---
_id: '64865'
abstract:
- lang: eng
text: We provide a method to systematically construct vector fields for which the
dynamics display transitions corresponding to a desired hierarchical connection
structure. This structure is given as a finite set of directed graphs $\mathbf{G}_1,\dotsc,\mathbf{G}_N$
(the lower level), together with another digraph $\mathbfΓ$ on $N$ vertices (the
top level). The dynamic realizations of $\mathbf{G}_1,\dotsc,\mathbf{G}_N$ are
heteroclinic networks and they can be thought of as individual connection patterns
on a given set of states. Edges in $\mathbfΓ$ correspond to transitions between
these different patterns. In our construction, the connections given through $\mathbfΓ$
are not heteroclinic, but excitable with zero threshold. This describes a dynamical
transition between two invariant sets where every $δ$-neighborhood of the first
set contains an initial condition with $ω$-limit in the second set. Thus, we prove
a theorem that allows the systematic creation of hierarchical networks that are
excitable on the top level, and heteroclinic on the lower level. Our results modify
and extend the simplex realization method by Ashwin & Postlethwaite.
author:
- first_name: Sören
full_name: von der Gracht, Sören
id: '97359'
last_name: von der Gracht
orcid: 0000-0002-8054-2058
- first_name: Alexander
full_name: Lohse, Alexander
last_name: Lohse
citation:
ama: von der Gracht S, Lohse A. Design of Hierarchical Excitable Networks. arXiv:260306157.
Published online 2026.
apa: von der Gracht, S., & Lohse, A. (2026). Design of Hierarchical Excitable
Networks. In arXiv:2603.06157.
bibtex: '@article{von der Gracht_Lohse_2026, title={Design of Hierarchical Excitable
Networks}, journal={arXiv:2603.06157}, author={von der Gracht, Sören and Lohse,
Alexander}, year={2026} }'
chicago: Gracht, Sören von der, and Alexander Lohse. “Design of Hierarchical Excitable
Networks.” ArXiv:2603.06157, 2026.
ieee: S. von der Gracht and A. Lohse, “Design of Hierarchical Excitable Networks,”
arXiv:2603.06157. 2026.
mla: von der Gracht, Sören, and Alexander Lohse. “Design of Hierarchical Excitable
Networks.” ArXiv:2603.06157, 2026.
short: S. von der Gracht, A. Lohse, ArXiv:2603.06157 (2026).
date_created: 2026-03-09T08:22:58Z
date_updated: 2026-03-09T08:26:49Z
ddc:
- '510'
department:
- _id: '101'
- _id: '841'
external_id:
arxiv:
- '2603.06157'
file:
- access_level: closed
content_type: application/pdf
creator: svdg
date_created: 2026-03-09T08:26:04Z
date_updated: 2026-03-09T08:26:04Z
file_id: '64866'
file_name: design-of-hierarchical-excitable-networks.pdf
file_size: 5179491
relation: main_file
success: 1
file_date_updated: 2026-03-09T08:26:04Z
has_accepted_license: '1'
language:
- iso: eng
publication: arXiv:2603.06157
related_material:
link:
- relation: research_paper
url: https://s-vdg.github.io/publication/design-of-hierarchical-excitable-networks/design-of-hierarchical-excitable-networks.pdf
status: public
title: Design of Hierarchical Excitable Networks
type: preprint
user_id: '97359'
year: '2026'
...
---
_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: '60048'
author:
- first_name: Raphael
full_name: Gerlach, Raphael
id: '32655'
last_name: Gerlach
orcid: 0009-0002-4750-2051
- first_name: Sören
full_name: von der Gracht, Sören
id: '97359'
last_name: von der Gracht
orcid: 0000-0002-8054-2058
- first_name: Michael
full_name: Dellnitz, Michael
last_name: Dellnitz
citation:
ama: 'Gerlach R, von der Gracht S, Dellnitz M. On the Dynamical Hierarchy in Gathering
Protocols with Circulant Topologies. In: Lecture Notes in Computer Science.
Springer Nature Switzerland; 2025. doi:10.1007/978-3-031-91736-3_19'
apa: Gerlach, R., von der Gracht, S., & Dellnitz, M. (2025). On the Dynamical
Hierarchy in Gathering Protocols with Circulant Topologies. In Lecture Notes
in Computer Science. Springer Nature Switzerland. https://doi.org/10.1007/978-3-031-91736-3_19
bibtex: '@inbook{Gerlach_von der Gracht_Dellnitz_2025, place={Cham}, title={On the Dynamical
Hierarchy in Gathering Protocols with Circulant Topologies}, DOI={10.1007/978-3-031-91736-3_19},
booktitle={Lecture Notes in Computer Science}, publisher={Springer Nature Switzerland},
author={Gerlach, Raphael and von der Gracht, Sören and Dellnitz, Michael}, year={2025}
}'
chicago: 'Gerlach, Raphael, Sören von der Gracht, and Michael Dellnitz. “On the Dynamical
Hierarchy in Gathering Protocols with Circulant Topologies.” In Lecture Notes
in Computer Science. Cham: Springer Nature Switzerland, 2025. https://doi.org/10.1007/978-3-031-91736-3_19.'
ieee: 'R. Gerlach, S. von der Gracht, and M. Dellnitz, “On the Dynamical Hierarchy
in Gathering Protocols with Circulant Topologies,” in Lecture Notes in Computer
Science, Cham: Springer Nature Switzerland, 2025.'
mla: Gerlach, Raphael, et al. “On the Dynamical Hierarchy in Gathering Protocols
with Circulant Topologies.” Lecture Notes in Computer Science, Springer
Nature Switzerland, 2025, doi:10.1007/978-3-031-91736-3_19.
short: 'R. Gerlach, S. von der Gracht, M. Dellnitz, in: Lecture Notes in Computer
Science, Springer Nature Switzerland, Cham, 2025.'
date_created: 2025-05-27T08:17:03Z
date_updated: 2025-05-27T08:22:42Z
department:
- _id: '101'
doi: 10.1007/978-3-031-91736-3_19
external_id:
arxiv:
- '2503.07576'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: ' ArXiv:2503.07576'
oa: '1'
place: Cham
project:
- _id: '106'
grant_number: '453112019'
name: 'Algorithmen für Schwarmrobotik: Verteiltes Rechnen trifft Dynamische Systeme'
publication: Lecture Notes in Computer Science
publication_identifier:
isbn:
- '9783031917356'
- '9783031917363'
issn:
- 0302-9743
- 1611-3349
publication_status: published
publisher: Springer Nature Switzerland
status: public
title: On the Dynamical Hierarchy in Gathering Protocols with Circulant Topologies
type: book_chapter
user_id: '32655'
year: '2025'
...
---
_id: '56298'
abstract:
- lang: eng
text: "In the general pattern formation (GPF) problem, a swarm of simple autonomous,\r\ndisoriented
robots must form a given pattern. The robots' simplicity imply a\r\nstrong limitation:
When the initial configuration is rotationally symmetric,\r\nonly patterns with
a similar symmetry can be formed [Yamashita, Suzyuki; TCS\r\n2010]. The only known
algorithm to form large patterns with limited visibility\r\nand without memory
requires the robots to start in a near-gathering (a swarm of\r\nconstant diameter)
[Hahn et al.; SAND 2024]. However, not only do we not know\r\nany near-gathering
algorithm guaranteed to preserve symmetry but most natural\r\ngathering strategies
trivially increase symmetries [Castenow et al.; OPODIS\r\n2022].\r\n Thus, we
study near-gathering without changing the swarm's rotational\r\nsymmetry for disoriented,
oblivious robots with limited visibility (the\r\nOBLOT-model, see [Flocchini et
al.; 2019]). We introduce a technique based on\r\nthe theory of dynamical systems
to analyze how a given algorithm affects\r\nsymmetry and provide sufficient conditions
for symmetry preservation. Until\r\nnow, it was unknown whether the considered
OBLOT-model allows for any\r\nnon-trivial algorithm that always preserves symmetry.
Our first result shows\r\nthat a variant of Go-to-the-Average always preserves
symmetry but may sometimes\r\nlead to multiple, unconnected near-gathering clusters.
Our second result is a\r\nsymmetry-preserving near-gathering algorithm that works
on swarms with a convex\r\nboundary (the outer boundary of the unit disc graph)
and without holes (circles\r\nof diameter 1 inside the boundary without any robots)."
author:
- first_name: Raphael
full_name: Gerlach, Raphael
id: '32655'
last_name: Gerlach
orcid: 0009-0002-4750-2051
- first_name: Sören
full_name: von der Gracht, Sören
id: '97359'
last_name: von der Gracht
orcid: 0000-0002-8054-2058
- first_name: Christopher
full_name: Hahn, Christopher
last_name: Hahn
- first_name: Jonas
full_name: Harbig, Jonas
id: '47213'
last_name: Harbig
- first_name: Peter
full_name: Kling, Peter
last_name: Kling
citation:
ama: 'Gerlach R, von der Gracht S, Hahn C, Harbig J, Kling P. Symmetry Preservation
in Swarms of Oblivious Robots with Limited Visibility. In: Bonomi S, Galletta
L, Rivière Etienne, Schiavoni Valerio, eds. 28th International Conference
on Principles of Distributed Systems (OPODIS 2024). Vol 324. Leibniz International
Proceedings in Informatics (LIPIcs). Schloss Dagstuhl -- Leibniz-Zentrum für Informatik;
2025. doi:10.4230/LIPIcs.OPODIS.2024.13'
apa: Gerlach, R., von der Gracht, S., Hahn, C., Harbig, J., & Kling, P. (2025).
Symmetry Preservation in Swarms of Oblivious Robots with Limited Visibility.
In S. Bonomi, L. Galletta, Etienne Rivière, & Valerio Schiavoni (Eds.),
28th International Conference on Principles of Distributed Systems (OPODIS
2024) (Vol. 324). Schloss Dagstuhl -- Leibniz-Zentrum für Informatik. https://doi.org/10.4230/LIPIcs.OPODIS.2024.13
bibtex: '@inproceedings{Gerlach_von der Gracht_Hahn_Harbig_Kling_2025, series={Leibniz
International Proceedings in Informatics (LIPIcs)}, title={Symmetry Preservation
in Swarms of Oblivious Robots with Limited Visibility}, volume={324}, DOI={10.4230/LIPIcs.OPODIS.2024.13},
booktitle={28th International Conference on Principles of Distributed Systems
(OPODIS 2024)}, publisher={Schloss Dagstuhl -- Leibniz-Zentrum für Informatik},
author={Gerlach, Raphael and von der Gracht, Sören and Hahn, Christopher and Harbig,
Jonas and Kling, Peter}, editor={Bonomi, Silvia and Galletta, Letterio and Rivière, Etienne
and Schiavoni, Valerio}, year={2025}, collection={Leibniz International Proceedings
in Informatics (LIPIcs)} }'
chicago: Gerlach, Raphael, Sören von der Gracht, Christopher Hahn, Jonas Harbig,
and Peter Kling. “Symmetry Preservation in Swarms of Oblivious Robots with Limited
Visibility.” In 28th International Conference on Principles of Distributed
Systems (OPODIS 2024), edited by Silvia Bonomi, Letterio Galletta, Etienne
Rivière, and Valerio Schiavoni, Vol. 324. Leibniz International Proceedings in
Informatics (LIPIcs). Schloss Dagstuhl -- Leibniz-Zentrum für Informatik, 2025.
https://doi.org/10.4230/LIPIcs.OPODIS.2024.13.
ieee: 'R. Gerlach, S. von der Gracht, C. Hahn, J. Harbig, and P. Kling, “Symmetry
Preservation in Swarms of Oblivious Robots with Limited Visibility,” in 28th
International Conference on Principles of Distributed Systems (OPODIS 2024),
Lucca, Italy, 2025, vol. 324, doi: 10.4230/LIPIcs.OPODIS.2024.13.'
mla: Gerlach, Raphael, et al. “Symmetry Preservation in Swarms of Oblivious Robots
with Limited Visibility.” 28th International Conference on Principles of Distributed
Systems (OPODIS 2024), edited by Silvia Bonomi et al., vol. 324, Schloss Dagstuhl
-- Leibniz-Zentrum für Informatik, 2025, doi:10.4230/LIPIcs.OPODIS.2024.13.
short: 'R. Gerlach, S. von der Gracht, C. Hahn, J. Harbig, P. Kling, in: S. Bonomi,
L. Galletta, Etienne Rivière, Valerio Schiavoni (Eds.), 28th International Conference
on Principles of Distributed Systems (OPODIS 2024), Schloss Dagstuhl -- Leibniz-Zentrum
für Informatik, 2025.'
conference:
end_date: 2024-12-13
location: Lucca, Italy
name: 28th International Conference on Principles of Distributed Systems (OPODIS
2024)
start_date: 2024-12-11
date_created: 2024-10-01T13:29:43Z
date_updated: 2025-01-09T11:39:19Z
department:
- _id: '101'
doi: 10.4230/LIPIcs.OPODIS.2024.13
editor:
- first_name: Silvia
full_name: Bonomi, Silvia
last_name: Bonomi
- first_name: Letterio
full_name: Galletta, Letterio
last_name: Galletta
- first_name: ' Etienne'
full_name: Rivière, Etienne
last_name: Rivière
- first_name: ' Valerio'
full_name: Schiavoni, Valerio
last_name: Schiavoni
external_id:
arxiv:
- '2409.19277'
intvolume: ' 324'
keyword:
- Swarm Algorithm
- Swarm Robots
- Distributed Algorithm
- Pattern Formation
- Limited Visibility
- Oblivious
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/abs/2409.19277
oa: '1'
project:
- _id: '106'
grant_number: '453112019'
name: 'Algorithmen für Schwarmrobotik: Verteiltes Rechnen trifft Dynamische Systeme'
publication: 28th International Conference on Principles of Distributed Systems (OPODIS
2024)
publication_identifier:
isbn:
- 978-3-95977-360-7
issn:
- 1868-8969
publication_status: published
publisher: Schloss Dagstuhl -- Leibniz-Zentrum für Informatik
series_title: Leibniz International Proceedings in Informatics (LIPIcs)
status: public
title: Symmetry Preservation in Swarms of Oblivious Robots with Limited Visibility
type: conference
user_id: '97359'
volume: 324
year: '2025'
...
---
_id: '58953'
abstract:
- lang: eng
text: In this article, we investigate symmetry properties of distributed systems
of mobile robots. We consider a swarm of n robots in the OBLOT model and analyze
their collective Fsync dynamics using of equivariant dynamical systems theory.
To this end, we show that the corresponding evolution function commutes with rotational
and reflective transformations of R^2. These form a group that is isomorphic to
O(2) x S_n, the product group of the orthogonal group and the permutation on n
elements. The theory of equivariant dynamical systems is used to deduce a hierarchy
along which symmetries of a robot swarm can potentially increase following an
arbitrary protocol. By decoupling the Look phase from the Compute and Move phases
in the mathematical description of an LCM cycle, this hierarchy can be characterized
in terms of automorphisms of connectivity graphs. In particular, we find all possible
types of symmetry increase, if the decoupled Compute and Move phase is invertible.
Finally, we apply our results to protocols which induce state-dependent linear
dynamics, where the reduced system consisting of only the Compute and Move phase
is linear.
author:
- first_name: Raphael
full_name: Gerlach, Raphael
id: '32655'
last_name: Gerlach
orcid: 0009-0002-4750-2051
- first_name: Sören
full_name: von der Gracht, Sören
id: '97359'
last_name: von der Gracht
orcid: 0000-0002-8054-2058
citation:
ama: Gerlach R, von der Gracht S. Analyzing Symmetries of Swarms of Mobile Robots
Using Equivariant Dynamical Systems. arXiv:250307576. Published online
2025.
apa: Gerlach, R., & von der Gracht, S. (2025). Analyzing Symmetries of Swarms
of Mobile Robots Using Equivariant Dynamical Systems. In arXiv:2503.07576.
bibtex: '@article{Gerlach_von der Gracht_2025, title={Analyzing Symmetries of Swarms
of Mobile Robots Using Equivariant Dynamical Systems}, journal={arXiv:2503.07576},
author={Gerlach, Raphael and von der Gracht, Sören}, year={2025} }'
chicago: Gerlach, Raphael, and Sören von der Gracht. “Analyzing Symmetries of Swarms
of Mobile Robots Using Equivariant Dynamical Systems.” ArXiv:2503.07576,
2025.
ieee: R. Gerlach and S. von der Gracht, “Analyzing Symmetries of Swarms of Mobile
Robots Using Equivariant Dynamical Systems,” arXiv:2503.07576. 2025.
mla: Gerlach, Raphael, and Sören von der Gracht. “Analyzing Symmetries of Swarms
of Mobile Robots Using Equivariant Dynamical Systems.” ArXiv:2503.07576,
2025.
short: R. Gerlach, S. von der Gracht, ArXiv:2503.07576 (2025).
date_created: 2025-03-11T08:21:05Z
date_updated: 2025-03-11T08:53:02Z
ddc:
- '004'
department:
- _id: '101'
external_id:
arxiv:
- '2503.07576'
file:
- access_level: open_access
content_type: application/pdf
creator: svdg
date_created: 2025-03-11T08:27:32Z
date_updated: 2025-03-11T08:27:32Z
file_id: '58954'
file_name: Analyzing_Symmetries_of_Swarms_of_Mobile_Robots_Using_Equivariant_Dynamical_Systems.pdf
file_size: 812198
relation: main_file
file_date_updated: 2025-03-11T08:27:32Z
has_accepted_license: '1'
keyword:
- dynamical systems
- coupled systems
- distributed computing
- robot swarms
- autonomous mobile robots
- symmetry
- equivariant dynamics
language:
- iso: eng
oa: '1'
page: '23'
project:
- _id: '106'
grant_number: '453112019'
name: 'Algorithmen für Schwarmrobotik: Verteiltes Rechnen trifft Dynamische Systeme'
publication: arXiv:2503.07576
status: public
title: Analyzing Symmetries of Swarms of Mobile Robots Using Equivariant Dynamical
Systems
type: preprint
user_id: '97359'
year: '2025'
...
---
_id: '52726'
abstract:
- lang: eng
text: Heteroclinic structures organize global features of dynamical systems. We
analyse whether heteroclinic structures can arise in network dynamics with higher-order
interactions which describe the nonlinear interactions between three or more units.
We find that while commonly analysed model equations such as network dynamics
on undirected hypergraphs may be useful to describe local dynamics such as cluster
synchronization, they give rise to obstructions that allow to design of heteroclinic
structures in phase space. By contrast, directed hypergraphs break the homogeneity
and lead to vector fields that support heteroclinic structures.
article_type: original
author:
- first_name: Christian
full_name: Bick, Christian
last_name: Bick
- first_name: Sören
full_name: von der Gracht, Sören
id: '97359'
last_name: von der Gracht
orcid: 0000-0002-8054-2058
citation:
ama: Bick C, von der Gracht S. Heteroclinic dynamics in network dynamical systems
with higher-order interactions. Journal of Complex Networks. 2024;12(2).
doi:10.1093/comnet/cnae009
apa: Bick, C., & von der Gracht, S. (2024). Heteroclinic dynamics in network
dynamical systems with higher-order interactions. Journal of Complex Networks,
12(2). https://doi.org/10.1093/comnet/cnae009
bibtex: '@article{Bick_von der Gracht_2024, title={Heteroclinic dynamics in network
dynamical systems with higher-order interactions}, volume={12}, DOI={10.1093/comnet/cnae009},
number={2}, journal={Journal of Complex Networks}, publisher={Oxford University
Press (OUP)}, author={Bick, Christian and von der Gracht, Sören}, year={2024}
}'
chicago: Bick, Christian, and Sören von der Gracht. “Heteroclinic Dynamics in Network
Dynamical Systems with Higher-Order Interactions.” Journal of Complex Networks
12, no. 2 (2024). https://doi.org/10.1093/comnet/cnae009.
ieee: 'C. Bick and S. von der Gracht, “Heteroclinic dynamics in network dynamical
systems with higher-order interactions,” Journal of Complex Networks, vol.
12, no. 2, 2024, doi: 10.1093/comnet/cnae009.'
mla: Bick, Christian, and Sören von der Gracht. “Heteroclinic Dynamics in Network
Dynamical Systems with Higher-Order Interactions.” Journal of Complex Networks,
vol. 12, no. 2, Oxford University Press (OUP), 2024, doi:10.1093/comnet/cnae009.
short: C. Bick, S. von der Gracht, Journal of Complex Networks 12 (2024).
date_created: 2024-03-22T09:04:57Z
date_updated: 2024-03-22T09:11:53Z
ddc:
- '510'
department:
- _id: '101'
doi: 10.1093/comnet/cnae009
external_id:
arxiv:
- '2309.02006'
file:
- access_level: closed
content_type: application/pdf
creator: svdg
date_created: 2024-03-22T09:06:07Z
date_updated: 2024-03-22T09:06:07Z
file_id: '52728'
file_name: heteroclinic-dynamics-in-network-dynamical-systems-with-higher-order-interactions.pdf
file_size: 649155
relation: main_file
success: 1
file_date_updated: 2024-03-22T09:06:07Z
has_accepted_license: '1'
intvolume: ' 12'
issue: '2'
keyword:
- Applied Mathematics
- Computational Mathematics
- Control and Optimization
- Management Science and Operations Research
- Computer Networks and Communications
language:
- iso: eng
license: https://creativecommons.org/licenses/by-nc/4.0/
main_file_link:
- open_access: '1'
url: https://academic.oup.com/comnet/article-pdf/12/2/cnae009/56832119/cnae009.pdf
oa: '1'
publication: Journal of Complex Networks
publication_identifier:
issn:
- 2051-1329
publication_status: published
publisher: Oxford University Press (OUP)
status: public
title: Heteroclinic dynamics in network dynamical systems with higher-order interactions
type: journal_article
user_id: '97359'
volume: 12
year: '2024'
...
---
_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. Proceedings of the Royal Society A: Mathematical, Physical
and Engineering Sciences. 2024;480(2301). doi:10.1098/rspa.2023.0945'
apa: 'von der Gracht, S., Nijholt, E., & Rink, B. (2024). Higher-order interactions
lead to ‘reluctant’ synchrony breaking. Proceedings of the Royal Society A:
Mathematical, Physical and Engineering Sciences, 480(2301). https://doi.org/10.1098/rspa.2023.0945'
bibtex: '@article{von der Gracht_Nijholt_Rink_2024, title={Higher-order interactions
lead to ‘reluctant’ synchrony breaking}, volume={480}, DOI={10.1098/rspa.2023.0945},
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.” Proceedings of the Royal Society A:
Mathematical, Physical and Engineering Sciences 480, no. 2301 (2024). https://doi.org/10.1098/rspa.2023.0945.'
ieee: 'S. von der Gracht, E. Nijholt, and B. Rink, “Higher-order interactions lead
to ‘reluctant’ synchrony breaking,” Proceedings of the Royal Society A: Mathematical,
Physical and Engineering Sciences, vol. 480, no. 2301, 2024, doi: 10.1098/rspa.2023.0945.'
mla: 'von der Gracht, Sören, et al. “Higher-Order Interactions Lead to ‘Reluctant’
Synchrony Breaking.” Proceedings of the Royal Society A: Mathematical, Physical
and Engineering Sciences, vol. 480, no. 2301, The Royal Society, 2024, doi:10.1098/rspa.2023.0945.'
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'
...
---
_id: '49326'
abstract:
- lang: eng
text: Many networked systems are governed by non-pairwise interactions between nodes.
The resulting higher-order interaction structure can then be encoded by means
of a hypernetwork. In this paper we consider dynamical systems on hypernetworks
by defining a class of admissible maps for every such hypernetwork. We explain
how to classify robust cluster synchronization patterns on hypernetworks by finding
balanced partitions, and we generalize the concept of a graph fibration to the
hypernetwork context. We also show that robust synchronization patterns are only
fully determined by polynomial admissible maps of high order. This means that,
unlike in dyadic networks, cluster synchronization on hypernetworks is a higher-order,
i.e., nonlinear, effect. We give a formula, in terms of the order of the hypernetwork,
for the degree of the polynomial admissible maps that determine robust synchronization
patterns. We also demonstrate that this degree is optimal by investigating a class
of examples. We conclude by demonstrating how this effect may cause remarkable
synchrony breaking bifurcations that occur at high polynomial degree.
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. Hypernetworks: Cluster Synchronization
Is a Higher-Order Effect. SIAM Journal on Applied Mathematics. 2023;83(6):2329-2353.
doi:10.1137/23m1561075'
apa: 'von der Gracht, S., Nijholt, E., & Rink, B. (2023). Hypernetworks: Cluster
Synchronization Is a Higher-Order Effect. SIAM Journal on Applied Mathematics,
83(6), 2329–2353. https://doi.org/10.1137/23m1561075'
bibtex: '@article{von der Gracht_Nijholt_Rink_2023, title={Hypernetworks: Cluster
Synchronization Is a Higher-Order Effect}, volume={83}, DOI={10.1137/23m1561075},
number={6}, journal={SIAM Journal on Applied Mathematics}, publisher={Society
for Industrial & Applied Mathematics (SIAM)}, author={von der Gracht, Sören
and Nijholt, Eddie and Rink, Bob}, year={2023}, pages={2329–2353} }'
chicago: 'Gracht, Sören von der, Eddie Nijholt, and Bob Rink. “Hypernetworks: Cluster
Synchronization Is a Higher-Order Effect.” SIAM Journal on Applied Mathematics
83, no. 6 (2023): 2329–53. https://doi.org/10.1137/23m1561075.'
ieee: 'S. von der Gracht, E. Nijholt, and B. Rink, “Hypernetworks: Cluster Synchronization
Is a Higher-Order Effect,” SIAM Journal on Applied Mathematics, vol. 83,
no. 6, pp. 2329–2353, 2023, doi: 10.1137/23m1561075.'
mla: 'von der Gracht, Sören, et al. “Hypernetworks: Cluster Synchronization Is a
Higher-Order Effect.” SIAM Journal on Applied Mathematics, vol. 83, no.
6, Society for Industrial & Applied Mathematics (SIAM), 2023, pp. 2329–53,
doi:10.1137/23m1561075.'
short: S. von der Gracht, E. Nijholt, B. Rink, SIAM Journal on Applied Mathematics
83 (2023) 2329–2353.
date_created: 2023-11-29T10:50:05Z
date_updated: 2023-11-29T10:52:23Z
department:
- _id: '101'
doi: 10.1137/23m1561075
external_id:
arxiv:
- '2302.08974'
intvolume: ' 83'
issue: '6'
keyword:
- Applied Mathematics
language:
- iso: eng
page: 2329-2353
publication: SIAM Journal on Applied Mathematics
publication_identifier:
issn:
- 0036-1399
- 1095-712X
publication_status: published
publisher: Society for Industrial & Applied Mathematics (SIAM)
status: public
title: 'Hypernetworks: Cluster Synchronization Is a Higher-Order Effect'
type: journal_article
user_id: '97359'
volume: 83
year: '2023'
...
---
_id: '45498'
abstract:
- lang: eng
text: "We present a novel method for high-order phase reduction in networks of\r\nweakly
coupled oscillators and, more generally, perturbations of reducible\r\nnormally
hyperbolic (quasi-)periodic tori. Our method works by computing an\r\nasymptotic
expansion for an embedding of the perturbed invariant torus, as well\r\nas for
the reduced phase dynamics in local coordinates. Both can be determined\r\nto
arbitrary degrees of accuracy, and we show that the phase dynamics may\r\ndirectly
be obtained in normal form. We apply the method to predict remote\r\nsynchronisation
in a chain of coupled Stuart-Landau oscillators."
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. A parametrisation method for high-order
phase reduction in coupled oscillator networks. arXiv:230603320.
apa: von der Gracht, S., Nijholt, E., & Rink, B. (n.d.). A parametrisation method
for high-order phase reduction in coupled oscillator networks. In arXiv:2306.03320.
bibtex: '@article{von der Gracht_Nijholt_Rink, title={A parametrisation method for
high-order phase reduction in coupled oscillator networks}, journal={arXiv:2306.03320},
author={von der Gracht, Sören and Nijholt, Eddie and Rink, Bob} }'
chicago: Gracht, Sören von der, Eddie Nijholt, and Bob Rink. “A Parametrisation
Method for High-Order Phase Reduction in Coupled Oscillator Networks.” ArXiv:2306.03320,
n.d.
ieee: S. von der Gracht, E. Nijholt, and B. Rink, “A parametrisation method for
high-order phase reduction in coupled oscillator networks,” arXiv:2306.03320.
.
mla: von der Gracht, Sören, et al. “A Parametrisation Method for High-Order Phase
Reduction in Coupled Oscillator Networks.” ArXiv:2306.03320.
short: S. von der Gracht, E. Nijholt, B. Rink, ArXiv:2306.03320 (n.d.).
date_created: 2023-06-07T07:57:28Z
date_updated: 2023-06-07T07:59:06Z
department:
- _id: '101'
external_id:
arxiv:
- '2306.03320'
language:
- iso: eng
main_file_link:
- url: https://arxiv.org/pdf/2306.03320
page: '29'
publication: arXiv:2306.03320
publication_status: submitted
status: public
title: A parametrisation method for high-order phase reduction in coupled oscillator
networks
type: preprint
user_id: '97359'
year: '2023'
...
---
_id: '33264'
abstract:
- lang: eng
text: We investigate bifurcations in feedforward coupled cell networks. Feedforward
structure (the absence of feedback) can be defined by a partial order on the cells.
We use this property to study generic one-parameter steady state bifurcations
for such networks. Branching solutions and their asymptotics are described in
terms of Taylor coefficients of the internal dynamics. They can be determined
via an algorithm that only exploits the network structure. Similar to previous
results on feedforward chains, we observe amplifications of the growth rates of
steady state branches induced by the feedforward structure. However, contrary
to these earlier results, as the interaction scenarios can be more complicated
in general feedforward networks, different branching patterns and different amplifications
can occur for different regions in the space of Taylor coefficients.
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. Amplified steady state bifurcations in
feedforward networks. Nonlinearity. 2022;35(4):2073-2120. doi:10.1088/1361-6544/ac5463
apa: von der Gracht, S., Nijholt, E., & Rink, B. (2022). Amplified steady state
bifurcations in feedforward networks. Nonlinearity, 35(4), 2073–2120.
https://doi.org/10.1088/1361-6544/ac5463
bibtex: '@article{von der Gracht_Nijholt_Rink_2022, title={Amplified steady state
bifurcations in feedforward networks}, volume={35}, DOI={10.1088/1361-6544/ac5463},
number={4}, journal={Nonlinearity}, publisher={IOP Publishing}, author={von der
Gracht, Sören and Nijholt, Eddie and Rink, Bob}, year={2022}, pages={2073–2120}
}'
chicago: 'Gracht, Sören von der, Eddie Nijholt, and Bob Rink. “Amplified Steady
State Bifurcations in Feedforward Networks.” Nonlinearity 35, no. 4 (2022):
2073–2120. https://doi.org/10.1088/1361-6544/ac5463.'
ieee: 'S. von der Gracht, E. Nijholt, and B. Rink, “Amplified steady state bifurcations
in feedforward networks,” Nonlinearity, vol. 35, no. 4, pp. 2073–2120,
2022, doi: 10.1088/1361-6544/ac5463.'
mla: von der Gracht, Sören, et al. “Amplified Steady State Bifurcations in Feedforward
Networks.” Nonlinearity, vol. 35, no. 4, IOP Publishing, 2022, pp. 2073–120,
doi:10.1088/1361-6544/ac5463.
short: S. von der Gracht, E. Nijholt, B. Rink, Nonlinearity 35 (2022) 2073–2120.
date_created: 2022-09-06T11:38:15Z
date_updated: 2022-09-07T08:36:46Z
doi: 10.1088/1361-6544/ac5463
extern: '1'
external_id:
arxiv:
- '2105.02547'
intvolume: ' 35'
issue: '4'
keyword:
- Applied Mathematics
- General Physics and Astronomy
- Mathematical Physics
- Statistical and Nonlinear Physics
language:
- iso: eng
page: 2073-2120
publication: Nonlinearity
publication_identifier:
issn:
- 0951-7715
- 1361-6544
publication_status: published
publisher: IOP Publishing
status: public
title: Amplified steady state bifurcations in feedforward networks
type: journal_article
user_id: '97359'
volume: 35
year: '2022'
...
---
_id: '33273'
abstract:
- lang: ger
text: "Dieses Lernangebot widmet sich der linearen Algebra als dem Teil der Mathematik,
der neben der Optimierung und der Stochastik die Grundlage für praktisch alle
Entwicklungen im Bereich Künstliche Intelligenz (KI) darstellt. Das Fach ist jedoch
für Anfänger meist ungewohnt abstrakt und wird daher oft als besonders schwierig
und unanschaulich empfunden. In diesem Kurs wird das Erlernen mathematischer Kenntnisse
in linearer Algebra verknüpft mit dem aktuellen und faszinierenden Anwendungsfeld
der künstlichen neuronalen Netze (KNN). Daraus ergeben sich in natürlicher Weise
Anwendungsbeispiele, an denen die wesentlichen Konzepte der linearen Algebra erklärt
werden können.\r\n\r\nBehandelte Themen sind:\r\n\r\n Der Vektorraum der reellen
Zahlentupel, reelle Vektorräume allgemein\r\n Lineare Abbildungen\r\n Matrizen\r\n
\ Koordinaten und darstellende Matrizen\r\n Lineare Gleichungssysteme, Gaußalgorithmus\r\n
\ Determinante\r\n Ein Ausblick auf nichtlineare Techniken, die für neuronale
Netzwerke relevant sind."
author:
- first_name: Thomas
full_name: Schramm, Thomas
last_name: Schramm
- first_name: Ingenuin
full_name: Gasser, Ingenuin
last_name: Gasser
- first_name: Sören
full_name: Schwenker, Sören
id: '97359'
last_name: Schwenker
orcid: 0000-0002-8054-2058
- first_name: Ruedi
full_name: Seiler, Ruedi
last_name: Seiler
- first_name: Alexander
full_name: Lohse, Alexander
last_name: Lohse
- first_name: Kay
full_name: Zobel, Kay
last_name: Zobel
citation:
ama: Schramm T, Gasser I, Schwenker S, Seiler R, Lohse A, Zobel K. Linear Algebra
Driven by Data Science. Hamburg Open Online University; 2020.
apa: Schramm, T., Gasser, I., Schwenker, S., Seiler, R., Lohse, A., & Zobel,
K. (2020). Linear Algebra driven by Data Science. Hamburg Open Online University.
bibtex: '@book{Schramm_Gasser_Schwenker_Seiler_Lohse_Zobel_2020, title={Linear Algebra
driven by Data Science}, publisher={Hamburg Open Online University}, author={Schramm,
Thomas and Gasser, Ingenuin and Schwenker, Sören and Seiler, Ruedi and Lohse,
Alexander and Zobel, Kay}, year={2020} }'
chicago: Schramm, Thomas, Ingenuin Gasser, Sören Schwenker, Ruedi Seiler, Alexander
Lohse, and Kay Zobel. Linear Algebra Driven by Data Science. Hamburg Open
Online University, 2020.
ieee: T. Schramm, I. Gasser, S. Schwenker, R. Seiler, A. Lohse, and K. Zobel, Linear
Algebra driven by Data Science. Hamburg Open Online University, 2020.
mla: Schramm, Thomas, et al. Linear Algebra Driven by Data Science. Hamburg
Open Online University, 2020.
short: T. Schramm, I. Gasser, S. Schwenker, R. Seiler, A. Lohse, K. Zobel, Linear
Algebra Driven by Data Science, Hamburg Open Online University, 2020.
date_created: 2022-09-06T12:06:41Z
date_updated: 2022-09-06T14:05:13Z
department:
- _id: '94'
extern: '1'
language:
- iso: eng
main_file_link:
- url: https://www.hoou.de/projects/linear-algebra-driven-by-data-science/
publisher: Hamburg Open Online University
status: public
title: Linear Algebra driven by Data Science
type: misc
user_id: '97359'
year: '2020'
...
---
_id: '33272'
author:
- first_name: Eddie
full_name: Nijholt, Eddie
last_name: Nijholt
- first_name: Bob
full_name: Rink, Bob
last_name: Rink
- first_name: Sören
full_name: Schwenker, Sören
id: '97359'
last_name: Schwenker
orcid: 0000-0002-8054-2058
citation:
ama: Nijholt E, Rink B, Schwenker S. Generalised Symmetry in Network Dynamics. DSWeb.
2020.
apa: Nijholt, E., Rink, B., & Schwenker, S. (2020). Generalised Symmetry in
Network Dynamics. DSWeb, April.
bibtex: '@article{Nijholt_Rink_Schwenker_2020, title={Generalised Symmetry in Network
Dynamics}, number={April}, journal={DSWeb}, author={Nijholt, Eddie and Rink, Bob
and Schwenker, Sören}, year={2020} }'
chicago: Nijholt, Eddie, Bob Rink, and Sören Schwenker. “Generalised Symmetry in
Network Dynamics.” DSWeb, 2020.
ieee: E. Nijholt, B. Rink, and S. Schwenker, “Generalised Symmetry in Network Dynamics,”
DSWeb, no. April, 2020.
mla: Nijholt, Eddie, et al. “Generalised Symmetry in Network Dynamics.” DSWeb,
no. April, 2020.
short: E. Nijholt, B. Rink, S. Schwenker, DSWeb (2020).
date_created: 2022-09-06T11:46:58Z
date_updated: 2022-09-06T14:05:35Z
extern: '1'
issue: April
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://dsweb.siam.org/The-Magazine/Article/generalised-symmetry-in-network-dynamics
oa: '1'
publication: DSWeb
publication_date: 2020-04.30
status: public
title: Generalised Symmetry in Network Dynamics
type: newspaper_article
user_id: '97359'
year: '2020'
...
---
_id: '33262'
abstract:
- lang: eng
text: The authors of Berg et al. [J. Algebra 348 (2011) 446–461] provide an algorithm
for finding a complete system of primitive orthogonal idempotents for CM, where
M is any finite R-trivial monoid. Their method relies on a technical result stating
that R-trivial monoid are equivalent to so-called weakly ordered monoids. We provide
an alternative algorithm, based only on the simple observation that an R-trivial
monoid may be realized by upper triangular matrices. This approach is inspired
by results in the field of coupled cell network dynamical systems, where L-trivial
monoids (the opposite notion) correspond to so-called feed-forward networks. We
first show that our algorithm works for ZM, after which we prove that it also
works for RM where R is an arbitrary ring with a known complete system of primitive
orthogonal idempotents. In particular, our algorithm works if R is any field.
In this respect our result constitutes a considerable generalization of the results
in Berg et al. [J. Algebra 348 (2011) 446–461]. Moreover, the system of idempotents
for RM is obtained from the one our algorithm yields for ZM in a straightforward
manner. In other words, for any finite R-trivial monoid M our algorithm only has
to be performed for ZM, after which a system of idempotents follows for any ring
with a given system of idempotents.
author:
- first_name: Eddie
full_name: Nijholt, Eddie
last_name: Nijholt
- first_name: Bob
full_name: Rink, Bob
last_name: Rink
- first_name: Sören
full_name: Schwenker, Sören
id: '97359'
last_name: Schwenker
orcid: 0000-0002-8054-2058
citation:
ama: Nijholt E, Rink B, Schwenker S. A new algorithm for computing idempotents of
ℛ-trivial monoids. Journal of Algebra and Its Applications. 2020;20(12).
doi:10.1142/s0219498821502273
apa: Nijholt, E., Rink, B., & Schwenker, S. (2020). A new algorithm for computing
idempotents of ℛ-trivial monoids. Journal of Algebra and Its Applications,
20(12). https://doi.org/10.1142/s0219498821502273
bibtex: '@article{Nijholt_Rink_Schwenker_2020, title={A new algorithm for computing
idempotents of ℛ-trivial monoids}, volume={20}, DOI={10.1142/s0219498821502273},
number={12}, journal={Journal of Algebra and Its Applications}, publisher={World
Scientific Pub Co Pte Ltd}, author={Nijholt, Eddie and Rink, Bob and Schwenker,
Sören}, year={2020} }'
chicago: Nijholt, Eddie, Bob Rink, and Sören Schwenker. “A New Algorithm for Computing
Idempotents of ℛ-Trivial Monoids.” Journal of Algebra and Its Applications
20, no. 12 (2020). https://doi.org/10.1142/s0219498821502273.
ieee: 'E. Nijholt, B. Rink, and S. Schwenker, “A new algorithm for computing idempotents
of ℛ-trivial monoids,” Journal of Algebra and Its Applications, vol. 20,
no. 12, 2020, doi: 10.1142/s0219498821502273.'
mla: Nijholt, Eddie, et al. “A New Algorithm for Computing Idempotents of ℛ-Trivial
Monoids.” Journal of Algebra and Its Applications, vol. 20, no. 12, World
Scientific Pub Co Pte Ltd, 2020, doi:10.1142/s0219498821502273.
short: E. Nijholt, B. Rink, S. Schwenker, Journal of Algebra and Its Applications
20 (2020).
date_created: 2022-09-06T11:37:00Z
date_updated: 2022-09-07T08:35:24Z
doi: 10.1142/s0219498821502273
extern: '1'
external_id:
arxiv:
- '1906.02844'
intvolume: ' 20'
issue: '12'
keyword:
- Applied Mathematics
- Algebra and Number Theory
language:
- iso: eng
publication: Journal of Algebra and Its Applications
publication_identifier:
issn:
- 0219-4988
- 1793-6829
publication_status: published
publisher: World Scientific Pub Co Pte Ltd
status: public
title: A new algorithm for computing idempotents of ℛ-trivial monoids
type: journal_article
user_id: '97359'
volume: 20
year: '2020'
...
---
_id: '33263'
abstract:
- lang: eng
text: Dynamical systems often admit geometric properties that must be taken into
account when studying their behavior. We show that many such properties can be
encoded by means of quiver representations. These properties include classical
symmetry, hidden symmetry, and feedforward structure, as well as subnetwork and
quotient relations in network dynamical systems. A quiver equivariant dynamical
system consists of a collection of dynamical systems with maps between them that
send solutions to solutions. We prove that such quiver structures are preserved
under Lyapunov--Schmidt reduction, center manifold reduction, and normal form
reduction.
author:
- first_name: Eddie
full_name: Nijholt, Eddie
last_name: Nijholt
- first_name: Bob W.
full_name: Rink, Bob W.
last_name: Rink
- first_name: Sören
full_name: Schwenker, Sören
id: '97359'
last_name: Schwenker
orcid: 0000-0002-8054-2058
citation:
ama: Nijholt E, Rink BW, Schwenker S. Quiver Representations and Dimension Reduction
in Dynamical Systems. SIAM Journal on Applied Dynamical Systems. 2020;19(4):2428-2468.
doi:10.1137/20m1345670
apa: Nijholt, E., Rink, B. W., & Schwenker, S. (2020). Quiver Representations
and Dimension Reduction in Dynamical Systems. SIAM Journal on Applied Dynamical
Systems, 19(4), 2428–2468. https://doi.org/10.1137/20m1345670
bibtex: '@article{Nijholt_Rink_Schwenker_2020, title={Quiver Representations and
Dimension Reduction in Dynamical Systems}, volume={19}, DOI={10.1137/20m1345670},
number={4}, journal={SIAM Journal on Applied Dynamical Systems}, publisher={Society
for Industrial & Applied Mathematics (SIAM)}, author={Nijholt, Eddie and Rink,
Bob W. and Schwenker, Sören}, year={2020}, pages={2428–2468} }'
chicago: 'Nijholt, Eddie, Bob W. Rink, and Sören Schwenker. “Quiver Representations
and Dimension Reduction in Dynamical Systems.” SIAM Journal on Applied Dynamical
Systems 19, no. 4 (2020): 2428–68. https://doi.org/10.1137/20m1345670.'
ieee: 'E. Nijholt, B. W. Rink, and S. Schwenker, “Quiver Representations and Dimension
Reduction in Dynamical Systems,” SIAM Journal on Applied Dynamical Systems,
vol. 19, no. 4, pp. 2428–2468, 2020, doi: 10.1137/20m1345670.'
mla: Nijholt, Eddie, et al. “Quiver Representations and Dimension Reduction in Dynamical
Systems.” SIAM Journal on Applied Dynamical Systems, vol. 19, no. 4, Society
for Industrial & Applied Mathematics (SIAM), 2020, pp. 2428–68, doi:10.1137/20m1345670.
short: E. Nijholt, B.W. Rink, S. Schwenker, SIAM Journal on Applied Dynamical Systems
19 (2020) 2428–2468.
date_created: 2022-09-06T11:37:42Z
date_updated: 2022-09-07T08:36:03Z
doi: 10.1137/20m1345670
extern: '1'
external_id:
arxiv:
- '2006.08073'
intvolume: ' 19'
issue: '4'
keyword:
- Modeling and Simulation
- Analysis
language:
- iso: eng
page: 2428-2468
publication: SIAM Journal on Applied Dynamical Systems
publication_identifier:
issn:
- 1536-0040
publication_status: published
publisher: Society for Industrial & Applied Mathematics (SIAM)
status: public
title: Quiver Representations and Dimension Reduction in Dynamical Systems
type: journal_article
user_id: '97359'
volume: 19
year: '2020'
...
---
_id: '33265'
abstract:
- lang: eng
text: "This thesis deals with the investigation of dynamical properties – in particular
generic synchrony breaking bifurcations – that are inherent to the structure of
a semigroup network as well the numerous algebraic structures that are related
to these types of networks. Most notably we investigate the interplay between
network dynamics and monoid representation theory as induced by the fundamental
network construction in terms of hidden symmetry as introduced by RINK and SANDERS.\r\n\r\nAfter
providing a brief survey of the field of network dynamics in Part I, we thoroughly
introduce the formalism of semigroup networks, the customized dynamical systems
theory, and the necessary background from monoid representation theory in Chapters
3 and 4. The remainder of Part II investigates generic synchrony breaking bifurcations
and contains three major results. The first is Theorem 5.11, which shows that
generic symmetry breaking steady state bifurcations in monoid equivariant dynamics
occur along absolutely indecomposable subrepresentations – a natural generalization
of the corresponding statement for group equivariant dynamics. Then Theorem 7.12
relates the decomposition of a representation given by a network with high-dimensional
internal phase spaces to that induced by the same network with one-dimensional
internal phase spaces. This result is used to show that there is a smallest dimension
of internal dynamics in which all generic l-parameter bifurcations of a fundamental
network can be observed (Theorem 7.24).\r\n\r\nIn Part III, we employ the machinery
that was summarized and further developed in Part II to feedforward networks.
We propose a general definition of this structural feature of a network and show
that it can equivalently be characterized in different algebraic notions in Theorem
8.35. These are then exploited to fully classify the corresponding monoid representation
for any feedforward network and to classify generic synchrony breaking steady
state bifurcations with one- or highdimensional internal dynamics."
author:
- first_name: Sören
full_name: Schwenker, Sören
id: '97359'
last_name: Schwenker
orcid: 0000-0002-8054-2058
citation:
ama: Schwenker S. Genericity in Network Dynamics. Universität Hamburg; 2019.
apa: Schwenker, S. (2019). Genericity in Network Dynamics. Universität Hamburg.
bibtex: '@book{Schwenker_2019, place={Hamburg}, title={Genericity in Network Dynamics},
publisher={Universität Hamburg}, author={Schwenker, Sören}, year={2019} }'
chicago: 'Schwenker, Sören. Genericity in Network Dynamics. Hamburg: Universität
Hamburg, 2019.'
ieee: 'S. Schwenker, Genericity in Network Dynamics. Hamburg: Universität
Hamburg, 2019.'
mla: Schwenker, Sören. Genericity in Network Dynamics. Universität Hamburg,
2019.
short: S. Schwenker, Genericity in Network Dynamics, Universität Hamburg, Hamburg,
2019.
date_created: 2022-09-06T11:41:13Z
date_updated: 2022-09-07T08:32:14Z
extern: '1'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://ediss.sub.uni-hamburg.de/handle/ediss/6159
oa: '1'
place: Hamburg
publisher: Universität Hamburg
status: public
supervisor:
- first_name: Reiner
full_name: Lauterbach, Reiner
last_name: Lauterbach
title: Genericity in Network Dynamics
type: dissertation
user_id: '97359'
year: '2019'
...
---
_id: '33261'
abstract:
- lang: eng
text: We prove that steady state bifurcations in finite-dimensional dynamical systems
that are symmetric with respect to a monoid representation generically occur along
an absolutely indecomposable subrepresentation. This is stated as a conjecture
in [B. Rink and J. Sanders, SIAM J. Math. Anal., 46 (2014), pp. 1577--1609]. It
is a generalization of the well-known fact that generic steady state bifurcations
in equivariant dynamical systems occur along an absolutely irreducible subrepresentation
if the symmetries form a group---finite or compact Lie. Our generalization also
includes noncompact symmetry groups. The result has applications in bifurcation
theory of homogeneous coupled cell networks as they can be embedded (under mild
additional assumptions) into monoid equivariant systems.
author:
- first_name: Sören
full_name: Schwenker, Sören
id: '97359'
last_name: Schwenker
orcid: 0000-0002-8054-2058
citation:
ama: Schwenker S. Generic Steady State Bifurcations in Monoid Equivariant Dynamics
with Applications in Homogeneous Coupled Cell Systems. SIAM Journal on Mathematical
Analysis. 2018;50(3):2466-2485. doi:10.1137/17m116118x
apa: Schwenker, S. (2018). Generic Steady State Bifurcations in Monoid Equivariant
Dynamics with Applications in Homogeneous Coupled Cell Systems. SIAM Journal
on Mathematical Analysis, 50(3), 2466–2485. https://doi.org/10.1137/17m116118x
bibtex: '@article{Schwenker_2018, title={Generic Steady State Bifurcations in Monoid
Equivariant Dynamics with Applications in Homogeneous Coupled Cell Systems}, volume={50},
DOI={10.1137/17m116118x}, number={3},
journal={SIAM Journal on Mathematical Analysis}, publisher={Society for Industrial
& Applied Mathematics (SIAM)}, author={Schwenker, Sören}, year={2018}, pages={2466–2485}
}'
chicago: 'Schwenker, Sören. “Generic Steady State Bifurcations in Monoid Equivariant
Dynamics with Applications in Homogeneous Coupled Cell Systems.” SIAM Journal
on Mathematical Analysis 50, no. 3 (2018): 2466–85. https://doi.org/10.1137/17m116118x.'
ieee: 'S. Schwenker, “Generic Steady State Bifurcations in Monoid Equivariant Dynamics
with Applications in Homogeneous Coupled Cell Systems,” SIAM Journal on Mathematical
Analysis, vol. 50, no. 3, pp. 2466–2485, 2018, doi: 10.1137/17m116118x.'
mla: Schwenker, Sören. “Generic Steady State Bifurcations in Monoid Equivariant
Dynamics with Applications in Homogeneous Coupled Cell Systems.” SIAM Journal
on Mathematical Analysis, vol. 50, no. 3, Society for Industrial & Applied
Mathematics (SIAM), 2018, pp. 2466–85, doi:10.1137/17m116118x.
short: S. Schwenker, SIAM Journal on Mathematical Analysis 50 (2018) 2466–2485.
date_created: 2022-09-06T11:24:18Z
date_updated: 2022-09-07T08:32:56Z
doi: 10.1137/17m116118x
extern: '1'
external_id:
arxiv:
- '1802.08490'
intvolume: ' 50'
issue: '3'
keyword:
- Applied Mathematics
- Computational Mathematics
- Analysis
language:
- iso: eng
page: 2466-2485
publication: SIAM Journal on Mathematical Analysis
publication_identifier:
issn:
- 0036-1410
- 1095-7154
publication_status: published
publisher: Society for Industrial & Applied Mathematics (SIAM)
status: public
title: Generic Steady State Bifurcations in Monoid Equivariant Dynamics with Applications
in Homogeneous Coupled Cell Systems
type: journal_article
user_id: '97359'
volume: 50
year: '2018'
...
---
_id: '33260'
abstract:
- lang: eng
text: In this paper we continue the study of group representations which are counterexamples
to the Ize conjecture. As in previous papers we find new infinite series of finite
groups leading to such counterexamples. These new series are quite different from
the previous ones, for example the group orders do not form an arithmetic progression.
However, as before we find Lie groups which contain all these groups. This additional
structure was observed, but not used in the previous studies of this problem.
Here we also investigate the related bifurcations. To a large extent, these are
closely related to the presence of mentioned compact Lie group containing the
finite groups. This might give a tool to study the bifurcations related to all
low dimensional counterexamples of the Ize conjecture. It also gives an indication
of where we can expect to find examples where the bifurcation behaviour is different
from what we have seen in the known examples.
author:
- first_name: Reiner
full_name: Lauterbach, Reiner
last_name: Lauterbach
- first_name: Sören
full_name: Schwenker, Sören
id: '97359'
last_name: Schwenker
orcid: 0000-0002-8054-2058
citation:
ama: Lauterbach R, Schwenker S. Equivariant bifurcations in four-dimensional fixed
point spaces. Dynamical Systems. 2016;32(1):117-147. doi:10.1080/14689367.2016.1219696
apa: Lauterbach, R., & Schwenker, S. (2016). Equivariant bifurcations in four-dimensional
fixed point spaces. Dynamical Systems, 32(1), 117–147. https://doi.org/10.1080/14689367.2016.1219696
bibtex: '@article{Lauterbach_Schwenker_2016, title={Equivariant bifurcations in
four-dimensional fixed point spaces}, volume={32}, DOI={10.1080/14689367.2016.1219696},
number={1}, journal={Dynamical Systems}, publisher={Informa UK Limited}, author={Lauterbach,
Reiner and Schwenker, Sören}, year={2016}, pages={117–147} }'
chicago: 'Lauterbach, Reiner, and Sören Schwenker. “Equivariant Bifurcations in
Four-Dimensional Fixed Point Spaces.” Dynamical Systems 32, no. 1 (2016):
117–47. https://doi.org/10.1080/14689367.2016.1219696.'
ieee: 'R. Lauterbach and S. Schwenker, “Equivariant bifurcations in four-dimensional
fixed point spaces,” Dynamical Systems, vol. 32, no. 1, pp. 117–147, 2016,
doi: 10.1080/14689367.2016.1219696.'
mla: Lauterbach, Reiner, and Sören Schwenker. “Equivariant Bifurcations in Four-Dimensional
Fixed Point Spaces.” Dynamical Systems, vol. 32, no. 1, Informa UK Limited,
2016, pp. 117–47, doi:10.1080/14689367.2016.1219696.
short: R. Lauterbach, S. Schwenker, Dynamical Systems 32 (2016) 117–147.
date_created: 2022-09-06T11:22:12Z
date_updated: 2022-09-07T08:33:36Z
doi: 10.1080/14689367.2016.1219696
extern: '1'
external_id:
arxiv:
- '1511.00545'
intvolume: ' 32'
issue: '1'
keyword:
- Computer Science Applications
- General Mathematics
language:
- iso: eng
page: 117-147
publication: Dynamical Systems
publication_identifier:
issn:
- 1468-9367
- 1468-9375
publication_status: published
publisher: Informa UK Limited
status: public
title: Equivariant bifurcations in four-dimensional fixed point spaces
type: journal_article
user_id: '97359'
volume: 32
year: '2016'
...