---
_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. Chaos, Solitons & Fractals. 2026;208.
doi:10.1016/j.chaos.2026.118196
apa: von der Gracht, S., Nijholt, E., & Rink, B. (2026). Homogeneous coupled
cell systems with high-dimensional internal dynamics. Chaos, Solitons &
Fractals, 208, Article 118196. https://doi.org/10.1016/j.chaos.2026.118196
bibtex: '@article{von der Gracht_Nijholt_Rink_2026, title={Homogeneous coupled cell
systems with high-dimensional internal dynamics}, volume={208}, DOI={10.1016/j.chaos.2026.118196},
number={118196}, journal={Chaos, Solitons & 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.” Chaos, Solitons &
Fractals 208 (2026). https://doi.org/10.1016/j.chaos.2026.118196.
ieee: 'S. von der Gracht, E. Nijholt, and B. Rink, “Homogeneous coupled cell systems
with high-dimensional internal dynamics,” Chaos, Solitons & Fractals,
vol. 208, Art. no. 118196, 2026, doi: 10.1016/j.chaos.2026.118196.'
mla: von der Gracht, Sören, et al. “Homogeneous Coupled Cell Systems with High-Dimensional
Internal Dynamics.” Chaos, Solitons & Fractals, vol. 208, 118196, Elsevier
BV, 2026, doi:10.1016/j.chaos.2026.118196.
short: S. von der Gracht, E. Nijholt, B. Rink, Chaos, Solitons & 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: '8918'
abstract:
- lang: eng
text: The paper deals with cyclic periodic structures modelling bladed disk assemblies
of blades with friction elements for vibration damping. These elements placed
between adjacent blades reduce the vibration amplitudes by means of dry friction
resulting from centrifugal forces acting on the elements and relative displacements
of the blades. However, the application of these friction elements results in
an additional dynamical coupling which together with mistuning of some system
parameters (e.g., blade eigenfrequency or contact parameters) may cause localization
of vibration. In the present paper a linear approximation of such a system is
investigated. The structure composed of cyclic periodic cells modelled each as
a clamped-free beam interacting with each other by means of viscoelastic elements
of complex stiffness is applied for dynamic system analysis. In case of free vibrations
as well as in case of steady-state dynamic response to a harmonic pressure field,
a perfect periodic structure and the structures with periodically mistuned parameters
(blade eigenfrequencies and contact parameters) are studied. Some regularities
in the dynamic response of the systems with mistuning have been noticed. Despite
the fact that only a linear approximation has been used, the results and conclusions
can be applied for models which describe the blade interaction in a nonlinear
way.
author:
- first_name: Tomasz
full_name: Krzyzynski, Tomasz
last_name: Krzyzynski
- first_name: Karl
full_name: Popp, Karl
last_name: Popp
- first_name: Walter
full_name: Sextro, Walter
id: '21220'
last_name: Sextro
citation:
ama: Krzyzynski T, Popp K, Sextro W. On some regularities in dynamic response of
cyclic periodic structures. Chaos, Solitons \& Fractals. 2000;11(10):1597-1609.
doi:10.1016/S0960-0779(99)00080-6
apa: Krzyzynski, T., Popp, K., & Sextro, W. (2000). On some regularities in
dynamic response of cyclic periodic structures. Chaos, Solitons \& Fractals,
11(10), 1597–1609. https://doi.org/10.1016/S0960-0779(99)00080-6
bibtex: '@article{Krzyzynski_Popp_Sextro_2000, title={On some regularities in dynamic
response of cyclic periodic structures}, volume={11}, DOI={10.1016/S0960-0779(99)00080-6},
number={10}, journal={Chaos, Solitons \& Fractals}, author={Krzyzynski, Tomasz
and Popp, Karl and Sextro, Walter}, year={2000}, pages={1597–1609} }'
chicago: 'Krzyzynski, Tomasz, Karl Popp, and Walter Sextro. “On Some Regularities
in Dynamic Response of Cyclic Periodic Structures.” Chaos, Solitons \&
Fractals 11, no. 10 (2000): 1597–1609. https://doi.org/10.1016/S0960-0779(99)00080-6.'
ieee: T. Krzyzynski, K. Popp, and W. Sextro, “On some regularities in dynamic response
of cyclic periodic structures,” Chaos, Solitons \& Fractals, vol. 11,
no. 10, pp. 1597–1609, 2000.
mla: Krzyzynski, Tomasz, et al. “On Some Regularities in Dynamic Response of Cyclic
Periodic Structures.” Chaos, Solitons \& Fractals, vol. 11, no. 10,
2000, pp. 1597–609, doi:10.1016/S0960-0779(99)00080-6.
short: T. Krzyzynski, K. Popp, W. Sextro, Chaos, Solitons \& Fractals 11 (2000)
1597–1609.
date_created: 2019-04-15T09:58:17Z
date_updated: 2022-01-06T07:04:05Z
department:
- _id: '151'
doi: 10.1016/S0960-0779(99)00080-6
intvolume: ' 11'
issue: '10'
language:
- iso: eng
page: 1597 - 1609
publication: Chaos, Solitons \& Fractals
publication_identifier:
issn:
- 0960-0779
status: public
title: On some regularities in dynamic response of cyclic periodic structures
type: journal_article
user_id: '55222'
volume: 11
year: '2000'
...
---
_id: '16614'
author:
- first_name: Rabbijah
full_name: Guder, Rabbijah
last_name: Guder
- first_name: Michael
full_name: Dellnitz, Michael
last_name: Dellnitz
- first_name: Edwin
full_name: Kreuzer, Edwin
last_name: Kreuzer
citation:
ama: Guder R, Dellnitz M, Kreuzer E. An adaptive method for the approximation of
the generalized cell mapping. Chaos, Solitons & Fractals. 1997:525-534.
doi:10.1016/s0960-0779(96)00118-x
apa: Guder, R., Dellnitz, M., & Kreuzer, E. (1997). An adaptive method for the
approximation of the generalized cell mapping. Chaos, Solitons & Fractals,
525–534. https://doi.org/10.1016/s0960-0779(96)00118-x
bibtex: '@article{Guder_Dellnitz_Kreuzer_1997, title={An adaptive method for the
approximation of the generalized cell mapping}, DOI={10.1016/s0960-0779(96)00118-x},
journal={Chaos, Solitons & Fractals}, author={Guder, Rabbijah and Dellnitz,
Michael and Kreuzer, Edwin}, year={1997}, pages={525–534} }'
chicago: Guder, Rabbijah, Michael Dellnitz, and Edwin Kreuzer. “An Adaptive Method
for the Approximation of the Generalized Cell Mapping.” Chaos, Solitons &
Fractals, 1997, 525–34. https://doi.org/10.1016/s0960-0779(96)00118-x.
ieee: R. Guder, M. Dellnitz, and E. Kreuzer, “An adaptive method for the approximation
of the generalized cell mapping,” Chaos, Solitons & Fractals, pp. 525–534,
1997.
mla: Guder, Rabbijah, et al. “An Adaptive Method for the Approximation of the Generalized
Cell Mapping.” Chaos, Solitons & Fractals, 1997, pp. 525–34, doi:10.1016/s0960-0779(96)00118-x.
short: R. Guder, M. Dellnitz, E. Kreuzer, Chaos, Solitons & Fractals (1997)
525–534.
date_created: 2020-04-16T08:05:37Z
date_updated: 2022-01-06T06:52:53Z
department:
- _id: '101'
doi: 10.1016/s0960-0779(96)00118-x
language:
- iso: eng
page: 525-534
publication: Chaos, Solitons & Fractals
publication_identifier:
issn:
- 0960-0779
publication_status: published
status: public
title: An adaptive method for the approximation of the generalized cell mapping
type: journal_article
user_id: '15701'
year: '1997'
...