[{"year":"2020","issue":"4","title":"Quiver Representations and Dimension Reduction in Dynamical Systems","date_created":"2022-09-06T11:37:42Z","publisher":"Society for Industrial & Applied Mathematics (SIAM)","abstract":[{"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.","lang":"eng"}],"publication":"SIAM Journal on Applied Dynamical Systems","language":[{"iso":"eng"}],"keyword":["Modeling and Simulation","Analysis"],"external_id":{"arxiv":["2006.08073"]},"intvolume":"        19","page":"2428-2468","citation":{"chicago":"Nijholt, Eddie, Bob W. Rink, and Sören Schwenker. “Quiver Representations and Dimension Reduction in Dynamical Systems.” <i>SIAM Journal on Applied Dynamical Systems</i> 19, no. 4 (2020): 2428–68. <a href=\"https://doi.org/10.1137/20m1345670\">https://doi.org/10.1137/20m1345670</a>.","ieee":"E. Nijholt, B. W. Rink, and S. Schwenker, “Quiver Representations and Dimension Reduction in Dynamical Systems,” <i>SIAM Journal on Applied Dynamical Systems</i>, vol. 19, no. 4, pp. 2428–2468, 2020, doi: <a href=\"https://doi.org/10.1137/20m1345670\">10.1137/20m1345670</a>.","ama":"Nijholt E, Rink BW, Schwenker S. Quiver Representations and Dimension Reduction in Dynamical Systems. <i>SIAM Journal on Applied Dynamical Systems</i>. 2020;19(4):2428-2468. doi:<a href=\"https://doi.org/10.1137/20m1345670\">10.1137/20m1345670</a>","apa":"Nijholt, E., Rink, B. W., &#38; Schwenker, S. (2020). Quiver Representations and Dimension Reduction in Dynamical Systems. <i>SIAM Journal on Applied Dynamical Systems</i>, <i>19</i>(4), 2428–2468. <a href=\"https://doi.org/10.1137/20m1345670\">https://doi.org/10.1137/20m1345670</a>","mla":"Nijholt, Eddie, et al. “Quiver Representations and Dimension Reduction in Dynamical Systems.” <i>SIAM Journal on Applied Dynamical Systems</i>, vol. 19, no. 4, Society for Industrial &#38; Applied Mathematics (SIAM), 2020, pp. 2428–68, doi:<a href=\"https://doi.org/10.1137/20m1345670\">10.1137/20m1345670</a>.","short":"E. Nijholt, B.W. Rink, S. Schwenker, SIAM Journal on Applied Dynamical Systems 19 (2020) 2428–2468.","bibtex":"@article{Nijholt_Rink_Schwenker_2020, title={Quiver Representations and Dimension Reduction in Dynamical Systems}, volume={19}, DOI={<a href=\"https://doi.org/10.1137/20m1345670\">10.1137/20m1345670</a>}, number={4}, journal={SIAM Journal on Applied Dynamical Systems}, publisher={Society for Industrial &#38; Applied Mathematics (SIAM)}, author={Nijholt, Eddie and Rink, Bob W. and Schwenker, Sören}, year={2020}, pages={2428–2468} }"},"publication_identifier":{"issn":["1536-0040"]},"publication_status":"published","doi":"10.1137/20m1345670","volume":19,"author":[{"first_name":"Eddie","full_name":"Nijholt, Eddie","last_name":"Nijholt"},{"first_name":"Bob W.","full_name":"Rink, Bob W.","last_name":"Rink"},{"last_name":"Schwenker","orcid":"0000-0002-8054-2058","full_name":"Schwenker, Sören","id":"97359","first_name":"Sören"}],"date_updated":"2022-09-07T08:36:03Z","status":"public","type":"journal_article","extern":"1","user_id":"97359","_id":"33263"},{"language":[{"iso":"eng"}],"department":[{"_id":"101"}],"user_id":"32655","_id":"16710","status":"public","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"}],"publication":"SIAM Journal on Applied Dynamical Systems","type":"journal_article","doi":"10.1137/19m1247139","main_file_link":[{"url":"https://epubs.siam.org/doi/epdf/10.1137/19M1247139"}],"title":"A Set-Oriented Path Following Method for the Approximation of Parameter Dependent Attractors","author":[{"first_name":"Raphael","full_name":"Gerlach, Raphael","id":"32655","last_name":"Gerlach"},{"last_name":"Ziessler","full_name":"Ziessler, Adrian","first_name":"Adrian"},{"first_name":"Bruno","last_name":"Eckhardt","full_name":"Eckhardt, Bruno"},{"full_name":"Dellnitz, Michael","last_name":"Dellnitz","first_name":"Michael"}],"date_created":"2020-04-16T14:05:41Z","date_updated":"2024-10-01T13:37:43Z","page":"705-723","citation":{"ama":"Gerlach R, Ziessler A, Eckhardt B, Dellnitz M. A Set-Oriented Path Following Method for the Approximation of Parameter Dependent Attractors. <i>SIAM Journal on Applied Dynamical Systems</i>. Published online 2020:705-723. doi:<a href=\"https://doi.org/10.1137/19m1247139\">10.1137/19m1247139</a>","chicago":"Gerlach, Raphael, Adrian Ziessler, Bruno Eckhardt, and Michael Dellnitz. “A Set-Oriented Path Following Method for the Approximation of Parameter Dependent Attractors.” <i>SIAM Journal on Applied Dynamical Systems</i>, 2020, 705–23. <a href=\"https://doi.org/10.1137/19m1247139\">https://doi.org/10.1137/19m1247139</a>.","ieee":"R. Gerlach, A. Ziessler, B. Eckhardt, and M. Dellnitz, “A Set-Oriented Path Following Method for the Approximation of Parameter Dependent Attractors,” <i>SIAM Journal on Applied Dynamical Systems</i>, pp. 705–723, 2020, doi: <a href=\"https://doi.org/10.1137/19m1247139\">10.1137/19m1247139</a>.","apa":"Gerlach, R., Ziessler, A., Eckhardt, B., &#38; Dellnitz, M. (2020). A Set-Oriented Path Following Method for the Approximation of Parameter Dependent Attractors. <i>SIAM Journal on Applied Dynamical Systems</i>, 705–723. <a href=\"https://doi.org/10.1137/19m1247139\">https://doi.org/10.1137/19m1247139</a>","bibtex":"@article{Gerlach_Ziessler_Eckhardt_Dellnitz_2020, title={A Set-Oriented Path Following Method for the Approximation of Parameter Dependent Attractors}, DOI={<a href=\"https://doi.org/10.1137/19m1247139\">10.1137/19m1247139</a>}, journal={SIAM Journal on Applied Dynamical Systems}, author={Gerlach, Raphael and Ziessler, Adrian and Eckhardt, Bruno and Dellnitz, Michael}, year={2020}, pages={705–723} }","short":"R. Gerlach, A. Ziessler, B. Eckhardt, M. Dellnitz, SIAM Journal on Applied Dynamical Systems (2020) 705–723.","mla":"Gerlach, Raphael, et al. “A Set-Oriented Path Following Method for the Approximation of Parameter Dependent Attractors.” <i>SIAM Journal on Applied Dynamical Systems</i>, 2020, pp. 705–23, doi:<a href=\"https://doi.org/10.1137/19m1247139\">10.1137/19m1247139</a>."},"year":"2020","publication_identifier":{"issn":["1536-0040"]},"publication_status":"published"},{"language":[{"iso":"eng"}],"department":[{"_id":"101"}],"user_id":"32655","_id":"16708","status":"public","abstract":[{"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","lang":"eng"}],"publication":"SIAM Journal on Applied Dynamical Systems","type":"journal_article","doi":"10.1137/18m1204395","main_file_link":[{"url":"https://epubs.siam.org/doi/epdf/10.1137/18M1204395"}],"title":"The Numerical Computation of Unstable Manifolds for Infinite Dimensional Dynamical Systems by Embedding Techniques","volume":18,"author":[{"last_name":"Ziessler","full_name":"Ziessler, Adrian","first_name":"Adrian"},{"full_name":"Dellnitz, Michael","last_name":"Dellnitz","first_name":"Michael"},{"first_name":"Raphael","last_name":"Gerlach","id":"32655","full_name":"Gerlach, Raphael"}],"date_created":"2020-04-16T14:04:20Z","date_updated":"2023-11-17T13:13:09Z","page":"1265-1292","intvolume":"        18","citation":{"apa":"Ziessler, A., Dellnitz, M., &#38; Gerlach, R. (2019). The Numerical Computation of Unstable Manifolds for Infinite Dimensional Dynamical Systems by Embedding Techniques. <i>SIAM Journal on Applied Dynamical Systems</i>, <i>18</i>(3), 1265–1292. <a href=\"https://doi.org/10.1137/18m1204395\">https://doi.org/10.1137/18m1204395</a>","bibtex":"@article{Ziessler_Dellnitz_Gerlach_2019, title={The Numerical Computation of Unstable Manifolds for Infinite Dimensional Dynamical Systems by Embedding Techniques}, volume={18}, DOI={<a href=\"https://doi.org/10.1137/18m1204395\">10.1137/18m1204395</a>}, number={3}, journal={SIAM Journal on Applied Dynamical Systems}, author={Ziessler, Adrian and Dellnitz, Michael and Gerlach, Raphael}, year={2019}, pages={1265–1292} }","mla":"Ziessler, Adrian, et al. “The Numerical Computation of Unstable Manifolds for Infinite Dimensional Dynamical Systems by Embedding Techniques.” <i>SIAM Journal on Applied Dynamical Systems</i>, vol. 18, no. 3, 2019, pp. 1265–92, doi:<a href=\"https://doi.org/10.1137/18m1204395\">10.1137/18m1204395</a>.","short":"A. Ziessler, M. Dellnitz, R. Gerlach, SIAM Journal on Applied Dynamical Systems 18 (2019) 1265–1292.","chicago":"Ziessler, Adrian, Michael Dellnitz, and Raphael Gerlach. “The Numerical Computation of Unstable Manifolds for Infinite Dimensional Dynamical Systems by Embedding Techniques.” <i>SIAM Journal on Applied Dynamical Systems</i> 18, no. 3 (2019): 1265–92. <a href=\"https://doi.org/10.1137/18m1204395\">https://doi.org/10.1137/18m1204395</a>.","ieee":"A. Ziessler, M. Dellnitz, and R. Gerlach, “The Numerical Computation of Unstable Manifolds for Infinite Dimensional Dynamical Systems by Embedding Techniques,” <i>SIAM Journal on Applied Dynamical Systems</i>, vol. 18, no. 3, pp. 1265–1292, 2019, doi: <a href=\"https://doi.org/10.1137/18m1204395\">10.1137/18m1204395</a>.","ama":"Ziessler A, Dellnitz M, Gerlach R. The Numerical Computation of Unstable Manifolds for Infinite Dimensional Dynamical Systems by Embedding Techniques. <i>SIAM Journal on Applied Dynamical Systems</i>. 2019;18(3):1265-1292. doi:<a href=\"https://doi.org/10.1137/18m1204395\">10.1137/18m1204395</a>"},"year":"2019","issue":"3","publication_identifier":{"issn":["1536-0040"]},"publication_status":"published"},{"status":"public","type":"journal_article","publication":"SIAM Journal on Applied Dynamical Systems","language":[{"iso":"eng"}],"user_id":"15701","department":[{"_id":"101"}],"_id":"16581","citation":{"ieee":"M. Dellnitz, S. Klus, and A. Ziessler, “A Set-Oriented Numerical Approach for Dynamical Systems with Parameter Uncertainty,” <i>SIAM Journal on Applied Dynamical Systems</i>, pp. 120–138, 2017.","chicago":"Dellnitz, Michael, Stefan Klus, and Adrian Ziessler. “A Set-Oriented Numerical Approach for Dynamical Systems with Parameter Uncertainty.” <i>SIAM Journal on Applied Dynamical Systems</i>, 2017, 120–38. <a href=\"https://doi.org/10.1137/16m1072735\">https://doi.org/10.1137/16m1072735</a>.","ama":"Dellnitz M, Klus S, Ziessler A. A Set-Oriented Numerical Approach for Dynamical Systems with Parameter Uncertainty. <i>SIAM Journal on Applied Dynamical Systems</i>. 2017:120-138. doi:<a href=\"https://doi.org/10.1137/16m1072735\">10.1137/16m1072735</a>","apa":"Dellnitz, M., Klus, S., &#38; Ziessler, A. (2017). A Set-Oriented Numerical Approach for Dynamical Systems with Parameter Uncertainty. <i>SIAM Journal on Applied Dynamical Systems</i>, 120–138. <a href=\"https://doi.org/10.1137/16m1072735\">https://doi.org/10.1137/16m1072735</a>","bibtex":"@article{Dellnitz_Klus_Ziessler_2017, title={A Set-Oriented Numerical Approach for Dynamical Systems with Parameter Uncertainty}, DOI={<a href=\"https://doi.org/10.1137/16m1072735\">10.1137/16m1072735</a>}, journal={SIAM Journal on Applied Dynamical Systems}, author={Dellnitz, Michael and Klus, Stefan and Ziessler, Adrian}, year={2017}, pages={120–138} }","mla":"Dellnitz, Michael, et al. “A Set-Oriented Numerical Approach for Dynamical Systems with Parameter Uncertainty.” <i>SIAM Journal on Applied Dynamical Systems</i>, 2017, pp. 120–38, doi:<a href=\"https://doi.org/10.1137/16m1072735\">10.1137/16m1072735</a>.","short":"M. Dellnitz, S. Klus, A. Ziessler, SIAM Journal on Applied Dynamical Systems (2017) 120–138."},"page":"120-138","year":"2017","publication_status":"published","publication_identifier":{"issn":["1536-0040"]},"doi":"10.1137/16m1072735","title":"A Set-Oriented Numerical Approach for Dynamical Systems with Parameter Uncertainty","date_created":"2020-04-16T05:39:17Z","author":[{"full_name":"Dellnitz, Michael","last_name":"Dellnitz","first_name":"Michael"},{"last_name":"Klus","full_name":"Klus, Stefan","first_name":"Stefan"},{"first_name":"Adrian","last_name":"Ziessler","full_name":"Ziessler, Adrian"}],"date_updated":"2022-01-06T06:52:52Z"},{"doi":"10.1137/030600210","title":"A Rigorous Numerical Method for the Global Analysis of Infinite-Dimensional Discrete Dynamical Systems","author":[{"full_name":"Day, S.","last_name":"Day","first_name":"S."},{"last_name":"Junge","full_name":"Junge, O.","first_name":"O."},{"last_name":"Mischaikow","full_name":"Mischaikow, K.","first_name":"K."}],"date_created":"2020-04-15T08:17:18Z","date_updated":"2022-01-06T06:52:52Z","citation":{"apa":"Day, S., Junge, O., &#38; Mischaikow, K. (2004). A Rigorous Numerical Method for the Global Analysis of Infinite-Dimensional Discrete Dynamical Systems. <i>SIAM Journal on Applied Dynamical Systems</i>, 117–160. <a href=\"https://doi.org/10.1137/030600210\">https://doi.org/10.1137/030600210</a>","mla":"Day, S., et al. “A Rigorous Numerical Method for the Global Analysis of Infinite-Dimensional Discrete Dynamical Systems.” <i>SIAM Journal on Applied Dynamical Systems</i>, 2004, pp. 117–60, doi:<a href=\"https://doi.org/10.1137/030600210\">10.1137/030600210</a>.","bibtex":"@article{Day_Junge_Mischaikow_2004, title={A Rigorous Numerical Method for the Global Analysis of Infinite-Dimensional Discrete Dynamical Systems}, DOI={<a href=\"https://doi.org/10.1137/030600210\">10.1137/030600210</a>}, journal={SIAM Journal on Applied Dynamical Systems}, author={Day, S. and Junge, O. and Mischaikow, K.}, year={2004}, pages={117–160} }","short":"S. Day, O. Junge, K. Mischaikow, SIAM Journal on Applied Dynamical Systems (2004) 117–160.","ama":"Day S, Junge O, Mischaikow K. A Rigorous Numerical Method for the Global Analysis of Infinite-Dimensional Discrete Dynamical Systems. <i>SIAM Journal on Applied Dynamical Systems</i>. 2004:117-160. doi:<a href=\"https://doi.org/10.1137/030600210\">10.1137/030600210</a>","ieee":"S. Day, O. Junge, and K. Mischaikow, “A Rigorous Numerical Method for the Global Analysis of Infinite-Dimensional Discrete Dynamical Systems,” <i>SIAM Journal on Applied Dynamical Systems</i>, pp. 117–160, 2004.","chicago":"Day, S., O. Junge, and K. Mischaikow. “A Rigorous Numerical Method for the Global Analysis of Infinite-Dimensional Discrete Dynamical Systems.” <i>SIAM Journal on Applied Dynamical Systems</i>, 2004, 117–60. <a href=\"https://doi.org/10.1137/030600210\">https://doi.org/10.1137/030600210</a>."},"page":"117-160","year":"2004","publication_status":"published","publication_identifier":{"issn":["1536-0040"]},"language":[{"iso":"eng"}],"user_id":"15701","department":[{"_id":"101"}],"_id":"16527","status":"public","type":"journal_article","publication":"SIAM Journal on Applied Dynamical Systems"}]