---
_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: '11870'
abstract:
- lang: eng
  text: We derive a class of computationally inexpensive linear dimension reduction
    criteria by introducing a weighted variant of the well-known K-class Fisher criterion
    associated with linear discriminant analysis (LDA). It can be seen that LDA weights
    contributions of individual class pairs according to the Euclidean distance of
    the respective class means. We generalize upon LDA by introducing a different
    weighting function
author:
- first_name: M.
  full_name: Loog, M.
  last_name: Loog
- first_name: R.P.W.
  full_name: Duin, R.P.W.
  last_name: Duin
- first_name: Reinhold
  full_name: Haeb-Umbach, Reinhold
  id: '242'
  last_name: Haeb-Umbach
citation:
  ama: Loog M, Duin RPW, Haeb-Umbach R. Multiclass linear dimension reduction by weighted
    pairwise Fisher criteria. <i>IEEE Transactions on Pattern Analysis and Machine
    Intelligence</i>. 2001;23(7):762-766. doi:<a href="https://doi.org/10.1109/34.935849">10.1109/34.935849</a>
  apa: Loog, M., Duin, R. P. W., &#38; Haeb-Umbach, R. (2001). Multiclass linear dimension
    reduction by weighted pairwise Fisher criteria. <i>IEEE Transactions on Pattern
    Analysis and Machine Intelligence</i>, <i>23</i>(7), 762–766. <a href="https://doi.org/10.1109/34.935849">https://doi.org/10.1109/34.935849</a>
  bibtex: '@article{Loog_Duin_Haeb-Umbach_2001, title={Multiclass linear dimension
    reduction by weighted pairwise Fisher criteria}, volume={23}, DOI={<a href="https://doi.org/10.1109/34.935849">10.1109/34.935849</a>},
    number={7}, journal={IEEE Transactions on Pattern Analysis and Machine Intelligence},
    author={Loog, M. and Duin, R.P.W. and Haeb-Umbach, Reinhold}, year={2001}, pages={762–766}
    }'
  chicago: 'Loog, M., R.P.W. Duin, and Reinhold Haeb-Umbach. “Multiclass Linear Dimension
    Reduction by Weighted Pairwise Fisher Criteria.” <i>IEEE Transactions on Pattern
    Analysis and Machine Intelligence</i> 23, no. 7 (2001): 762–66. <a href="https://doi.org/10.1109/34.935849">https://doi.org/10.1109/34.935849</a>.'
  ieee: M. Loog, R. P. W. Duin, and R. Haeb-Umbach, “Multiclass linear dimension reduction
    by weighted pairwise Fisher criteria,” <i>IEEE Transactions on Pattern Analysis
    and Machine Intelligence</i>, vol. 23, no. 7, pp. 762–766, 2001.
  mla: Loog, M., et al. “Multiclass Linear Dimension Reduction by Weighted Pairwise
    Fisher Criteria.” <i>IEEE Transactions on Pattern Analysis and Machine Intelligence</i>,
    vol. 23, no. 7, 2001, pp. 762–66, doi:<a href="https://doi.org/10.1109/34.935849">10.1109/34.935849</a>.
  short: M. Loog, R.P.W. Duin, R. Haeb-Umbach, IEEE Transactions on Pattern Analysis
    and Machine Intelligence 23 (2001) 762–766.
date_created: 2019-07-12T05:29:51Z
date_updated: 2022-01-06T06:51:11Z
department:
- _id: '54'
doi: 10.1109/34.935849
intvolume: '        23'
issue: '7'
keyword:
- approximate pairwise accuracy
- Bayes error
- Bayes methods
- error statistics
- Euclidean distance
- Fisher criterion
- linear dimension reduction
- linear discriminant analysis
- pattern classification
- statistical analysis
- statistical pattern classification
- weighting function
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://groups.uni-paderborn.de/nt/pubs/2001/LoDuHa01.pdf
oa: '1'
page: 762-766
publication: IEEE Transactions on Pattern Analysis and Machine Intelligence
status: public
title: Multiclass linear dimension reduction by weighted pairwise Fisher criteria
type: journal_article
user_id: '44006'
volume: 23
year: '2001'
...