---
_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: '16710'
abstract:
- lang: eng
text: "In this work we present a set-oriented path following method for the computation
of relative global\r\nattractors of parameter-dependent dynamical systems. We
start with an initial approximation of the\r\nrelative global attractor for a
fixed parameter λ0 computed by a set-oriented subdivision method.\r\nBy using
previously obtained approximations of the parameter-dependent relative global
attractor\r\nwe can track it with respect to a one-dimensional parameter λ > λ0
without restarting the whole\r\nsubdivision procedure. We illustrate the feasibility
of the set-oriented path following method by\r\nexploring the dynamics in low-dimensional
models for shear flows during the transition to turbulence\r\nand of large-scale
atmospheric regime changes .\r\n"
author:
- first_name: Raphael
full_name: Gerlach, Raphael
id: '32655'
last_name: Gerlach
- first_name: Adrian
full_name: Ziessler, Adrian
last_name: Ziessler
- first_name: Bruno
full_name: Eckhardt, Bruno
last_name: Eckhardt
- first_name: Michael
full_name: Dellnitz, Michael
last_name: Dellnitz
citation:
ama: Gerlach R, Ziessler A, Eckhardt B, Dellnitz M. A Set-Oriented Path Following
Method for the Approximation of Parameter Dependent Attractors. SIAM Journal
on Applied Dynamical Systems. Published online 2020:705-723. doi:10.1137/19m1247139
apa: Gerlach, R., Ziessler, A., Eckhardt, B., & Dellnitz, M. (2020). A Set-Oriented
Path Following Method for the Approximation of Parameter Dependent Attractors.
SIAM Journal on Applied Dynamical Systems, 705–723. https://doi.org/10.1137/19m1247139
bibtex: '@article{Gerlach_Ziessler_Eckhardt_Dellnitz_2020, title={A Set-Oriented
Path Following Method for the Approximation of Parameter Dependent Attractors},
DOI={10.1137/19m1247139}, journal={SIAM
Journal on Applied Dynamical Systems}, author={Gerlach, Raphael and Ziessler,
Adrian and Eckhardt, Bruno and Dellnitz, Michael}, year={2020}, pages={705–723}
}'
chicago: Gerlach, Raphael, Adrian Ziessler, Bruno Eckhardt, and Michael Dellnitz.
“A Set-Oriented Path Following Method for the Approximation of Parameter Dependent
Attractors.” SIAM Journal on Applied Dynamical Systems, 2020, 705–23. https://doi.org/10.1137/19m1247139.
ieee: 'R. Gerlach, A. Ziessler, B. Eckhardt, and M. Dellnitz, “A Set-Oriented Path
Following Method for the Approximation of Parameter Dependent Attractors,” SIAM
Journal on Applied Dynamical Systems, pp. 705–723, 2020, doi: 10.1137/19m1247139.'
mla: Gerlach, Raphael, et al. “A Set-Oriented Path Following Method for the Approximation
of Parameter Dependent Attractors.” SIAM Journal on Applied Dynamical Systems,
2020, pp. 705–23, doi:10.1137/19m1247139.
short: R. Gerlach, A. Ziessler, B. Eckhardt, M. Dellnitz, SIAM Journal on Applied
Dynamical Systems (2020) 705–723.
date_created: 2020-04-16T14:05:41Z
date_updated: 2024-10-01T13:37:43Z
department:
- _id: '101'
doi: 10.1137/19m1247139
language:
- iso: eng
main_file_link:
- url: https://epubs.siam.org/doi/epdf/10.1137/19M1247139
page: 705-723
publication: SIAM Journal on Applied Dynamical Systems
publication_identifier:
issn:
- 1536-0040
publication_status: published
status: public
title: A Set-Oriented Path Following Method for the Approximation of Parameter Dependent
Attractors
type: journal_article
user_id: '32655'
year: '2020'
...
---
_id: '16708'
abstract:
- lang: eng
text: " In this work we extend the novel framework developed by Dellnitz, Hessel-von
Molo, and Ziessler to\r\nthe computation of finite dimensional unstable manifolds
of infinite dimensional dynamical systems.\r\nTo this end, we adapt a set-oriented
continuation technique developed by Dellnitz and Hohmann for\r\nthe computation
of such objects of finite dimensional systems with the results obtained in the
work\r\nof Dellnitz, Hessel-von Molo, and Ziessler. We show how to implement this
approach for the analysis\r\nof partial differential equations and illustrate
its feasibility by computing unstable manifolds of the\r\none-dimensional Kuramoto--Sivashinsky
equation as well as for the Mackey--Glass delay differential\r\nequation.\r\n"
author:
- first_name: Adrian
full_name: Ziessler, Adrian
last_name: Ziessler
- first_name: Michael
full_name: Dellnitz, Michael
last_name: Dellnitz
- first_name: Raphael
full_name: Gerlach, Raphael
id: '32655'
last_name: Gerlach
citation:
ama: Ziessler A, Dellnitz M, Gerlach R. The Numerical Computation of Unstable Manifolds
for Infinite Dimensional Dynamical Systems by Embedding Techniques. SIAM Journal
on Applied Dynamical Systems. 2019;18(3):1265-1292. doi:10.1137/18m1204395
apa: Ziessler, A., Dellnitz, M., & Gerlach, R. (2019). The Numerical Computation
of Unstable Manifolds for Infinite Dimensional Dynamical Systems by Embedding
Techniques. SIAM Journal on Applied Dynamical Systems, 18(3), 1265–1292.
https://doi.org/10.1137/18m1204395
bibtex: '@article{Ziessler_Dellnitz_Gerlach_2019, title={The Numerical Computation
of Unstable Manifolds for Infinite Dimensional Dynamical Systems by Embedding
Techniques}, volume={18}, DOI={10.1137/18m1204395},
number={3}, journal={SIAM Journal on Applied Dynamical Systems}, author={Ziessler,
Adrian and Dellnitz, Michael and Gerlach, Raphael}, year={2019}, pages={1265–1292}
}'
chicago: 'Ziessler, Adrian, Michael Dellnitz, and Raphael Gerlach. “The Numerical
Computation of Unstable Manifolds for Infinite Dimensional Dynamical Systems by
Embedding Techniques.” SIAM Journal on Applied Dynamical Systems 18, no.
3 (2019): 1265–92. https://doi.org/10.1137/18m1204395.'
ieee: 'A. Ziessler, M. Dellnitz, and R. Gerlach, “The Numerical Computation of Unstable
Manifolds for Infinite Dimensional Dynamical Systems by Embedding Techniques,”
SIAM Journal on Applied Dynamical Systems, vol. 18, no. 3, pp. 1265–1292,
2019, doi: 10.1137/18m1204395.'
mla: Ziessler, Adrian, et al. “The Numerical Computation of Unstable Manifolds for
Infinite Dimensional Dynamical Systems by Embedding Techniques.” SIAM Journal
on Applied Dynamical Systems, vol. 18, no. 3, 2019, pp. 1265–92, doi:10.1137/18m1204395.
short: A. Ziessler, M. Dellnitz, R. Gerlach, SIAM Journal on Applied Dynamical Systems
18 (2019) 1265–1292.
date_created: 2020-04-16T14:04:20Z
date_updated: 2023-11-17T13:13:09Z
department:
- _id: '101'
doi: 10.1137/18m1204395
intvolume: ' 18'
issue: '3'
language:
- iso: eng
main_file_link:
- url: https://epubs.siam.org/doi/epdf/10.1137/18M1204395
page: 1265-1292
publication: SIAM Journal on Applied Dynamical Systems
publication_identifier:
issn:
- 1536-0040
publication_status: published
status: public
title: The Numerical Computation of Unstable Manifolds for Infinite Dimensional Dynamical
Systems by Embedding Techniques
type: journal_article
user_id: '32655'
volume: 18
year: '2019'
...
---
_id: '16581'
author:
- first_name: Michael
full_name: Dellnitz, Michael
last_name: Dellnitz
- first_name: Stefan
full_name: Klus, Stefan
last_name: Klus
- first_name: Adrian
full_name: Ziessler, Adrian
last_name: Ziessler
citation:
ama: Dellnitz M, Klus S, Ziessler A. A Set-Oriented Numerical Approach for Dynamical
Systems with Parameter Uncertainty. SIAM Journal on Applied Dynamical Systems.
2017:120-138. doi:10.1137/16m1072735
apa: Dellnitz, M., Klus, S., & Ziessler, A. (2017). A Set-Oriented Numerical
Approach for Dynamical Systems with Parameter Uncertainty. SIAM Journal on
Applied Dynamical Systems, 120–138. https://doi.org/10.1137/16m1072735
bibtex: '@article{Dellnitz_Klus_Ziessler_2017, title={A Set-Oriented Numerical Approach
for Dynamical Systems with Parameter Uncertainty}, DOI={10.1137/16m1072735},
journal={SIAM Journal on Applied Dynamical Systems}, author={Dellnitz, Michael
and Klus, Stefan and Ziessler, Adrian}, year={2017}, pages={120–138} }'
chicago: Dellnitz, Michael, Stefan Klus, and Adrian Ziessler. “A Set-Oriented Numerical
Approach for Dynamical Systems with Parameter Uncertainty.” SIAM Journal on
Applied Dynamical Systems, 2017, 120–38. https://doi.org/10.1137/16m1072735.
ieee: M. Dellnitz, S. Klus, and A. Ziessler, “A Set-Oriented Numerical Approach
for Dynamical Systems with Parameter Uncertainty,” SIAM Journal on Applied
Dynamical Systems, pp. 120–138, 2017.
mla: Dellnitz, Michael, et al. “A Set-Oriented Numerical Approach for Dynamical
Systems with Parameter Uncertainty.” SIAM Journal on Applied Dynamical Systems,
2017, pp. 120–38, doi:10.1137/16m1072735.
short: M. Dellnitz, S. Klus, A. Ziessler, SIAM Journal on Applied Dynamical Systems
(2017) 120–138.
date_created: 2020-04-16T05:39:17Z
date_updated: 2022-01-06T06:52:52Z
department:
- _id: '101'
doi: 10.1137/16m1072735
language:
- iso: eng
page: 120-138
publication: SIAM Journal on Applied Dynamical Systems
publication_identifier:
issn:
- 1536-0040
publication_status: published
status: public
title: A Set-Oriented Numerical Approach for Dynamical Systems with Parameter Uncertainty
type: journal_article
user_id: '15701'
year: '2017'
...
---
_id: '16527'
author:
- first_name: S.
full_name: Day, S.
last_name: Day
- first_name: O.
full_name: Junge, O.
last_name: Junge
- first_name: K.
full_name: Mischaikow, K.
last_name: Mischaikow
citation:
ama: Day S, Junge O, Mischaikow K. A Rigorous Numerical Method for the Global Analysis
of Infinite-Dimensional Discrete Dynamical Systems. SIAM Journal on Applied
Dynamical Systems. 2004:117-160. doi:10.1137/030600210
apa: Day, S., Junge, O., & Mischaikow, K. (2004). A Rigorous Numerical Method
for the Global Analysis of Infinite-Dimensional Discrete Dynamical Systems. SIAM
Journal on Applied Dynamical Systems, 117–160. https://doi.org/10.1137/030600210
bibtex: '@article{Day_Junge_Mischaikow_2004, title={A Rigorous Numerical Method
for the Global Analysis of Infinite-Dimensional Discrete Dynamical Systems}, DOI={10.1137/030600210}, journal={SIAM
Journal on Applied Dynamical Systems}, author={Day, S. and Junge, O. and Mischaikow,
K.}, year={2004}, pages={117–160} }'
chicago: Day, S., O. Junge, and K. Mischaikow. “A Rigorous Numerical Method for
the Global Analysis of Infinite-Dimensional Discrete Dynamical Systems.” SIAM
Journal on Applied Dynamical Systems, 2004, 117–60. https://doi.org/10.1137/030600210.
ieee: S. Day, O. Junge, and K. Mischaikow, “A Rigorous Numerical Method for the
Global Analysis of Infinite-Dimensional Discrete Dynamical Systems,” SIAM Journal
on Applied Dynamical Systems, pp. 117–160, 2004.
mla: Day, S., et al. “A Rigorous Numerical Method for the Global Analysis of Infinite-Dimensional
Discrete Dynamical Systems.” SIAM Journal on Applied Dynamical Systems,
2004, pp. 117–60, doi:10.1137/030600210.
short: S. Day, O. Junge, K. Mischaikow, SIAM Journal on Applied Dynamical Systems
(2004) 117–160.
date_created: 2020-04-15T08:17:18Z
date_updated: 2022-01-06T06:52:52Z
department:
- _id: '101'
doi: 10.1137/030600210
language:
- iso: eng
page: 117-160
publication: SIAM Journal on Applied Dynamical Systems
publication_identifier:
issn:
- 1536-0040
publication_status: published
status: public
title: A Rigorous Numerical Method for the Global Analysis of Infinite-Dimensional
Discrete Dynamical Systems
type: journal_article
user_id: '15701'
year: '2004'
...