[{"external_id":{"arxiv":["2503.07576"]},"language":[{"iso":"eng"}],"publication":"Lecture Notes in Computer Science","publisher":"Springer Nature Switzerland","date_created":"2025-05-27T08:17:03Z","title":"On the Dynamical Hierarchy in Gathering Protocols with Circulant Topologies","year":"2025","project":[{"grant_number":"453112019","name":"Algorithmen für Schwarmrobotik: Verteiltes Rechnen trifft Dynamische Systeme","_id":"106"}],"_id":"60048","user_id":"32655","department":[{"_id":"101"}],"type":"book_chapter","status":"public","oa":"1","date_updated":"2025-05-27T08:22:42Z","author":[{"first_name":"Raphael","orcid":"0009-0002-4750-2051","last_name":"Gerlach","id":"32655","full_name":"Gerlach, Raphael"},{"first_name":"Sören","orcid":"0000-0002-8054-2058","last_name":"von der Gracht","id":"97359","full_name":"von der Gracht, Sören"},{"last_name":"Dellnitz","full_name":"Dellnitz, Michael","first_name":"Michael"}],"main_file_link":[{"open_access":"1","url":" ArXiv:2503.07576"}],"doi":"10.1007/978-3-031-91736-3_19","publication_status":"published","publication_identifier":{"issn":["0302-9743","1611-3349"],"isbn":["9783031917356","9783031917363"]},"place":"Cham","citation":{"ama":"Gerlach R, von der Gracht S, Dellnitz M. On the Dynamical Hierarchy in Gathering Protocols with Circulant Topologies. In: <i>Lecture Notes in Computer Science</i>. Springer Nature Switzerland; 2025. doi:<a href=\"https://doi.org/10.1007/978-3-031-91736-3_19\">10.1007/978-3-031-91736-3_19</a>","ieee":"R. Gerlach, S. von der Gracht, and M. Dellnitz, “On the Dynamical Hierarchy in Gathering Protocols with Circulant Topologies,” in <i>Lecture Notes in Computer Science</i>, Cham: Springer Nature Switzerland, 2025.","chicago":"Gerlach, Raphael, Sören von der Gracht, and Michael Dellnitz. “On the Dynamical Hierarchy in Gathering Protocols with Circulant Topologies.” In <i>Lecture Notes in Computer Science</i>. Cham: Springer Nature Switzerland, 2025. <a href=\"https://doi.org/10.1007/978-3-031-91736-3_19\">https://doi.org/10.1007/978-3-031-91736-3_19</a>.","apa":"Gerlach, R., von der Gracht, S., &#38; Dellnitz, M. (2025). On the Dynamical Hierarchy in Gathering Protocols with Circulant Topologies. In <i>Lecture Notes in Computer Science</i>. Springer Nature Switzerland. <a href=\"https://doi.org/10.1007/978-3-031-91736-3_19\">https://doi.org/10.1007/978-3-031-91736-3_19</a>","mla":"Gerlach, Raphael, et al. “On the Dynamical Hierarchy in Gathering Protocols with Circulant Topologies.” <i>Lecture Notes in Computer Science</i>, Springer Nature Switzerland, 2025, doi:<a href=\"https://doi.org/10.1007/978-3-031-91736-3_19\">10.1007/978-3-031-91736-3_19</a>.","short":"R. Gerlach, S. von der Gracht, M. Dellnitz, in: Lecture Notes in Computer Science, Springer Nature Switzerland, Cham, 2025.","bibtex":"@inbook{Gerlach_von der Gracht_Dellnitz_2025, place={Cham}, title={On the Dynamical Hierarchy in Gathering Protocols with Circulant Topologies}, DOI={<a href=\"https://doi.org/10.1007/978-3-031-91736-3_19\">10.1007/978-3-031-91736-3_19</a>}, booktitle={Lecture Notes in Computer Science}, publisher={Springer Nature Switzerland}, author={Gerlach, Raphael and von der Gracht, Sören and Dellnitz, Michael}, year={2025} }"}},{"_id":"56298","project":[{"grant_number":"453112019","name":"Algorithmen für Schwarmrobotik: Verteiltes Rechnen trifft Dynamische Systeme","_id":"106"}],"department":[{"_id":"101"}],"user_id":"97359","series_title":"Leibniz International Proceedings in Informatics (LIPIcs)","type":"conference","editor":[{"first_name":"Silvia","full_name":"Bonomi, Silvia","last_name":"Bonomi"},{"first_name":"Letterio","last_name":"Galletta","full_name":"Galletta, Letterio"},{"last_name":"Rivière","full_name":"Rivière,  Etienne","first_name":" Etienne"},{"first_name":" Valerio","last_name":"Schiavoni","full_name":"Schiavoni,  Valerio"}],"status":"public","oa":"1","date_updated":"2025-01-09T11:39:19Z","volume":324,"author":[{"first_name":"Raphael","orcid":"0009-0002-4750-2051","last_name":"Gerlach","full_name":"Gerlach, Raphael","id":"32655"},{"first_name":"Sören","last_name":"von der Gracht","orcid":"0000-0002-8054-2058","full_name":"von der Gracht, Sören","id":"97359"},{"first_name":"Christopher","last_name":"Hahn","full_name":"Hahn, Christopher"},{"last_name":"Harbig","full_name":"Harbig, Jonas","id":"47213","first_name":"Jonas"},{"first_name":"Peter","last_name":"Kling","full_name":"Kling, Peter"}],"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"},"doi":"10.4230/LIPIcs.OPODIS.2024.13","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/2409.19277"}],"publication_identifier":{"issn":["1868-8969"],"isbn":["978-3-95977-360-7"]},"publication_status":"published","intvolume":"       324","citation":{"apa":"Gerlach, R., von der Gracht, S., Hahn, C., Harbig, J., &#38; Kling, P. (2025). Symmetry Preservation in Swarms of Oblivious Robots with Limited  Visibility. In S. Bonomi, L. Galletta,  Etienne Rivière, &#38;  Valerio Schiavoni (Eds.), <i>28th International Conference on Principles of Distributed Systems (OPODIS 2024)</i> (Vol. 324). Schloss Dagstuhl -- Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPIcs.OPODIS.2024.13\">https://doi.org/10.4230/LIPIcs.OPODIS.2024.13</a>","mla":"Gerlach, Raphael, et al. “Symmetry Preservation in Swarms of Oblivious Robots with Limited  Visibility.” <i>28th International Conference on Principles of Distributed Systems (OPODIS 2024)</i>, edited by Silvia Bonomi et al., vol. 324, Schloss Dagstuhl -- Leibniz-Zentrum für Informatik, 2025, doi:<a href=\"https://doi.org/10.4230/LIPIcs.OPODIS.2024.13\">10.4230/LIPIcs.OPODIS.2024.13</a>.","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.","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={<a href=\"https://doi.org/10.4230/LIPIcs.OPODIS.2024.13\">10.4230/LIPIcs.OPODIS.2024.13</a>}, 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)} }","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. <i>28th International Conference on Principles of Distributed Systems (OPODIS 2024)</i>. Vol 324. Leibniz International Proceedings in Informatics (LIPIcs). Schloss Dagstuhl -- Leibniz-Zentrum für Informatik; 2025. doi:<a href=\"https://doi.org/10.4230/LIPIcs.OPODIS.2024.13\">10.4230/LIPIcs.OPODIS.2024.13</a>","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 <i>28th International Conference on Principles of Distributed Systems (OPODIS 2024)</i>, Lucca, Italy, 2025, vol. 324, doi: <a href=\"https://doi.org/10.4230/LIPIcs.OPODIS.2024.13\">10.4230/LIPIcs.OPODIS.2024.13</a>.","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 <i>28th International Conference on Principles of Distributed Systems (OPODIS 2024)</i>, 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. <a href=\"https://doi.org/10.4230/LIPIcs.OPODIS.2024.13\">https://doi.org/10.4230/LIPIcs.OPODIS.2024.13</a>."},"external_id":{"arxiv":["2409.19277"]},"keyword":["Swarm Algorithm","Swarm Robots","Distributed Algorithm","Pattern Formation","Limited Visibility","Oblivious"],"language":[{"iso":"eng"}],"publication":"28th International Conference on Principles of Distributed Systems (OPODIS 2024)","abstract":[{"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).","lang":"eng"}],"publisher":"Schloss Dagstuhl -- Leibniz-Zentrum für Informatik","date_created":"2024-10-01T13:29:43Z","title":"Symmetry Preservation in Swarms of Oblivious Robots with Limited  Visibility","year":"2025"},{"author":[{"id":"32655","full_name":"Gerlach, Raphael","orcid":"0009-0002-4750-2051","last_name":"Gerlach","first_name":"Raphael"},{"first_name":"Sören","id":"97359","full_name":"von der Gracht, Sören","orcid":"0000-0002-8054-2058","last_name":"von der Gracht"}],"date_updated":"2025-03-11T08:53:02Z","oa":"1","citation":{"ieee":"R. Gerlach and S. von der Gracht, “Analyzing Symmetries of Swarms of Mobile Robots Using Equivariant  Dynamical Systems,” <i>arXiv:2503.07576</i>. 2025.","chicago":"Gerlach, Raphael, and Sören von der Gracht. “Analyzing Symmetries of Swarms of Mobile Robots Using Equivariant  Dynamical Systems.” <i>ArXiv:2503.07576</i>, 2025.","ama":"Gerlach R, von der Gracht S. Analyzing Symmetries of Swarms of Mobile Robots Using Equivariant  Dynamical Systems. <i>arXiv:250307576</i>. Published online 2025.","apa":"Gerlach, R., &#38; von der Gracht, S. (2025). Analyzing Symmetries of Swarms of Mobile Robots Using Equivariant  Dynamical Systems. In <i>arXiv:2503.07576</i>.","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} }","mla":"Gerlach, Raphael, and Sören von der Gracht. “Analyzing Symmetries of Swarms of Mobile Robots Using Equivariant  Dynamical Systems.” <i>ArXiv:2503.07576</i>, 2025.","short":"R. Gerlach, S. von der Gracht, ArXiv:2503.07576 (2025)."},"page":"23","has_accepted_license":"1","file_date_updated":"2025-03-11T08:27:32Z","user_id":"97359","department":[{"_id":"101"}],"project":[{"_id":"106","name":"Algorithmen für Schwarmrobotik: Verteiltes Rechnen trifft Dynamische Systeme","grant_number":"453112019"}],"_id":"58953","status":"public","type":"preprint","title":"Analyzing Symmetries of Swarms of Mobile Robots Using Equivariant  Dynamical Systems","date_created":"2025-03-11T08:21:05Z","year":"2025","language":[{"iso":"eng"}],"ddc":["004"],"keyword":["dynamical systems","coupled systems","distributed computing","robot swarms","autonomous mobile robots","symmetry","equivariant dynamics"],"external_id":{"arxiv":["2503.07576"]},"file":[{"relation":"main_file","content_type":"application/pdf","file_size":812198,"file_name":"Analyzing_Symmetries_of_Swarms_of_Mobile_Robots_Using_Equivariant_Dynamical_Systems.pdf","access_level":"open_access","file_id":"58954","date_updated":"2025-03-11T08:27:32Z","date_created":"2025-03-11T08:27:32Z","creator":"svdg"}],"abstract":[{"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.","lang":"eng"}],"publication":"arXiv:2503.07576"},{"status":"public","abstract":[{"lang":"ger","text":"Ein zentraler Aspekt bei der Untersuchung dynamischer Systeme ist die Analyse ihrer invarianten Mengen wie des globalen Attraktors und (in)stabiler Mannigfaltigkeiten. Insbesondere wenn das zugrunde liegende System von einem Parameter abhängt, ist es entscheidend, sie im Bezug auf diesen Parameter effizient zu verfolgen. Für die Berechnung invarianter Mengen stützen wir uns für ihre Approximation auf numerische Algorithmen. Typischerweise können diese Methoden jedoch nur auf endlich-dimensionale dynamische Systeme angewendet werden. In dieser Arbeit präsentieren wir daher einen numerischen Rahmen für die globale dynamische Analyse unendlich-dimensionaler Systeme. Wir werden Einbettungstechniken verwenden, um das core dynamical system (CDS) zu definieren, welches ein dynamisch äquivalentes endlich-dimensionales System ist.Das CDS wird dann verwendet, um eingebettete invariante Mengen, also eins-zu-eins Bilder, mittels Mengen-orientierten numerischen Methoden zu approximieren. Bei der Konstruktion des CDS ist es entscheidend, eine geeignete Beobachtungsabbildung auszuwählen und die geeignete inverse Abbildung zu entwerfen. Dazu werden wir geeignete numerische Implementierungen des CDS für DDEs und PDEs vorstellen. Für eine nachfolgende geometrische Analyse der eingebetteten invarianten Menge betrachten wir eine Lerntechnik namens diffusion maps, die ihre intrinsische Geometrie enthüllt sowie ihre Dimension schätzt. Schließlich wenden wir unsere entwickelten numerischen Methoden an einigen bekannten unendlich-dimensionale dynamischen Systeme an, wie die Mackey-Glass-Gleichung, die Kuramoto-Sivashinsky-Gleichung und die Navier-Stokes-Gleichung."},{"lang":"eng","text":"One central aspect in the study of dynamical systems is the analysis of its invariant sets such as the global attractor and (un)stable manifolds. In particular, when the underlying system depends on a parameter it is crucial to efficiently track those set with respect to this parameter. For the computation of invariant sets we rely on numerical algorithms for their approximation but typically those tools can only be applied to finite-dimensional dynamical systems. Thus, in thesis we present a numerical framework for the global dynamical analysis of infinite-dimensional systems. We will use embedding techniques for the definition of the core dynamical system (CDS) which is a dynamically equivalent finite-dimensional system. The CDS is then used for the approximation of related embedded invariant sets, i.e, one-to-one images, by set-oriented numerical methods. For the construction of the CDS it is crucial to choose an appropriate observation map and to design its corresponding inverse. Therefore, we will present suitable numerical realizations of the CDS for DDEs and PDEs. For a subsequent geometric analysis of the embedded invariant set we will consider a manifold learning technique called diffusion maps which reveals its intrinsic geometry and estimates its dimension. Finally, we apply our develop numerical tools on some well-known infinite-dimensional dynamical systems such as the Mackey-Glass equation, the Kuramoto-Sivashinsky equation and the Navier-Stokes equation."}],"type":"dissertation","language":[{"iso":"eng"}],"user_id":"32643","department":[{"_id":"101"}],"_id":"32057","citation":{"ieee":"R. Gerlach, <i>The Computation and Analysis of Invariant Sets of Infinite-Dimensional Systems</i>. 2021.","chicago":"Gerlach, Raphael. <i>The Computation and Analysis of Invariant Sets of Infinite-Dimensional Systems</i>, 2021. <a href=\"https://doi.org/10.17619/UNIPB/1-1278\">https://doi.org/10.17619/UNIPB/1-1278</a>.","ama":"Gerlach R. <i>The Computation and Analysis of Invariant Sets of Infinite-Dimensional Systems</i>.; 2021. doi:<a href=\"https://doi.org/10.17619/UNIPB/1-1278\">10.17619/UNIPB/1-1278</a>","bibtex":"@book{Gerlach_2021, title={The Computation and Analysis of Invariant Sets of Infinite-Dimensional Systems}, DOI={<a href=\"https://doi.org/10.17619/UNIPB/1-1278\">10.17619/UNIPB/1-1278</a>}, author={Gerlach, Raphael}, year={2021} }","mla":"Gerlach, Raphael. <i>The Computation and Analysis of Invariant Sets of Infinite-Dimensional Systems</i>. 2021, doi:<a href=\"https://doi.org/10.17619/UNIPB/1-1278\">10.17619/UNIPB/1-1278</a>.","short":"R. Gerlach, The Computation and Analysis of Invariant Sets of Infinite-Dimensional Systems, 2021.","apa":"Gerlach, R. (2021). <i>The Computation and Analysis of Invariant Sets of Infinite-Dimensional Systems</i>. <a href=\"https://doi.org/10.17619/UNIPB/1-1278\">https://doi.org/10.17619/UNIPB/1-1278</a>"},"year":"2021","main_file_link":[{"url":"https://digital.ub.uni-paderborn.de/hs/download/pdf/6214949","open_access":"1"}],"doi":"10.17619/UNIPB/1-1278","title":"The Computation and Analysis of Invariant Sets of Infinite-Dimensional Systems","supervisor":[{"first_name":"Michael","last_name":"Dellnitz ","full_name":"Dellnitz , Michael"},{"full_name":"Koltai, Péter","last_name":"Koltai","first_name":"Péter"}],"date_created":"2022-06-20T09:54:24Z","author":[{"full_name":"Gerlach, Raphael","id":"32655","last_name":"Gerlach","first_name":"Raphael"}],"oa":"1","date_updated":"2022-06-20T13:40:30Z"},{"series_title":"Studies in Systems, Decision and Control","user_id":"32655","department":[{"_id":"101"}],"_id":"17994","status":"public","editor":[{"first_name":"Oliver","full_name":"Junge, Oliver","last_name":"Junge"},{"first_name":"Oliver","last_name":"Schütze","full_name":"Schütze, Oliver"},{"full_name":"Ober-Blöbaum, Sina","last_name":"Ober-Blöbaum","first_name":"Sina"},{"first_name":"Kathrin","full_name":"Padberg-Gehle, Kathrin","last_name":"Padberg-Gehle"}],"type":"book_chapter","main_file_link":[{"url":"https://link.springer.com/chapter/10.1007/978-3-030-51264-4_3"}],"doi":"10.1007/978-3-030-51264-4_3","author":[{"full_name":"Gerlach, Raphael","id":"32655","last_name":"Gerlach","first_name":"Raphael"},{"first_name":"Adrian","last_name":"Ziessler","full_name":"Ziessler, Adrian"}],"volume":304,"date_updated":"2023-11-17T13:13:25Z","citation":{"chicago":"Gerlach, Raphael, and Adrian Ziessler. “The Approximation of Invariant Sets in Infinite Dimensional Dynamical Systems.” In <i>Advances in Dynamics, Optimization and Computation</i>, edited by Oliver Junge, Oliver Schütze, Sina Ober-Blöbaum, and Kathrin Padberg-Gehle, 304:66–85. Studies in Systems, Decision and Control. Cham: Springer International Publishing, 2020. <a href=\"https://doi.org/10.1007/978-3-030-51264-4_3\">https://doi.org/10.1007/978-3-030-51264-4_3</a>.","ieee":"R. Gerlach and A. Ziessler, “The Approximation of Invariant Sets in Infinite Dimensional Dynamical Systems,” in <i>Advances in Dynamics, Optimization and Computation</i>, vol. 304, O. Junge, O. Schütze, S. Ober-Blöbaum, and K. Padberg-Gehle, Eds. Cham: Springer International Publishing, 2020, pp. 66–85.","ama":"Gerlach R, Ziessler A. The Approximation of Invariant Sets in Infinite Dimensional Dynamical Systems. In: Junge O, Schütze O, Ober-Blöbaum S, Padberg-Gehle K, eds. <i>Advances in Dynamics, Optimization and Computation</i>. Vol 304. Studies in Systems, Decision and Control. Springer International Publishing; 2020:66-85. doi:<a href=\"https://doi.org/10.1007/978-3-030-51264-4_3\">10.1007/978-3-030-51264-4_3</a>","apa":"Gerlach, R., &#38; Ziessler, A. (2020). The Approximation of Invariant Sets in Infinite Dimensional Dynamical Systems. In O. Junge, O. Schütze, S. Ober-Blöbaum, &#38; K. Padberg-Gehle (Eds.), <i>Advances in Dynamics, Optimization and Computation</i> (Vol. 304, pp. 66–85). Springer International Publishing. <a href=\"https://doi.org/10.1007/978-3-030-51264-4_3\">https://doi.org/10.1007/978-3-030-51264-4_3</a>","mla":"Gerlach, Raphael, and Adrian Ziessler. “The Approximation of Invariant Sets in Infinite Dimensional Dynamical Systems.” <i>Advances in Dynamics, Optimization and Computation</i>, edited by Oliver Junge et al., vol. 304, Springer International Publishing, 2020, pp. 66–85, doi:<a href=\"https://doi.org/10.1007/978-3-030-51264-4_3\">10.1007/978-3-030-51264-4_3</a>.","bibtex":"@inbook{Gerlach_Ziessler_2020, place={Cham}, series={Studies in Systems, Decision and Control}, title={The Approximation of Invariant Sets in Infinite Dimensional Dynamical Systems}, volume={304}, DOI={<a href=\"https://doi.org/10.1007/978-3-030-51264-4_3\">10.1007/978-3-030-51264-4_3</a>}, booktitle={Advances in Dynamics, Optimization and Computation}, publisher={Springer International Publishing}, author={Gerlach, Raphael and Ziessler, Adrian}, editor={Junge, Oliver and Schütze, Oliver and Ober-Blöbaum, Sina and Padberg-Gehle, Kathrin}, year={2020}, pages={66–85}, collection={Studies in Systems, Decision and Control} }","short":"R. Gerlach, A. Ziessler, in: O. Junge, O. Schütze, S. Ober-Blöbaum, K. Padberg-Gehle (Eds.), Advances in Dynamics, Optimization and Computation, Springer International Publishing, Cham, 2020, pp. 66–85."},"intvolume":"       304","page":"66-85","place":"Cham","publication_status":"published","publication_identifier":{"issn":["2198-4182","2198-4190"],"isbn":["9783030512637","9783030512644"]},"language":[{"iso":"eng"}],"abstract":[{"lang":"eng","text":"In this work we review the novel framework for the computation of finite dimensional invariant sets of infinite dimensional dynamical systems developed in [6] and [36]. By utilizing results on embedding techniques for infinite dimensional systems we extend a classical subdivision scheme [8] as well as a continuation algorithm [7] for the computation of attractors and invariant manifolds of finite dimensional systems to the infinite dimensional case. We show how to implement this approach for the analysis of delay differential equations and partial differential equations and illustrate the feasibility of our implementation by computing the attractor of the Mackey-Glass equation and the unstable manifold of the one-dimensional Kuramoto-Sivashinsky equation."}],"publication":"Advances in Dynamics, Optimization and Computation","title":"The Approximation of Invariant Sets in Infinite Dimensional Dynamical Systems","date_created":"2020-08-14T15:02:22Z","publisher":"Springer International Publishing","year":"2020"},{"doi":"10.1080/14689367.2019.1661355","main_file_link":[{"url":"https://doi.org/10.1080/14689367.2019.1661355"}],"title":"On the equivariance properties of self-adjoint matrices","volume":35,"author":[{"full_name":"Dellnitz, Michael","last_name":"Dellnitz","first_name":"Michael"},{"first_name":"Bennet","last_name":"Gebken","full_name":"Gebken, Bennet","id":"32643"},{"first_name":"Raphael","id":"32655","full_name":"Gerlach, Raphael","last_name":"Gerlach"},{"first_name":"Stefan","last_name":"Klus","full_name":"Klus, Stefan"}],"date_created":"2020-04-16T14:07:25Z","date_updated":"2023-11-17T13:12:59Z","page":"197-215","intvolume":"        35","citation":{"apa":"Dellnitz, M., Gebken, B., Gerlach, R., &#38; Klus, S. (2020). On the equivariance properties of self-adjoint matrices. <i>Dynamical Systems</i>, <i>35</i>(2), 197–215. <a href=\"https://doi.org/10.1080/14689367.2019.1661355\">https://doi.org/10.1080/14689367.2019.1661355</a>","short":"M. Dellnitz, B. Gebken, R. Gerlach, S. Klus, Dynamical Systems 35 (2020) 197–215.","mla":"Dellnitz, Michael, et al. “On the Equivariance Properties of Self-Adjoint Matrices.” <i>Dynamical Systems</i>, vol. 35, no. 2, 2020, pp. 197–215, doi:<a href=\"https://doi.org/10.1080/14689367.2019.1661355\">10.1080/14689367.2019.1661355</a>.","bibtex":"@article{Dellnitz_Gebken_Gerlach_Klus_2020, title={On the equivariance properties of self-adjoint matrices}, volume={35}, DOI={<a href=\"https://doi.org/10.1080/14689367.2019.1661355\">10.1080/14689367.2019.1661355</a>}, number={2}, journal={Dynamical Systems}, author={Dellnitz, Michael and Gebken, Bennet and Gerlach, Raphael and Klus, Stefan}, year={2020}, pages={197–215} }","ieee":"M. Dellnitz, B. Gebken, R. Gerlach, and S. Klus, “On the equivariance properties of self-adjoint matrices,” <i>Dynamical Systems</i>, vol. 35, no. 2, pp. 197–215, 2020, doi: <a href=\"https://doi.org/10.1080/14689367.2019.1661355\">10.1080/14689367.2019.1661355</a>.","chicago":"Dellnitz, Michael, Bennet Gebken, Raphael Gerlach, and Stefan Klus. “On the Equivariance Properties of Self-Adjoint Matrices.” <i>Dynamical Systems</i> 35, no. 2 (2020): 197–215. <a href=\"https://doi.org/10.1080/14689367.2019.1661355\">https://doi.org/10.1080/14689367.2019.1661355</a>.","ama":"Dellnitz M, Gebken B, Gerlach R, Klus S. On the equivariance properties of self-adjoint matrices. <i>Dynamical Systems</i>. 2020;35(2):197-215. doi:<a href=\"https://doi.org/10.1080/14689367.2019.1661355\">10.1080/14689367.2019.1661355</a>"},"year":"2020","issue":"2","publication_identifier":{"issn":["1468-9367","1468-9375"]},"publication_status":"published","language":[{"iso":"eng"}],"department":[{"_id":"101"}],"user_id":"32655","_id":"16712","status":"public","abstract":[{"text":"We investigate self-adjoint matrices A∈Rn,n with respect to their equivariance properties. We show in particular that a matrix is self-adjoint if and only if it is equivariant with respect to the action of a group Γ2(A)⊂O(n) which is isomorphic to ⊗nk=1Z2. If the self-adjoint matrix possesses multiple eigenvalues – this may, for instance, be induced by symmetry properties of an underlying dynamical system – then A is even equivariant with respect to the action of a group Γ(A)≃∏ki=1O(mi) where m1,…,mk are the multiplicities of the eigenvalues λ1,…,λk of A. We discuss implications of this result for equivariant bifurcation problems, and we briefly address further applications for the Procrustes problem, graph symmetries and Taylor expansions.","lang":"eng"}],"publication":"Dynamical Systems","type":"journal_article"},{"year":"2020","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>.","short":"R. Gerlach, A. Ziessler, B. Eckhardt, M. Dellnitz, SIAM Journal on Applied Dynamical Systems (2020) 705–723.","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} }","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>.","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>"},"publication_identifier":{"issn":["1536-0040"]},"publication_status":"published","title":"A Set-Oriented Path Following Method for the Approximation of Parameter Dependent Attractors","doi":"10.1137/19m1247139","main_file_link":[{"url":"https://epubs.siam.org/doi/epdf/10.1137/19M1247139"}],"date_updated":"2024-10-01T13:37:43Z","date_created":"2020-04-16T14:05:41Z","author":[{"first_name":"Raphael","last_name":"Gerlach","id":"32655","full_name":"Gerlach, Raphael"},{"full_name":"Ziessler, Adrian","last_name":"Ziessler","first_name":"Adrian"},{"first_name":"Bruno","full_name":"Eckhardt, Bruno","last_name":"Eckhardt"},{"first_name":"Michael","full_name":"Dellnitz, Michael","last_name":"Dellnitz"}],"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"}],"status":"public","publication":"SIAM Journal on Applied Dynamical Systems","type":"journal_article","language":[{"iso":"eng"}],"_id":"16710","department":[{"_id":"101"}],"user_id":"32655"},{"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"}],"status":"public","publication":"SIAM Journal on Applied Dynamical Systems","type":"journal_article","language":[{"iso":"eng"}],"_id":"16708","department":[{"_id":"101"}],"user_id":"32655","year":"2019","intvolume":"        18","page":"1265-1292","citation":{"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>.","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} }","short":"A. Ziessler, M. Dellnitz, R. Gerlach, SIAM Journal on Applied Dynamical Systems 18 (2019) 1265–1292.","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>","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>"},"publication_identifier":{"issn":["1536-0040"]},"publication_status":"published","issue":"3","title":"The Numerical Computation of Unstable Manifolds for Infinite Dimensional Dynamical Systems by Embedding Techniques","doi":"10.1137/18m1204395","main_file_link":[{"url":"https://epubs.siam.org/doi/epdf/10.1137/18M1204395"}],"date_updated":"2023-11-17T13:13:09Z","volume":18,"date_created":"2020-04-16T14:04:20Z","author":[{"last_name":"Ziessler","full_name":"Ziessler, Adrian","first_name":"Adrian"},{"last_name":"Dellnitz","full_name":"Dellnitz, Michael","first_name":"Michael"},{"full_name":"Gerlach, Raphael","id":"32655","last_name":"Gerlach","first_name":"Raphael"}]},{"has_accepted_license":"1","year":"2019","citation":{"mla":"Gerlach, Raphael, et al. “Revealing the Intrinsic Geometry of Finite Dimensional Invariant Sets of  Infinite Dimensional Dynamical Systems.” <i>ArXiv:1902.08824</i>, 2019.","short":"R. Gerlach, P. Koltai, M. Dellnitz, ArXiv:1902.08824 (2019).","bibtex":"@article{Gerlach_Koltai_Dellnitz_2019, title={Revealing the intrinsic geometry of finite dimensional invariant sets of  infinite dimensional dynamical systems}, journal={arXiv:1902.08824}, author={Gerlach, Raphael and Koltai, Péter and Dellnitz, Michael}, year={2019} }","apa":"Gerlach, R., Koltai, P., &#38; Dellnitz, M. (2019). Revealing the intrinsic geometry of finite dimensional invariant sets of  infinite dimensional dynamical systems. In <i>arXiv:1902.08824</i>.","chicago":"Gerlach, Raphael, Péter Koltai, and Michael Dellnitz. “Revealing the Intrinsic Geometry of Finite Dimensional Invariant Sets of  Infinite Dimensional Dynamical Systems.” <i>ArXiv:1902.08824</i>, 2019.","ieee":"R. Gerlach, P. Koltai, and M. Dellnitz, “Revealing the intrinsic geometry of finite dimensional invariant sets of  infinite dimensional dynamical systems,” <i>arXiv:1902.08824</i>. 2019.","ama":"Gerlach R, Koltai P, Dellnitz M. Revealing the intrinsic geometry of finite dimensional invariant sets of  infinite dimensional dynamical systems. <i>arXiv:190208824</i>. Published online 2019."},"date_updated":"2024-09-24T12:09:27Z","oa":"1","date_created":"2020-04-16T14:06:21Z","author":[{"first_name":"Raphael","last_name":"Gerlach","id":"32655","full_name":"Gerlach, Raphael"},{"last_name":"Koltai","full_name":"Koltai, Péter","first_name":"Péter"},{"last_name":"Dellnitz","full_name":"Dellnitz, Michael","first_name":"Michael"}],"title":"Revealing the intrinsic geometry of finite dimensional invariant sets of  infinite dimensional dynamical systems","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1902.08824"}],"type":"preprint","publication":"arXiv:1902.08824","abstract":[{"text":"Embedding techniques allow the approximations of finite dimensional\r\nattractors and manifolds of infinite dimensional dynamical systems via\r\nsubdivision and continuation methods. These approximations give a topological\r\none-to-one image of the original set. In order to additionally reveal their\r\ngeometry we use diffusion mapst o find intrinsic coordinates. We illustrate our\r\nresults on the unstable manifold of the one-dimensional Kuramoto--Sivashinsky\r\nequation, as well as for the attractor of the Mackey-Glass delay differential\r\nequation.","lang":"eng"}],"status":"public","_id":"16711","external_id":{"arxiv":["1902.08824"]},"user_id":"32655","department":[{"_id":"101"}],"ddc":["510"],"language":[{"iso":"eng"}]}]