date_updated: 2026-03-16T08:40:04Z
  file_id: '64980'
  file_name: homogeneous-coupled-cell-systems-with-high-dimensional-internal-dynamics.pdf
  file_size: 1951746
  relation: main_file
  success: 1
file_date_updated: 2026-03-16T08:40:04Z
has_accepted_license: '1'
intvolume: '       208'
keyword:
- Coupled cell systems
- Network dynamics
- Dimension reduction
- Bifurcation theory
- Symmetry
- Monoid representation theory
language:
- iso: eng
publication: Chaos, Solitons & Fractals
publication_identifier:
  issn:
  - 0960-0779
publication_status: published
publisher: Elsevier BV
status: public
title: Homogeneous coupled cell systems with high-dimensional internal dynamics
type: journal_article
user_id: '97359'
volume: 208
year: '2026'
...
---
_id: '59171'
abstract:
- lang: eng
  text: To model dynamical systems on networks with higher-order (non-pairwise) interactions,
    we recently introduced a new class of ordinary differential equations (ODEs) on
    hypernetworks. Here, we consider one-parameter synchrony breaking bifurcations
    in such ODEs. We call a synchrony breaking steady-state branch ‘reluctant’ if
    it is tangent to a synchrony space, but does not lie inside it. We prove that
    reluctant synchrony breaking is ubiquitous in hypernetwork systems, by constructing
    a large class of examples that support it. We also give an explicit formula for
    the order of tangency to the synchrony space of a reluctant steady-state branch.
author:
- first_name: Sören
  full_name: von der Gracht, Sören
  id: '97359'
  last_name: von der Gracht
  orcid: 0000-0002-8054-2058
- first_name: Eddie
  full_name: Nijholt, Eddie
  last_name: Nijholt
- first_name: Bob
  full_name: Rink, Bob
  last_name: Rink
citation:
  ama: 'von der Gracht S, Nijholt E, Rink B. Higher-order interactions lead to ‘reluctant’
    synchrony breaking. <i>Proceedings of the Royal Society A: Mathematical, Physical
    and Engineering Sciences</i>. 2024;480(2301). doi:<a href="https://doi.org/10.1098/rspa.2023.0945">10.1098/rspa.2023.0945</a>'
  apa: 'von der Gracht, S., Nijholt, E., &#38; Rink, B. (2024). Higher-order interactions
    lead to ‘reluctant’ synchrony breaking. <i>Proceedings of the Royal Society A:
    Mathematical, Physical and Engineering Sciences</i>, <i>480</i>(2301). <a href="https://doi.org/10.1098/rspa.2023.0945">https://doi.org/10.1098/rspa.2023.0945</a>'
  bibtex: '@article{von der Gracht_Nijholt_Rink_2024, title={Higher-order interactions
    lead to ‘reluctant’ synchrony breaking}, volume={480}, DOI={<a href="https://doi.org/10.1098/rspa.2023.0945">10.1098/rspa.2023.0945</a>},
    number={2301}, journal={Proceedings of the Royal Society A: Mathematical, Physical
    and Engineering Sciences}, publisher={The Royal Society}, author={von der Gracht,
    Sören and Nijholt, Eddie and Rink, Bob}, year={2024} }'
  chicago: 'Gracht, Sören von der, Eddie Nijholt, and Bob Rink. “Higher-Order Interactions
    Lead to ‘Reluctant’ Synchrony Breaking.” <i>Proceedings of the Royal Society A:
    Mathematical, Physical and Engineering Sciences</i> 480, no. 2301 (2024). <a href="https://doi.org/10.1098/rspa.2023.0945">https://doi.org/10.1098/rspa.2023.0945</a>.'
  ieee: 'S. von der Gracht, E. Nijholt, and B. Rink, “Higher-order interactions lead
    to ‘reluctant’ synchrony breaking,” <i>Proceedings of the Royal Society A: Mathematical,
    Physical and Engineering Sciences</i>, vol. 480, no. 2301, 2024, doi: <a href="https://doi.org/10.1098/rspa.2023.0945">10.1098/rspa.2023.0945</a>.'
  mla: 'von der Gracht, Sören, et al. “Higher-Order Interactions Lead to ‘Reluctant’
    Synchrony Breaking.” <i>Proceedings of the Royal Society A: Mathematical, Physical
    and Engineering Sciences</i>, vol. 480, no. 2301, The Royal Society, 2024, doi:<a
    href="https://doi.org/10.1098/rspa.2023.0945">10.1098/rspa.2023.0945</a>.'
  short: 'S. von der Gracht, E. Nijholt, B. Rink, Proceedings of the Royal Society
    A: Mathematical, Physical and Engineering Sciences 480 (2024).'
date_created: 2025-03-27T10:15:06Z
date_updated: 2025-03-27T10:19:56Z
ddc:
- '510'
department:
- _id: '101'
doi: 10.1098/rspa.2023.0945
file:
- access_level: open_access
  content_type: application/pdf
  creator: svdg
  date_created: 2025-03-27T10:16:20Z
  date_updated: 2025-03-27T10:19:48Z
  file_id: '59172'
  file_name: higher-order-interactions-lead-to-reluctant-synchrony-breaking.pdf
  file_size: 820435
  relation: main_file
file_date_updated: 2025-03-27T10:19:48Z
has_accepted_license: '1'
intvolume: '       480'
issue: '2301'
keyword:
- higher-order interactions
- synchrony breaking
- network dynamics
- coupled cell systems
language:
- iso: eng
oa: '1'
publication: 'Proceedings of the Royal Society A: Mathematical, Physical and Engineering
  Sciences'
publication_identifier:
  issn:
  - 1364-5021
  - 1471-2946
publication_status: published
publisher: The Royal Society
status: public
title: Higher-order interactions lead to ‘reluctant’ synchrony breaking
type: journal_article
user_id: '97359'
volume: 480
year: '2024'
...