[{"language":[{"iso":"eng"}],"date_updated":"2024-01-03T17:00:48Z","project":[{"grant_number":"453112019","name":"Algorithmen für Schwarmrobotik: Verteiltes Rechnen trifft Dynamische Systeme","_id":"106"}],"publication_status":"submitted","department":[{"_id":"101"}],"related_material":{"link":[{"url":"https://math.uni-paderborn.de/en/ag/chair-of-applied-mathematics/research/research-projects/swarmdynamics","relation":"supplementary_material"}]},"title":"On the Dynamical Hierarchy in Gathering Protocols with Circulant Topologies","external_id":{"arxiv":["2305.06632"]},"citation":{"ieee":"R. Gerlach, S. von der Gracht, and M. Dellnitz, “On the Dynamical Hierarchy in Gathering Protocols with Circulant Topologies,” arXiv:2305.06632. .","short":"R. Gerlach, S. von der Gracht, M. Dellnitz, ArXiv:2305.06632 (n.d.).","bibtex":"@article{Gerlach_von der Gracht_Dellnitz, title={On the Dynamical Hierarchy in Gathering Protocols with Circulant Topologies}, journal={arXiv:2305.06632}, author={Gerlach, Raphael and von der Gracht, Sören and Dellnitz, Michael} }","mla":"Gerlach, Raphael, et al. “On the Dynamical Hierarchy in Gathering Protocols with Circulant Topologies.” ArXiv:2305.06632.","chicago":"Gerlach, Raphael, Sören von der Gracht, and Michael Dellnitz. “On the Dynamical Hierarchy in Gathering Protocols with Circulant Topologies.” ArXiv:2305.06632, n.d.","ama":"Gerlach R, von der Gracht S, Dellnitz M. On the Dynamical Hierarchy in Gathering Protocols with Circulant Topologies. arXiv:230506632.","apa":"Gerlach, R., von der Gracht, S., & Dellnitz, M. (n.d.). On the Dynamical Hierarchy in Gathering Protocols with Circulant Topologies. In arXiv:2305.06632."},"year":"2023","type":"preprint","page":"38","main_file_link":[{"url":"https://arxiv.org/abs/2305.06632"}],"_id":"44840","status":"public","date_created":"2023-05-12T10:16:06Z","author":[{"first_name":"Raphael","full_name":"Gerlach, Raphael","last_name":"Gerlach","id":"32655"},{"first_name":"Sören","orcid":"0000-0002-8054-2058","full_name":"von der Gracht, Sören","last_name":"von der Gracht","id":"97359"},{"first_name":"Michael","full_name":"Dellnitz, Michael","last_name":"Dellnitz"}],"keyword":["Dynamical Systems","Coupled Systems","Distributed Computing","Robot Swarms","Autonomous Mobile Robots","Gathering"],"publication":"arXiv:2305.06632","user_id":"32655","abstract":[{"lang":"eng","text":"In this article we investigate the convergence behavior of gathering protocols with fixed circulant topologies using tools form dynamical systems. Given a fixed number of mobile entities moving in the Euclidean plane, we model a gathering protocol as a system of ordinary differential equations whose equilibria are exactly all possible gathering points. Then, we find necessary and sufficient conditions for the structure of the underlying interaction graph such that the protocol is stable and converging, i.e., gathering, in the distributive computing sense by using tools from dynamical systems. Moreover, these tools allow for a more fine grained analysis in terms of speed of convergence in the dynamical systems sense. In fact, we derive a decomposition\r\nof the state space into stable invariant subspaces with different convergence\r\nrates. In particular, this decomposition is identical for every (linear)\r\ncirculant gathering protocol, whereas only the convergence rates depend on the\r\nweights in interaction graph itself."}]},{"date_updated":"2022-06-20T13:40:30Z","_id":"32057","oa":"1","doi":"10.17619/UNIPB/1-1278","main_file_link":[{"open_access":"1","url":"https://digital.ub.uni-paderborn.de/hs/download/pdf/6214949"}],"supervisor":[{"first_name":"Michael","full_name":"Dellnitz , Michael","last_name":"Dellnitz "},{"last_name":"Koltai","full_name":"Koltai, Péter","first_name":"Péter"}],"language":[{"iso":"eng"}],"citation":{"ieee":"R. Gerlach, The Computation and Analysis of Invariant Sets of Infinite-Dimensional Systems. 2021.","short":"R. Gerlach, The Computation and Analysis of Invariant Sets of Infinite-Dimensional Systems, 2021.","bibtex":"@book{Gerlach_2021, title={The Computation and Analysis of Invariant Sets of Infinite-Dimensional Systems}, DOI={10.17619/UNIPB/1-1278}, author={Gerlach, Raphael}, year={2021} }","mla":"Gerlach, Raphael. The Computation and Analysis of Invariant Sets of Infinite-Dimensional Systems. 2021, doi:10.17619/UNIPB/1-1278.","apa":"Gerlach, R. (2021). The Computation and Analysis of Invariant Sets of Infinite-Dimensional Systems. https://doi.org/10.17619/UNIPB/1-1278","ama":"Gerlach R. The Computation and Analysis of Invariant Sets of Infinite-Dimensional Systems.; 2021. doi:10.17619/UNIPB/1-1278","chicago":"Gerlach, Raphael. The Computation and Analysis of Invariant Sets of Infinite-Dimensional Systems, 2021. https://doi.org/10.17619/UNIPB/1-1278."},"type":"dissertation","year":"2021","abstract":[{"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":"ger"},{"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.","lang":"eng"}],"user_id":"32643","title":"The Computation and Analysis of Invariant Sets of Infinite-Dimensional Systems","author":[{"id":"32655","last_name":"Gerlach","full_name":"Gerlach, Raphael","first_name":"Raphael"}],"department":[{"_id":"101"}],"status":"public","date_created":"2022-06-20T09:54:24Z"},{"date_created":"2020-04-16T14:05:41Z","status":"public","publication_identifier":{"issn":["1536-0040"]},"publication_status":"published","department":[{"_id":"101"}],"publication":"SIAM Journal on Applied Dynamical Systems","author":[{"last_name":"Gerlach","id":"32655","first_name":"Raphael","full_name":"Gerlach, Raphael"},{"full_name":"Ziessler, Adrian","first_name":"Adrian","last_name":"Ziessler"},{"first_name":"Bruno","full_name":"Eckhardt, Bruno","last_name":"Eckhardt"},{"full_name":"Dellnitz, Michael","first_name":"Michael","last_name":"Dellnitz"}],"user_id":"32655","title":"A Set-Oriented Path Following Method for the Approximation of Parameter Dependent Attractors","abstract":[{"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","lang":"eng"}],"language":[{"iso":"eng"}],"page":"705-723","type":"journal_article","year":"2020","citation":{"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","short":"R. Gerlach, A. Ziessler, B. Eckhardt, M. Dellnitz, SIAM Journal on Applied Dynamical Systems (2020) 705–723.","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","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.","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} }","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."},"main_file_link":[{"url":"https://epubs.siam.org/doi/epdf/10.1137/19M1247139","open_access":"1"}],"oa":"1","doi":"10.1137/19m1247139","date_updated":"2022-06-21T09:18:46Z","_id":"16710"},{"title":"The Approximation of Invariant Sets in Infinite Dimensional Dynamical Systems","place":"Cham","editor":[{"last_name":"Junge","full_name":"Junge, Oliver","first_name":"Oliver"},{"last_name":"Schütze","first_name":"Oliver","full_name":"Schütze, Oliver"},{"full_name":"Ober-Blöbaum, Sina","first_name":"Sina","last_name":"Ober-Blöbaum"},{"last_name":"Padberg-Gehle","full_name":"Padberg-Gehle, Kathrin","first_name":"Kathrin"}],"publication_status":"published","publication_identifier":{"isbn":["9783030512637","9783030512644"],"issn":["2198-4182","2198-4190"]},"department":[{"_id":"101"}],"doi":"10.1007/978-3-030-51264-4_3","date_updated":"2023-11-17T13:13:25Z","language":[{"iso":"eng"}],"series_title":"Studies in Systems, Decision and Control","user_id":"32655","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."}],"status":"public","date_created":"2020-08-14T15:02:22Z","volume":304,"author":[{"full_name":"Gerlach, Raphael","first_name":"Raphael","id":"32655","last_name":"Gerlach"},{"full_name":"Ziessler, Adrian","first_name":"Adrian","last_name":"Ziessler"}],"publisher":"Springer International Publishing","publication":"Advances in Dynamics, Optimization and Computation","_id":"17994","intvolume":" 304","type":"book_chapter","year":"2020","citation":{"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={10.1007/978-3-030-51264-4_3}, 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} }","mla":"Gerlach, Raphael, and Adrian Ziessler. “The Approximation of Invariant Sets in Infinite Dimensional Dynamical Systems.” Advances in Dynamics, Optimization and Computation, edited by Oliver Junge et al., vol. 304, Springer International Publishing, 2020, pp. 66–85, doi:10.1007/978-3-030-51264-4_3.","apa":"Gerlach, R., & Ziessler, A. (2020). The Approximation of Invariant Sets in Infinite Dimensional Dynamical Systems. In O. Junge, O. Schütze, S. Ober-Blöbaum, & K. Padberg-Gehle (Eds.), Advances in Dynamics, Optimization and Computation (Vol. 304, pp. 66–85). Springer International Publishing. https://doi.org/10.1007/978-3-030-51264-4_3","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. Advances in Dynamics, Optimization and Computation. Vol 304. Studies in Systems, Decision and Control. Springer International Publishing; 2020:66-85. doi:10.1007/978-3-030-51264-4_3","chicago":"Gerlach, Raphael, and Adrian Ziessler. “The Approximation of Invariant Sets in Infinite Dimensional Dynamical Systems.” In Advances in Dynamics, Optimization and Computation, 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. https://doi.org/10.1007/978-3-030-51264-4_3.","ieee":"R. Gerlach and A. Ziessler, “The Approximation of Invariant Sets in Infinite Dimensional Dynamical Systems,” in Advances in Dynamics, Optimization and Computation, vol. 304, O. Junge, O. Schütze, S. Ober-Blöbaum, and K. Padberg-Gehle, Eds. Cham: Springer International Publishing, 2020, pp. 66–85.","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."},"page":"66-85","main_file_link":[{"url":"https://link.springer.com/chapter/10.1007/978-3-030-51264-4_3"}]},{"date_created":"2020-04-16T14:07:25Z","status":"public","volume":35,"publication":"Dynamical Systems","author":[{"first_name":"Michael","full_name":"Dellnitz, Michael","last_name":"Dellnitz"},{"first_name":"Bennet","full_name":"Gebken, Bennet","last_name":"Gebken","id":"32643"},{"last_name":"Gerlach","id":"32655","first_name":"Raphael","full_name":"Gerlach, Raphael"},{"last_name":"Klus","full_name":"Klus, Stefan","first_name":"Stefan"}],"user_id":"32655","abstract":[{"lang":"eng","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."}],"page":"197-215","citation":{"ieee":"M. Dellnitz, B. Gebken, R. Gerlach, and S. Klus, “On the equivariance properties of self-adjoint matrices,” Dynamical Systems, vol. 35, no. 2, pp. 197–215, 2020, doi: 10.1080/14689367.2019.1661355.","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.” Dynamical Systems, vol. 35, no. 2, 2020, pp. 197–215, doi:10.1080/14689367.2019.1661355.","bibtex":"@article{Dellnitz_Gebken_Gerlach_Klus_2020, title={On the equivariance properties of self-adjoint matrices}, volume={35}, DOI={10.1080/14689367.2019.1661355}, number={2}, journal={Dynamical Systems}, author={Dellnitz, Michael and Gebken, Bennet and Gerlach, Raphael and Klus, Stefan}, year={2020}, pages={197–215} }","chicago":"Dellnitz, Michael, Bennet Gebken, Raphael Gerlach, and Stefan Klus. “On the Equivariance Properties of Self-Adjoint Matrices.” Dynamical Systems 35, no. 2 (2020): 197–215. https://doi.org/10.1080/14689367.2019.1661355.","ama":"Dellnitz M, Gebken B, Gerlach R, Klus S. On the equivariance properties of self-adjoint matrices. Dynamical Systems. 2020;35(2):197-215. doi:10.1080/14689367.2019.1661355","apa":"Dellnitz, M., Gebken, B., Gerlach, R., & Klus, S. (2020). On the equivariance properties of self-adjoint matrices. Dynamical Systems, 35(2), 197–215. https://doi.org/10.1080/14689367.2019.1661355"},"type":"journal_article","year":"2020","main_file_link":[{"url":"https://doi.org/10.1080/14689367.2019.1661355"}],"issue":"2","_id":"16712","intvolume":" 35","publication_status":"published","publication_identifier":{"issn":["1468-9367","1468-9375"]},"department":[{"_id":"101"}],"title":"On the equivariance properties of self-adjoint matrices","language":[{"iso":"eng"}],"doi":"10.1080/14689367.2019.1661355","date_updated":"2023-11-17T13:12:59Z"},{"date_created":"2020-04-16T14:06:21Z","status":"public","has_accepted_license":"1","department":[{"_id":"101"}],"publication":"arXiv:1902.08824","author":[{"last_name":"Gerlach","id":"32655","first_name":"Raphael","full_name":"Gerlach, Raphael"},{"last_name":"Koltai","first_name":"Péter","full_name":"Koltai, Péter"},{"first_name":"Michael","full_name":"Dellnitz, Michael","last_name":"Dellnitz"}],"user_id":"32655","ddc":["510"],"title":"Revealing the intrinsic geometry of finite dimensional invariant sets of infinite dimensional dynamical systems","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"}],"language":[{"iso":"eng"}],"type":"preprint","year":"2019","citation":{"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} }","mla":"Gerlach, Raphael, et al. “Revealing the Intrinsic Geometry of Finite Dimensional Invariant Sets of Infinite Dimensional Dynamical Systems.” ArXiv:1902.08824, 2019.","ieee":"R. Gerlach, P. Koltai, and M. Dellnitz, “Revealing the intrinsic geometry of finite dimensional invariant sets of infinite dimensional dynamical systems,” arXiv:1902.08824. 2019.","chicago":"Gerlach, Raphael, Péter Koltai, and Michael Dellnitz. “Revealing the Intrinsic Geometry of Finite Dimensional Invariant Sets of Infinite Dimensional Dynamical Systems.” ArXiv:1902.08824, 2019.","short":"R. Gerlach, P. Koltai, M. Dellnitz, ArXiv:1902.08824 (2019).","ama":"Gerlach R, Koltai P, Dellnitz M. Revealing the intrinsic geometry of finite dimensional invariant sets of infinite dimensional dynamical systems. arXiv:190208824. Published online 2019.","apa":"Gerlach, R., Koltai, P., & Dellnitz, M. (2019). Revealing the intrinsic geometry of finite dimensional invariant sets of infinite dimensional dynamical systems. In arXiv:1902.08824."},"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1902.08824"}],"oa":"1","date_updated":"2022-06-20T13:17:53Z","_id":"16711"},{"issue":"3","_id":"16708","intvolume":" 18","page":"1265-1292","type":"journal_article","year":"2019","citation":{"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.","short":"A. Ziessler, M. Dellnitz, R. Gerlach, SIAM Journal on Applied Dynamical Systems 18 (2019) 1265–1292.","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.","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} }","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","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","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."},"main_file_link":[{"url":"https://epubs.siam.org/doi/epdf/10.1137/18M1204395"}],"user_id":"32655","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"}],"date_created":"2020-04-16T14:04:20Z","status":"public","volume":18,"publication":"SIAM Journal on Applied Dynamical Systems","author":[{"last_name":"Ziessler","first_name":"Adrian","full_name":"Ziessler, Adrian"},{"first_name":"Michael","full_name":"Dellnitz, Michael","last_name":"Dellnitz"},{"id":"32655","last_name":"Gerlach","full_name":"Gerlach, Raphael","first_name":"Raphael"}],"doi":"10.1137/18m1204395","date_updated":"2023-11-17T13:13:09Z","language":[{"iso":"eng"}],"title":"The Numerical Computation of Unstable Manifolds for Infinite Dimensional Dynamical Systems by Embedding Techniques","publication_status":"published","publication_identifier":{"issn":["1536-0040"]},"department":[{"_id":"101"}]}]