[{"author":[{"id":"97359","full_name":"von der Gracht, Sören","orcid":"0000-0002-8054-2058","last_name":"von der Gracht","first_name":"Sören"},{"full_name":"Lohse, Alexander","last_name":"Lohse","first_name":"Alexander"}],"year":"2026","title":"Design of Hierarchical Excitable Networks","date_updated":"2026-03-09T08:26:49Z","language":[{"iso":"eng"}],"publication":"arXiv:2603.06157","related_material":{"link":[{"url":"https://s-vdg.github.io/publication/design-of-hierarchical-excitable-networks/design-of-hierarchical-excitable-networks.pdf","relation":"research_paper"}]},"abstract":[{"lang":"eng","text":"We provide a method to systematically construct vector fields for which the dynamics display transitions corresponding to a desired hierarchical connection structure. This structure is given as a finite set of directed graphs $\\mathbf{G}_1,\\dotsc,\\mathbf{G}_N$ (the lower level), together with another digraph $\\mathbfΓ$ on $N$ vertices (the top level). The dynamic realizations of $\\mathbf{G}_1,\\dotsc,\\mathbf{G}_N$ are heteroclinic networks and they can be thought of as individual connection patterns on a given set of states. Edges in $\\mathbfΓ$ correspond to transitions between these different patterns. In our construction, the connections given through $\\mathbfΓ$ are not heteroclinic, but excitable with zero threshold. This describes a dynamical transition between two invariant sets where every $δ$-neighborhood of the first set contains an initial condition with $ω$-limit in the second set. Thus, we prove a theorem that allows the systematic creation of hierarchical networks that are excitable on the top level, and heteroclinic on the lower level. Our results modify and extend the simplex realization method by Ashwin & Postlethwaite."}],"date_created":"2026-03-09T08:22:58Z","file":[{"creator":"svdg","date_created":"2026-03-09T08:26:04Z","file_name":"design-of-hierarchical-excitable-networks.pdf","file_size":5179491,"access_level":"closed","relation":"main_file","date_updated":"2026-03-09T08:26:04Z","file_id":"64866","success":1,"content_type":"application/pdf"}],"department":[{"_id":"101"},{"_id":"841"}],"type":"preprint","status":"public","has_accepted_license":"1","_id":"64865","ddc":["510"],"user_id":"97359","citation":{"chicago":"Gracht, Sören von der, and Alexander Lohse. “Design of Hierarchical Excitable Networks.” <i>ArXiv:2603.06157</i>, 2026.","short":"S. von der Gracht, A. Lohse, ArXiv:2603.06157 (2026).","ieee":"S. von der Gracht and A. Lohse, “Design of Hierarchical Excitable Networks,” <i>arXiv:2603.06157</i>. 2026.","apa":"von der Gracht, S., &#38; Lohse, A. (2026). Design of Hierarchical Excitable Networks. In <i>arXiv:2603.06157</i>.","bibtex":"@article{von der Gracht_Lohse_2026, title={Design of Hierarchical Excitable Networks}, journal={arXiv:2603.06157}, author={von der Gracht, Sören and Lohse, Alexander}, year={2026} }","ama":"von der Gracht S, Lohse A. Design of Hierarchical Excitable Networks. <i>arXiv:260306157</i>. Published online 2026.","mla":"von der Gracht, Sören, and Alexander Lohse. “Design of Hierarchical Excitable Networks.” <i>ArXiv:2603.06157</i>, 2026."},"file_date_updated":"2026-03-09T08:26:04Z","external_id":{"arxiv":["2603.06157"]}},{"doi":"10.1016/j.chaos.2026.118196","language":[{"iso":"eng"}],"article_number":"118196","article_type":"original","intvolume":"       208","publication_status":"published","date_updated":"2026-03-16T08:42:56Z","author":[{"id":"97359","full_name":"von der Gracht, Sören","last_name":"von der Gracht","orcid":"0000-0002-8054-2058","first_name":"Sören"},{"full_name":"Nijholt, Eddie","last_name":"Nijholt","first_name":"Eddie"},{"last_name":"Rink","first_name":"Bob","full_name":"Rink, Bob"}],"publication_identifier":{"issn":["0960-0779"]},"title":"Homogeneous coupled cell systems with high-dimensional internal dynamics","year":"2026","department":[{"_id":"101"},{"_id":"841"}],"type":"journal_article","keyword":["Coupled cell systems","Network dynamics","Dimension reduction","Bifurcation theory","Symmetry","Monoid representation theory"],"date_created":"2026-03-16T08:39:07Z","file":[{"date_updated":"2026-03-16T08:40:04Z","relation":"main_file","access_level":"closed","file_size":1951746,"file_name":"homogeneous-coupled-cell-systems-with-high-dimensional-internal-dynamics.pdf","success":1,"content_type":"application/pdf","file_id":"64980","creator":"svdg","date_created":"2026-03-16T08:40:04Z"}],"abstract":[{"text":"We investigate homogeneous coupled cell systems with high-dimensional internal dynamics. In many studies on network dynamics, the analysis is restricted to networks with one-dimensional internal dynamics. Here, we show how symmetry explains the relation between dynamical behavior of systems with one-dimensional internal dynamics and with higher dimensional internal dynamics, when the underlying network topology is the same. Fundamental networks of homogeneous coupled cell systems (B. Rink, J. Sanders. Coupled Cell Networks and Their Hidden Symmetries. SIAM J. Math. Anal. 46.2 (2014)) can be expressed in terms of monoid representations, which uniquely decompose into indecomposable subrepresentations. In the high-dimensional internal dynamics case, these subrepresentations are isomorphic to multiple copies of those one computes in the one-dimensional internal dynamics case. This has interesting implications for possible center subspaces in bifurcation analysis. We describe the effect on steady state and Hopf bifurcations in l-parameter families of network vector fields. The main results in that regard are that (1) generic one-parameter steady state bifurcations are qualitatively independent of the dimension of the internal dynamics and that, (2) in order to observe all generic l-parameter bifurcations that may occur for internal dynamics of any dimension, the internal dynamics has to be at least l-dimensional for steady state bifurcations and 2l-dimensional for Hopf bifurcations. Furthermore, we illustrate how additional structure in the network can be exploited to obtain even greater understanding of bifurcation scenarios in the high-dimensional case beyond qualitative statements about the collective dynamics. One-parameter steady state bifurcations in feedforward networks exhibit an unusual amplification in the asymptotic growth rates of individual cells, when these are one-dimensional (S. von der Gracht, E. Nijholt, B. Rink. Amplified steady state bifurcations in feedforward networks. Nonlinearity 35.4 (2022)). As another main result, we prove that (3) the same cells exhibit this amplifying effect with the same growth rates when the internal dynamics is high-dimensional.","lang":"eng"}],"publication":"Chaos, Solitons & Fractals","volume":208,"user_id":"97359","ddc":["510"],"_id":"64979","publisher":"Elsevier BV","has_accepted_license":"1","status":"public","external_id":{"arxiv":["2510.06740"]},"citation":{"ieee":"S. von der Gracht, E. Nijholt, and B. Rink, “Homogeneous coupled cell systems with high-dimensional internal dynamics,” <i>Chaos, Solitons &#38; Fractals</i>, vol. 208, Art. no. 118196, 2026, doi: <a href=\"https://doi.org/10.1016/j.chaos.2026.118196\">10.1016/j.chaos.2026.118196</a>.","apa":"von der Gracht, S., Nijholt, E., &#38; Rink, B. (2026). Homogeneous coupled cell systems with high-dimensional internal dynamics. <i>Chaos, Solitons &#38; Fractals</i>, <i>208</i>, Article 118196. <a href=\"https://doi.org/10.1016/j.chaos.2026.118196\">https://doi.org/10.1016/j.chaos.2026.118196</a>","chicago":"Gracht, Sören von der, Eddie Nijholt, and Bob Rink. “Homogeneous Coupled Cell Systems with High-Dimensional Internal Dynamics.” <i>Chaos, Solitons &#38; Fractals</i> 208 (2026). <a href=\"https://doi.org/10.1016/j.chaos.2026.118196\">https://doi.org/10.1016/j.chaos.2026.118196</a>.","short":"S. von der Gracht, E. Nijholt, B. Rink, Chaos, Solitons &#38; Fractals 208 (2026).","mla":"von der Gracht, Sören, et al. “Homogeneous Coupled Cell Systems with High-Dimensional Internal Dynamics.” <i>Chaos, Solitons &#38; Fractals</i>, vol. 208, 118196, Elsevier BV, 2026, doi:<a href=\"https://doi.org/10.1016/j.chaos.2026.118196\">10.1016/j.chaos.2026.118196</a>.","bibtex":"@article{von der Gracht_Nijholt_Rink_2026, title={Homogeneous coupled cell systems with high-dimensional internal dynamics}, volume={208}, DOI={<a href=\"https://doi.org/10.1016/j.chaos.2026.118196\">10.1016/j.chaos.2026.118196</a>}, number={118196}, journal={Chaos, Solitons &#38; Fractals}, publisher={Elsevier BV}, author={von der Gracht, Sören and Nijholt, Eddie and Rink, Bob}, year={2026} }","ama":"von der Gracht S, Nijholt E, Rink B. Homogeneous coupled cell systems with high-dimensional internal dynamics. <i>Chaos, Solitons &#38; Fractals</i>. 2026;208. doi:<a href=\"https://doi.org/10.1016/j.chaos.2026.118196\">10.1016/j.chaos.2026.118196</a>"},"file_date_updated":"2026-03-16T08:40:04Z"},{"doi":"10.48550/arXiv.2606.25934","user_id":"103797","language":[{"iso":"eng"}],"_id":"66057","date_updated":"2026-06-26T06:09:29Z","author":[{"last_name":"Bullerjahn","first_name":"Nils","orcid":"https://orcid.org/0009-0003-8460-1574","full_name":"Bullerjahn, Nils","id":"103797"},{"id":"100441","last_name":"Kovács","orcid":"0000-0001-9872-3474","first_name":"Balázs","full_name":"Kovács, Balázs"}],"status":"public","year":"2026","title":"Error estimates for $A$-stable backward difference full discretizations of Willmore flow of closed surfaces","department":[{"_id":"841"}],"type":"journal_article","date_created":"2026-06-26T06:03:08Z","citation":{"bibtex":"@article{Bullerjahn_Kovács_2026, title={Error estimates for $A$-stable backward difference full discretizations of Willmore flow of closed surfaces}, DOI={<a href=\"https://doi.org/10.48550/arXiv.2606.25934\">10.48550/arXiv.2606.25934</a>}, journal={ArXiv}, author={Bullerjahn, Nils and Kovács, Balázs}, year={2026} }","ama":"Bullerjahn N, Kovács B. Error estimates for $A$-stable backward difference full discretizations of Willmore flow of closed surfaces. <i>ArXiv</i>. Published online 2026. doi:<a href=\"https://doi.org/10.48550/arXiv.2606.25934\">10.48550/arXiv.2606.25934</a>","mla":"Bullerjahn, Nils, and Balázs Kovács. “Error Estimates for $A$-Stable Backward Difference Full Discretizations of Willmore Flow of Closed Surfaces.” <i>ArXiv</i>, 2026, doi:<a href=\"https://doi.org/10.48550/arXiv.2606.25934\">10.48550/arXiv.2606.25934</a>.","short":"N. Bullerjahn, B. Kovács, ArXiv (2026).","chicago":"Bullerjahn, Nils, and Balázs Kovács. “Error Estimates for $A$-Stable Backward Difference Full Discretizations of Willmore Flow of Closed Surfaces.” <i>ArXiv</i>, 2026. <a href=\"https://doi.org/10.48550/arXiv.2606.25934\">https://doi.org/10.48550/arXiv.2606.25934</a>.","ieee":"N. Bullerjahn and B. Kovács, “Error estimates for $A$-stable backward difference full discretizations of Willmore flow of closed surfaces,” <i>ArXiv</i>, 2026, doi: <a href=\"https://doi.org/10.48550/arXiv.2606.25934\">10.48550/arXiv.2606.25934</a>.","apa":"Bullerjahn, N., &#38; Kovács, B. (2026). Error estimates for $A$-stable backward difference full discretizations of Willmore flow of closed surfaces. <i>ArXiv</i>. <a href=\"https://doi.org/10.48550/arXiv.2606.25934\">https://doi.org/10.48550/arXiv.2606.25934</a>"},"publication":"ArXiv"},{"citation":{"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>.","short":"R. Gerlach, S. von der Gracht, M. Dellnitz, in: Lecture Notes in Computer Science, Springer Nature Switzerland, Cham, 2025.","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>","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.","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>","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} }","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>."},"project":[{"name":"Algorithmen für Schwarmrobotik: Verteiltes Rechnen trifft Dynamische Systeme","_id":"106","grant_number":"453112019"}],"place":"Cham","external_id":{"arxiv":["2503.07576"]},"oa":"1","status":"public","publisher":"Springer Nature Switzerland","_id":"60048","user_id":"32655","publication":"Lecture Notes in Computer Science","date_created":"2025-05-27T08:17:03Z","department":[{"_id":"101"}],"type":"book_chapter","author":[{"id":"32655","orcid":"0009-0002-4750-2051","first_name":"Raphael","last_name":"Gerlach","full_name":"Gerlach, Raphael"},{"id":"97359","full_name":"von der Gracht, Sören","orcid":"0000-0002-8054-2058","first_name":"Sören","last_name":"von der Gracht"},{"full_name":"Dellnitz, Michael","last_name":"Dellnitz","first_name":"Michael"}],"publication_identifier":{"isbn":["9783031917356","9783031917363"],"issn":["0302-9743","1611-3349"]},"title":"On the Dynamical Hierarchy in Gathering Protocols with Circulant Topologies","year":"2025","date_updated":"2025-05-27T08:22:42Z","publication_status":"published","language":[{"iso":"eng"}],"main_file_link":[{"open_access":"1","url":" ArXiv:2503.07576"}],"doi":"10.1007/978-3-031-91736-3_19"},{"author":[{"id":"103797","first_name":"Nils","last_name":"Bullerjahn","orcid":"https://orcid.org/0009-0003-8460-1574","full_name":"Bullerjahn, Nils"},{"id":"100441","last_name":"Kovács","first_name":"Balázs","orcid":"0000-0001-9872-3474","full_name":"Kovács, Balázs"}],"title":"Error estimates for full discretization of Cahn--Hilliard equation with dynamic boundary conditions","status":"public","year":"2025","article_type":"original","date_updated":"2026-02-18T14:46:18Z","publication_status":"accepted","_id":"55459","language":[{"iso":"eng"}],"doi":"10.1093/imanum/draf009","user_id":"100441","citation":{"ama":"Bullerjahn N, Kovács B. Error estimates for full discretization of Cahn--Hilliard equation with dynamic boundary conditions. <i>IMA Journal of Numerical Analysis</i>. doi:<a href=\"https://doi.org/10.1093/imanum/draf009\">10.1093/imanum/draf009</a>","bibtex":"@article{Bullerjahn_Kovács, title={Error estimates for full discretization of Cahn--Hilliard equation with dynamic boundary conditions}, DOI={<a href=\"https://doi.org/10.1093/imanum/draf009\">10.1093/imanum/draf009</a>}, journal={IMA Journal of Numerical Analysis}, author={Bullerjahn, Nils and Kovács, Balázs} }","mla":"Bullerjahn, Nils, and Balázs Kovács. “Error Estimates for Full Discretization of Cahn--Hilliard Equation with Dynamic Boundary Conditions.” <i>IMA Journal of Numerical Analysis</i>, doi:<a href=\"https://doi.org/10.1093/imanum/draf009\">10.1093/imanum/draf009</a>.","short":"N. Bullerjahn, B. Kovács, IMA Journal of Numerical Analysis (n.d.).","chicago":"Bullerjahn, Nils, and Balázs Kovács. “Error Estimates for Full Discretization of Cahn--Hilliard Equation with Dynamic Boundary Conditions.” <i>IMA Journal of Numerical Analysis</i>, n.d. <a href=\"https://doi.org/10.1093/imanum/draf009\">https://doi.org/10.1093/imanum/draf009</a>.","apa":"Bullerjahn, N., &#38; Kovács, B. (n.d.). Error estimates for full discretization of Cahn--Hilliard equation with dynamic boundary conditions. <i>IMA Journal of Numerical Analysis</i>. <a href=\"https://doi.org/10.1093/imanum/draf009\">https://doi.org/10.1093/imanum/draf009</a>","ieee":"N. Bullerjahn and B. Kovács, “Error estimates for full discretization of Cahn--Hilliard equation with dynamic boundary conditions,” <i>IMA Journal of Numerical Analysis</i>, doi: <a href=\"https://doi.org/10.1093/imanum/draf009\">10.1093/imanum/draf009</a>."},"publication":"IMA Journal of Numerical Analysis","date_created":"2024-07-31T09:03:45Z","department":[{"_id":"841"}],"type":"journal_article"},{"author":[{"last_name":"Edelmann","first_name":"Dominik","full_name":"Edelmann, Dominik"},{"full_name":"Kovács, Balázs","first_name":"Balázs","orcid":"0000-0001-9872-3474","last_name":"Kovács","id":"100441"},{"last_name":"Lubich","first_name":"Christian","full_name":"Lubich, Christian"}],"year":"2025","title":"Numerical analysis of an evolving bulk--surface model of tumour growth","status":"public","intvolume":"        45","date_updated":"2026-02-18T14:44:54Z","_id":"53141","language":[{"iso":"eng"}],"page":"2581--2627","volume":45,"doi":"10.1093/imanum/drae077","user_id":"100441","citation":{"short":"D. Edelmann, B. Kovács, C. Lubich, IMA Journal of Numerical Analysis 45 (2025) 2581--2627.","chicago":"Edelmann, Dominik, Balázs Kovács, and Christian Lubich. “Numerical Analysis of an Evolving Bulk--Surface Model of Tumour Growth.” <i>IMA Journal of Numerical Analysis</i> 45, no. 5 (2025): 2581--2627. <a href=\"https://doi.org/10.1093/imanum/drae077\">https://doi.org/10.1093/imanum/drae077</a>.","apa":"Edelmann, D., Kovács, B., &#38; Lubich, C. (2025). Numerical analysis of an evolving bulk--surface model of tumour growth. <i>IMA Journal of Numerical Analysis</i>, <i>45</i>(5), 2581--2627. <a href=\"https://doi.org/10.1093/imanum/drae077\">https://doi.org/10.1093/imanum/drae077</a>","ieee":"D. Edelmann, B. Kovács, and C. Lubich, “Numerical analysis of an evolving bulk--surface model of tumour growth,” <i>IMA Journal of Numerical Analysis</i>, vol. 45, no. 5, pp. 2581--2627, 2025, doi: <a href=\"https://doi.org/10.1093/imanum/drae077\">10.1093/imanum/drae077</a>.","ama":"Edelmann D, Kovács B, Lubich C. Numerical analysis of an evolving bulk--surface model of tumour growth. <i>IMA Journal of Numerical Analysis</i>. 2025;45(5):2581--2627. doi:<a href=\"https://doi.org/10.1093/imanum/drae077\">10.1093/imanum/drae077</a>","bibtex":"@article{Edelmann_Kovács_Lubich_2025, title={Numerical analysis of an evolving bulk--surface model of tumour growth}, volume={45}, DOI={<a href=\"https://doi.org/10.1093/imanum/drae077\">10.1093/imanum/drae077</a>}, number={5}, journal={IMA Journal of Numerical Analysis}, author={Edelmann, Dominik and Kovács, Balázs and Lubich, Christian}, year={2025}, pages={2581--2627} }","mla":"Edelmann, Dominik, et al. “Numerical Analysis of an Evolving Bulk--Surface Model of Tumour Growth.” <i>IMA Journal of Numerical Analysis</i>, vol. 45, no. 5, 2025, pp. 2581--2627, doi:<a href=\"https://doi.org/10.1093/imanum/drae077\">10.1093/imanum/drae077</a>."},"publication":"IMA Journal of Numerical Analysis","issue":"5","date_created":"2024-04-03T09:11:36Z","department":[{"_id":"841"}],"type":"journal_article"},{"abstract":[{"text":"In this paper, we prove that spatially semi-discrete evolving finite element\r\nmethod for parabolic equations on a given evolving hypersurface of arbitrary\r\ndimensions preserves the maximal $L^p$-regularity at the discrete level. We\r\nfirst establish the results on a stationary surface and then extend them, via a\r\nperturbation argument, to the case where the underlying surface is evolving\r\nunder a prescribed velocity field. The proof combines techniques in evolving\r\nfinite element method, properties of Green's functions on (discretised) closed\r\nsurfaces, and local energy estimates for finite element methods","lang":"eng"}],"publication":"IMA Journal of Numerical Analysis","citation":{"short":"G. Bai, B. Kovács, B. Li, IMA Journal of Numerical Analysis (2025).","chicago":"Bai, Genming, Balázs Kovács, and Buyang Li. “Maximal Regularity of Evolving FEMs for Parabolic Equations on an  Evolving Surface.” <i>IMA Journal of Numerical Analysis</i>, 2025. <a href=\"https://doi.org/10.1093/imanum/draf082.\">https://doi.org/10.1093/imanum/draf082.</a>","ieee":"G. Bai, B. Kovács, and B. Li, “Maximal regularity of evolving FEMs for parabolic equations on an  evolving surface,” <i>IMA Journal of Numerical Analysis</i>, 2025, doi: <a href=\"https://doi.org/10.1093/imanum/draf082.\">10.1093/imanum/draf082.</a>","apa":"Bai, G., Kovács, B., &#38; Li, B. (2025). Maximal regularity of evolving FEMs for parabolic equations on an  evolving surface. <i>IMA Journal of Numerical Analysis</i>. <a href=\"https://doi.org/10.1093/imanum/draf082.\">https://doi.org/10.1093/imanum/draf082.</a>","bibtex":"@article{Bai_Kovács_Li_2025, title={Maximal regularity of evolving FEMs for parabolic equations on an  evolving surface}, DOI={<a href=\"https://doi.org/10.1093/imanum/draf082.\">10.1093/imanum/draf082.</a>}, journal={IMA Journal of Numerical Analysis}, author={Bai, Genming and Kovács, Balázs and Li, Buyang}, year={2025} }","ama":"Bai G, Kovács B, Li B. Maximal regularity of evolving FEMs for parabolic equations on an  evolving surface. <i>IMA Journal of Numerical Analysis</i>. Published online 2025. doi:<a href=\"https://doi.org/10.1093/imanum/draf082.\">10.1093/imanum/draf082.</a>","mla":"Bai, Genming, et al. “Maximal Regularity of Evolving FEMs for Parabolic Equations on an  Evolving Surface.” <i>IMA Journal of Numerical Analysis</i>, 2025, doi:<a href=\"https://doi.org/10.1093/imanum/draf082.\">10.1093/imanum/draf082.</a>"},"type":"journal_article","department":[{"_id":"841"}],"external_id":{"arxiv":["2408.14096"]},"date_created":"2024-08-27T07:37:39Z","date_updated":"2026-02-18T14:47:34Z","year":"2025","title":"Maximal regularity of evolving FEMs for parabolic equations on an  evolving surface","status":"public","author":[{"last_name":"Bai","first_name":"Genming","full_name":"Bai, Genming"},{"id":"100441","full_name":"Kovács, Balázs","last_name":"Kovács","first_name":"Balázs","orcid":"0000-0001-9872-3474"},{"full_name":"Li, Buyang","last_name":"Li","first_name":"Buyang"}],"doi":"10.1093/imanum/draf082.","user_id":"100441","language":[{"iso":"eng"}],"_id":"55781"},{"publication":"28th International Conference on Principles of Distributed Systems (OPODIS 2024)","abstract":[{"lang":"eng","text":"In the general pattern formation (GPF) problem, a swarm of simple autonomous,\r\ndisoriented robots must form a given pattern. The robots' simplicity imply a\r\nstrong limitation: When the initial configuration is rotationally symmetric,\r\nonly patterns with a similar symmetry can be formed [Yamashita, Suzyuki; TCS\r\n2010]. The only known algorithm to form large patterns with limited visibility\r\nand without memory requires the robots to start in a near-gathering (a swarm of\r\nconstant diameter) [Hahn et al.; SAND 2024]. However, not only do we not know\r\nany near-gathering algorithm guaranteed to preserve symmetry but most natural\r\ngathering strategies trivially increase symmetries [Castenow et al.; OPODIS\r\n2022].\r\n  Thus, we study near-gathering without changing the swarm's rotational\r\nsymmetry for disoriented, oblivious robots with limited visibility (the\r\nOBLOT-model, see [Flocchini et al.; 2019]). We introduce a technique based on\r\nthe theory of dynamical systems to analyze how a given algorithm affects\r\nsymmetry and provide sufficient conditions for symmetry preservation. Until\r\nnow, it was unknown whether the considered OBLOT-model allows for any\r\nnon-trivial algorithm that always preserves symmetry. Our first result shows\r\nthat a variant of Go-to-the-Average always preserves symmetry but may sometimes\r\nlead to multiple, unconnected near-gathering clusters. Our second result is a\r\nsymmetry-preserving near-gathering algorithm that works on swarms with a convex\r\nboundary (the outer boundary of the unit disc graph) and without holes (circles\r\nof diameter 1 inside the boundary without any robots)."}],"date_created":"2024-10-01T13:29:43Z","keyword":["Swarm Algorithm","Swarm Robots","Distributed Algorithm","Pattern Formation","Limited Visibility","Oblivious"],"type":"conference","department":[{"_id":"101"}],"year":"2025","title":"Symmetry Preservation in Swarms of Oblivious Robots with Limited  Visibility","publication_identifier":{"isbn":["978-3-95977-360-7"],"issn":["1868-8969"]},"author":[{"id":"32655","full_name":"Gerlach, Raphael","first_name":"Raphael","orcid":"0009-0002-4750-2051","last_name":"Gerlach"},{"id":"97359","first_name":"Sören","last_name":"von der Gracht","orcid":"0000-0002-8054-2058","full_name":"von der Gracht, Sören"},{"first_name":"Christopher","last_name":"Hahn","full_name":"Hahn, Christopher"},{"id":"47213","first_name":"Jonas","last_name":"Harbig","full_name":"Harbig, Jonas"},{"full_name":"Kling, Peter","last_name":"Kling","first_name":"Peter"}],"date_updated":"2025-01-09T11:39:19Z","publication_status":"published","intvolume":"       324","main_file_link":[{"url":"https://arxiv.org/abs/2409.19277","open_access":"1"}],"language":[{"iso":"eng"}],"series_title":"Leibniz International Proceedings in Informatics (LIPIcs)","doi":"10.4230/LIPIcs.OPODIS.2024.13","citation":{"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>.","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>.","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>","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>.","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."},"project":[{"name":"Algorithmen für Schwarmrobotik: Verteiltes Rechnen trifft Dynamische Systeme","grant_number":"453112019","_id":"106"}],"external_id":{"arxiv":["2409.19277"]},"oa":"1","status":"public","conference":{"end_date":"2024-12-13","start_date":"2024-12-11","name":"28th International Conference on Principles of Distributed Systems (OPODIS 2024)","location":"Lucca, Italy"},"publisher":"Schloss Dagstuhl -- Leibniz-Zentrum für Informatik","_id":"56298","user_id":"97359","editor":[{"first_name":"Silvia","last_name":"Bonomi","full_name":"Bonomi, Silvia"},{"full_name":"Galletta, Letterio","last_name":"Galletta","first_name":"Letterio"},{"first_name":" Etienne","last_name":"Rivière","full_name":"Rivière,  Etienne"},{"last_name":"Schiavoni","first_name":" Valerio","full_name":"Schiavoni,  Valerio"}],"volume":324},{"_id":"58532","language":[{"iso":"eng"}],"doi":"10.48550/ARXIV.2502.03847","user_id":"103797","author":[{"id":"103797","full_name":"Bullerjahn, Nils","first_name":"Nils","orcid":"https://orcid.org/0009-0003-8460-1574","last_name":"Bullerjahn"}],"title":"Error estimates for full discretization by an almost mass conservation technique for Cahn--Hilliard systems with dynamic boundary conditions","year":"2025","status":"public","date_updated":"2025-02-07T08:28:48Z","date_created":"2025-02-07T08:27:10Z","department":[{"_id":"841"}],"type":"journal_article","citation":{"chicago":"Bullerjahn, Nils. “Error Estimates for Full Discretization by an Almost Mass Conservation Technique for Cahn--Hilliard Systems with Dynamic Boundary Conditions.” <i>ArXiv</i>, 2025. <a href=\"https://doi.org/10.48550/ARXIV.2502.03847\">https://doi.org/10.48550/ARXIV.2502.03847</a>.","ama":"Bullerjahn N. Error estimates for full discretization by an almost mass conservation technique for Cahn--Hilliard systems with dynamic boundary conditions. <i>arXiv</i>. Published online 2025. doi:<a href=\"https://doi.org/10.48550/ARXIV.2502.03847\">10.48550/ARXIV.2502.03847</a>","short":"N. Bullerjahn, ArXiv (2025).","bibtex":"@article{Bullerjahn_2025, title={Error estimates for full discretization by an almost mass conservation technique for Cahn--Hilliard systems with dynamic boundary conditions}, DOI={<a href=\"https://doi.org/10.48550/ARXIV.2502.03847\">10.48550/ARXIV.2502.03847</a>}, journal={arXiv}, author={Bullerjahn, Nils}, year={2025} }","apa":"Bullerjahn, N. (2025). Error estimates for full discretization by an almost mass conservation technique for Cahn--Hilliard systems with dynamic boundary conditions. <i>ArXiv</i>. <a href=\"https://doi.org/10.48550/ARXIV.2502.03847\">https://doi.org/10.48550/ARXIV.2502.03847</a>","mla":"Bullerjahn, Nils. “Error Estimates for Full Discretization by an Almost Mass Conservation Technique for Cahn--Hilliard Systems with Dynamic Boundary Conditions.” <i>ArXiv</i>, 2025, doi:<a href=\"https://doi.org/10.48550/ARXIV.2502.03847\">10.48550/ARXIV.2502.03847</a>.","ieee":"N. Bullerjahn, “Error estimates for full discretization by an almost mass conservation technique for Cahn--Hilliard systems with dynamic boundary conditions,” <i>arXiv</i>, 2025, doi: <a href=\"https://doi.org/10.48550/ARXIV.2502.03847\">10.48550/ARXIV.2502.03847</a>."},"publication":"arXiv"},{"keyword":["dynamical systems","coupled systems","distributed computing","robot swarms","autonomous mobile robots","symmetry","equivariant dynamics"],"type":"preprint","department":[{"_id":"101"}],"file":[{"date_updated":"2025-03-11T08:27:32Z","relation":"main_file","access_level":"open_access","file_size":812198,"file_name":"Analyzing_Symmetries_of_Swarms_of_Mobile_Robots_Using_Equivariant_Dynamical_Systems.pdf","content_type":"application/pdf","file_id":"58954","creator":"svdg","date_created":"2025-03-11T08:27:32Z"}],"date_created":"2025-03-11T08:21:05Z","abstract":[{"lang":"eng","text":"In this article, we investigate symmetry properties of distributed systems of mobile robots. We consider a swarm of n robots in the OBLOT model and analyze their collective Fsync dynamics using of equivariant dynamical systems theory. To this end, we show that the corresponding evolution function commutes with rotational and reflective transformations of R^2. These form a group that is isomorphic to O(2) x S_n, the product group of the orthogonal group and the permutation on n elements. The theory of equivariant dynamical systems is used to deduce a hierarchy along which symmetries of a robot swarm can potentially increase following an arbitrary protocol. By decoupling the Look phase from the Compute and Move phases in the mathematical description of an LCM cycle, this hierarchy can be characterized in terms of automorphisms of connectivity graphs. In particular, we find all possible types of symmetry increase, if the decoupled Compute and Move phase is invertible. Finally, we apply our results to protocols which induce state-dependent linear dynamics, where the reduced system consisting of only the Compute and Move phase is linear."}],"publication":"arXiv:2503.07576","language":[{"iso":"eng"}],"date_updated":"2025-03-11T08:53:02Z","year":"2025","title":"Analyzing Symmetries of Swarms of Mobile Robots Using Equivariant  Dynamical Systems","author":[{"full_name":"Gerlach, Raphael","orcid":"0009-0002-4750-2051","first_name":"Raphael","last_name":"Gerlach","id":"32655"},{"id":"97359","first_name":"Sören","last_name":"von der Gracht","orcid":"0000-0002-8054-2058","full_name":"von der Gracht, Sören"}],"oa":"1","external_id":{"arxiv":["2503.07576"]},"project":[{"name":"Algorithmen für Schwarmrobotik: Verteiltes Rechnen trifft Dynamische Systeme","grant_number":"453112019","_id":"106"}],"file_date_updated":"2025-03-11T08:27:32Z","citation":{"bibtex":"@article{Gerlach_von der Gracht_2025, title={Analyzing Symmetries of Swarms of Mobile Robots Using Equivariant  Dynamical Systems}, journal={arXiv:2503.07576}, author={Gerlach, Raphael and von der Gracht, Sören}, year={2025} }","chicago":"Gerlach, Raphael, and Sören von der Gracht. “Analyzing Symmetries of Swarms of Mobile Robots Using Equivariant  Dynamical Systems.” <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.","short":"R. Gerlach, S. von der Gracht, ArXiv:2503.07576 (2025).","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.","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.","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>."},"user_id":"97359","ddc":["004"],"page":"23","_id":"58953","has_accepted_license":"1","status":"public"},{"file_date_updated":"2024-03-22T09:06:07Z","citation":{"ieee":"C. Bick and S. von der Gracht, “Heteroclinic dynamics in network dynamical systems with higher-order interactions,” <i>Journal of Complex Networks</i>, vol. 12, no. 2, 2024, doi: <a href=\"https://doi.org/10.1093/comnet/cnae009\">10.1093/comnet/cnae009</a>.","apa":"Bick, C., &#38; von der Gracht, S. (2024). Heteroclinic dynamics in network dynamical systems with higher-order interactions. <i>Journal of Complex Networks</i>, <i>12</i>(2). <a href=\"https://doi.org/10.1093/comnet/cnae009\">https://doi.org/10.1093/comnet/cnae009</a>","short":"C. Bick, S. von der Gracht, Journal of Complex Networks 12 (2024).","chicago":"Bick, Christian, and Sören von der Gracht. “Heteroclinic Dynamics in Network Dynamical Systems with Higher-Order Interactions.” <i>Journal of Complex Networks</i> 12, no. 2 (2024). <a href=\"https://doi.org/10.1093/comnet/cnae009\">https://doi.org/10.1093/comnet/cnae009</a>.","mla":"Bick, Christian, and Sören von der Gracht. “Heteroclinic Dynamics in Network Dynamical Systems with Higher-Order Interactions.” <i>Journal of Complex Networks</i>, vol. 12, no. 2, Oxford University Press (OUP), 2024, doi:<a href=\"https://doi.org/10.1093/comnet/cnae009\">10.1093/comnet/cnae009</a>.","bibtex":"@article{Bick_von der Gracht_2024, title={Heteroclinic dynamics in network dynamical systems with higher-order interactions}, volume={12}, DOI={<a href=\"https://doi.org/10.1093/comnet/cnae009\">10.1093/comnet/cnae009</a>}, number={2}, journal={Journal of Complex Networks}, publisher={Oxford University Press (OUP)}, author={Bick, Christian and von der Gracht, Sören}, year={2024} }","ama":"Bick C, von der Gracht S. Heteroclinic dynamics in network dynamical systems with higher-order interactions. <i>Journal of Complex Networks</i>. 2024;12(2). doi:<a href=\"https://doi.org/10.1093/comnet/cnae009\">10.1093/comnet/cnae009</a>"},"oa":"1","external_id":{"arxiv":["2309.02006"]},"has_accepted_license":"1","status":"public","ddc":["510"],"user_id":"97359","volume":12,"publisher":"Oxford University Press (OUP)","_id":"52726","abstract":[{"lang":"eng","text":"Heteroclinic structures organize global features of dynamical systems. We analyse whether heteroclinic structures can arise in network dynamics with higher-order interactions which describe the nonlinear interactions between three or more units. We find that while commonly analysed model equations such as network dynamics on undirected hypergraphs may be useful to describe local dynamics such as cluster synchronization, they give rise to obstructions that allow to design of heteroclinic structures in phase space. By contrast, directed hypergraphs break the homogeneity and lead to vector fields that support heteroclinic structures."}],"publication":"Journal of Complex Networks","issue":"2","keyword":["Applied Mathematics","Computational Mathematics","Control and Optimization","Management Science and Operations Research","Computer Networks and Communications"],"type":"journal_article","department":[{"_id":"101"}],"file":[{"date_created":"2024-03-22T09:06:07Z","creator":"svdg","success":1,"content_type":"application/pdf","file_id":"52728","file_size":649155,"access_level":"closed","file_name":"heteroclinic-dynamics-in-network-dynamical-systems-with-higher-order-interactions.pdf","date_updated":"2024-03-22T09:06:07Z","relation":"main_file"}],"date_created":"2024-03-22T09:04:57Z","date_updated":"2024-03-22T09:11:53Z","publication_status":"published","intvolume":"        12","article_type":"original","year":"2024","title":"Heteroclinic dynamics in network dynamical systems with higher-order interactions","author":[{"full_name":"Bick, Christian","last_name":"Bick","first_name":"Christian"},{"id":"97359","first_name":"Sören","last_name":"von der Gracht","orcid":"0000-0002-8054-2058","full_name":"von der Gracht, Sören"}],"publication_identifier":{"issn":["2051-1329"]},"doi":"10.1093/comnet/cnae009","main_file_link":[{"url":"https://academic.oup.com/comnet/article-pdf/12/2/cnae009/56832119/cnae009.pdf","open_access":"1"}],"language":[{"iso":"eng"}]},{"title":"Numerical surgery for mean curvature flow of surfaces","year":"2024","status":"public","author":[{"id":"100441","full_name":"Kovács, Balázs","last_name":"Kovács","first_name":"Balázs","orcid":"0000-0001-9872-3474"}],"date_updated":"2024-08-27T07:42:56Z","intvolume":"        46","page":"A645--A669","language":[{"iso":"eng"}],"_id":"45972","user_id":"100441","doi":"10.1137/22M1531919","volume":46,"publication":"SIAM Journal on Scientific Computing","issue":"2","citation":{"apa":"Kovács, B. (2024). Numerical surgery for mean curvature flow of surfaces. <i>SIAM Journal on Scientific Computing</i>, <i>46</i>(2), A645--A669. <a href=\"https://doi.org/10.1137/22M1531919\">https://doi.org/10.1137/22M1531919</a>","ieee":"B. Kovács, “Numerical surgery for mean curvature flow of surfaces,” <i>SIAM Journal on Scientific Computing</i>, vol. 46, no. 2, pp. A645--A669, 2024, doi: <a href=\"https://doi.org/10.1137/22M1531919\">10.1137/22M1531919</a>.","chicago":"Kovács, Balázs. “Numerical Surgery for Mean Curvature Flow of Surfaces.” <i>SIAM Journal on Scientific Computing</i> 46, no. 2 (2024): A645--A669. <a href=\"https://doi.org/10.1137/22M1531919\">https://doi.org/10.1137/22M1531919</a>.","short":"B. Kovács, SIAM Journal on Scientific Computing 46 (2024) A645--A669.","mla":"Kovács, Balázs. “Numerical Surgery for Mean Curvature Flow of Surfaces.” <i>SIAM Journal on Scientific Computing</i>, vol. 46, no. 2, 2024, pp. A645--A669, doi:<a href=\"https://doi.org/10.1137/22M1531919\">10.1137/22M1531919</a>.","ama":"Kovács B. Numerical surgery for mean curvature flow of surfaces. <i>SIAM Journal on Scientific Computing</i>. 2024;46(2):A645--A669. doi:<a href=\"https://doi.org/10.1137/22M1531919\">10.1137/22M1531919</a>","bibtex":"@article{Kovács_2024, title={Numerical surgery for mean curvature flow of surfaces}, volume={46}, DOI={<a href=\"https://doi.org/10.1137/22M1531919\">10.1137/22M1531919</a>}, number={2}, journal={SIAM Journal on Scientific Computing}, author={Kovács, Balázs}, year={2024}, pages={A645--A669} }"},"extern":"1","date_created":"2023-07-10T12:32:34Z","type":"journal_article","department":[{"_id":"841"}]},{"abstract":[{"lang":"eng","text":"This paper develops and discusses a residual-based a posteriori error\r\nestimate and a space--time adaptive algorithm for solving parabolic surface\r\npartial differential equations on closed stationary surfaces. The full\r\ndiscretization uses the surface finite element method in space and the backward\r\nEuler method in time. The proposed error indicator bounds the error quantities\r\nglobally in space from above and below, and globally in time from above and\r\nlocally from below. A space--time adaptive algorithm is proposed using the\r\nderived error indicator. Numerical experiments illustrate and complement the\r\ntheory."}],"citation":{"apa":"Kovács, B., &#38; Lantelme, M. F. R. (2024). A posteriori error estimates for parabolic partial differential equations on stationary surfaces. In <i>arXiv:2407.02101</i>.","ieee":"B. Kovács and M. F. R. Lantelme, “A posteriori error estimates for parabolic partial differential equations on stationary surfaces,” <i>arXiv:2407.02101</i>. 2024.","short":"B. Kovács, M.F.R. Lantelme, ArXiv:2407.02101 (2024).","chicago":"Kovács, Balázs, and Michael Frederik Raúl Lantelme. “A Posteriori Error Estimates for Parabolic Partial Differential Equations on Stationary Surfaces.” <i>ArXiv:2407.02101</i>, 2024.","mla":"Kovács, Balázs, and Michael Frederik Raúl Lantelme. “A Posteriori Error Estimates for Parabolic Partial Differential Equations on Stationary Surfaces.” <i>ArXiv:2407.02101</i>, 2024.","ama":"Kovács B, Lantelme MFR. A posteriori error estimates for parabolic partial differential equations on stationary surfaces. <i>arXiv:240702101</i>. Published online 2024.","bibtex":"@article{Kovács_Lantelme_2024, title={A posteriori error estimates for parabolic partial differential equations on stationary surfaces}, journal={arXiv:2407.02101}, author={Kovács, Balázs and Lantelme, Michael Frederik Raúl}, year={2024} }"},"publication":"arXiv:2407.02101","department":[{"_id":"841"}],"type":"preprint","date_created":"2024-07-04T12:53:47Z","external_id":{"arxiv":["2407.02101"]},"date_updated":"2026-02-18T14:45:36Z","author":[{"full_name":"Kovács, Balázs","first_name":"Balázs","orcid":"0000-0001-9872-3474","last_name":"Kovács","id":"100441"},{"full_name":"Lantelme, Michael Frederik Raúl","first_name":"Michael Frederik Raúl","last_name":"Lantelme","id":"102867"}],"year":"2024","status":"public","title":"A posteriori error estimates for parabolic partial differential equations on stationary surfaces","user_id":"100441","language":[{"iso":"eng"}],"_id":"55078"},{"department":[{"_id":"101"}],"keyword":["higher-order interactions","synchrony breaking","network dynamics","coupled cell systems"],"type":"journal_article","date_created":"2025-03-27T10:15:06Z","file":[{"date_updated":"2025-03-27T10:19:48Z","relation":"main_file","access_level":"open_access","file_size":820435,"file_name":"higher-order-interactions-lead-to-reluctant-synchrony-breaking.pdf","content_type":"application/pdf","file_id":"59172","creator":"svdg","date_created":"2025-03-27T10:16:20Z"}],"abstract":[{"text":"To model dynamical systems on networks with higher-order (non-pairwise) interactions, we recently introduced a new class of ordinary differential equations (ODEs) on hypernetworks. Here, we consider one-parameter synchrony breaking bifurcations in such ODEs. We call a synchrony breaking steady-state branch ‘reluctant’ if it is tangent to a synchrony space, but does not lie inside it. We prove that reluctant synchrony breaking is ubiquitous in hypernetwork systems, by constructing a large class of examples that support it. We also give an explicit formula for the order of tangency to the synchrony space of a reluctant steady-state branch.","lang":"eng"}],"publication":"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences","issue":"2301","doi":"10.1098/rspa.2023.0945","language":[{"iso":"eng"}],"intvolume":"       480","date_updated":"2025-03-27T10:19:56Z","publication_status":"published","author":[{"id":"97359","full_name":"von der Gracht, Sören","last_name":"von der Gracht","orcid":"0000-0002-8054-2058","first_name":"Sören"},{"first_name":"Eddie","last_name":"Nijholt","full_name":"Nijholt, Eddie"},{"last_name":"Rink","first_name":"Bob","full_name":"Rink, Bob"}],"publication_identifier":{"issn":["1364-5021","1471-2946"]},"title":"Higher-order interactions lead to ‘reluctant’ synchrony breaking","year":"2024","oa":"1","citation":{"ieee":"S. von der Gracht, E. Nijholt, and B. Rink, “Higher-order interactions lead to ‘reluctant’ synchrony breaking,” <i>Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences</i>, vol. 480, no. 2301, 2024, doi: <a href=\"https://doi.org/10.1098/rspa.2023.0945\">10.1098/rspa.2023.0945</a>.","apa":"von der Gracht, S., Nijholt, E., &#38; Rink, B. (2024). Higher-order interactions lead to ‘reluctant’ synchrony breaking. <i>Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences</i>, <i>480</i>(2301). <a href=\"https://doi.org/10.1098/rspa.2023.0945\">https://doi.org/10.1098/rspa.2023.0945</a>","chicago":"Gracht, Sören von der, Eddie Nijholt, and Bob Rink. “Higher-Order Interactions Lead to ‘Reluctant’ Synchrony Breaking.” <i>Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences</i> 480, no. 2301 (2024). <a href=\"https://doi.org/10.1098/rspa.2023.0945\">https://doi.org/10.1098/rspa.2023.0945</a>.","short":"S. von der Gracht, E. Nijholt, B. Rink, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 480 (2024).","mla":"von der Gracht, Sören, et al. “Higher-Order Interactions Lead to ‘Reluctant’ Synchrony Breaking.” <i>Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences</i>, vol. 480, no. 2301, The Royal Society, 2024, doi:<a href=\"https://doi.org/10.1098/rspa.2023.0945\">10.1098/rspa.2023.0945</a>.","bibtex":"@article{von der Gracht_Nijholt_Rink_2024, title={Higher-order interactions lead to ‘reluctant’ synchrony breaking}, volume={480}, DOI={<a href=\"https://doi.org/10.1098/rspa.2023.0945\">10.1098/rspa.2023.0945</a>}, number={2301}, journal={Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences}, publisher={The Royal Society}, author={von der Gracht, Sören and Nijholt, Eddie and Rink, Bob}, year={2024} }","ama":"von der Gracht S, Nijholt E, Rink B. Higher-order interactions lead to ‘reluctant’ synchrony breaking. <i>Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences</i>. 2024;480(2301). doi:<a href=\"https://doi.org/10.1098/rspa.2023.0945\">10.1098/rspa.2023.0945</a>"},"file_date_updated":"2025-03-27T10:19:48Z","volume":480,"ddc":["510"],"user_id":"97359","publisher":"The Royal Society","_id":"59171","has_accepted_license":"1","status":"public"},{"citation":{"bibtex":"@article{von der Gracht_Nijholt_Rink_2023, title={Hypernetworks: Cluster Synchronization Is a Higher-Order Effect}, volume={83}, DOI={<a href=\"https://doi.org/10.1137/23m1561075\">10.1137/23m1561075</a>}, number={6}, journal={SIAM Journal on Applied Mathematics}, publisher={Society for Industrial &#38; Applied Mathematics (SIAM)}, author={von der Gracht, Sören and Nijholt, Eddie and Rink, Bob}, year={2023}, pages={2329–2353} }","ama":"von der Gracht S, Nijholt E, Rink B. Hypernetworks: Cluster Synchronization Is a Higher-Order Effect. <i>SIAM Journal on Applied Mathematics</i>. 2023;83(6):2329-2353. doi:<a href=\"https://doi.org/10.1137/23m1561075\">10.1137/23m1561075</a>","mla":"von der Gracht, Sören, et al. “Hypernetworks: Cluster Synchronization Is a Higher-Order Effect.” <i>SIAM Journal on Applied Mathematics</i>, vol. 83, no. 6, Society for Industrial &#38; Applied Mathematics (SIAM), 2023, pp. 2329–53, doi:<a href=\"https://doi.org/10.1137/23m1561075\">10.1137/23m1561075</a>.","chicago":"Gracht, Sören von der, Eddie Nijholt, and Bob Rink. “Hypernetworks: Cluster Synchronization Is a Higher-Order Effect.” <i>SIAM Journal on Applied Mathematics</i> 83, no. 6 (2023): 2329–53. <a href=\"https://doi.org/10.1137/23m1561075\">https://doi.org/10.1137/23m1561075</a>.","short":"S. von der Gracht, E. Nijholt, B. Rink, SIAM Journal on Applied Mathematics 83 (2023) 2329–2353.","ieee":"S. von der Gracht, E. Nijholt, and B. Rink, “Hypernetworks: Cluster Synchronization Is a Higher-Order Effect,” <i>SIAM Journal on Applied Mathematics</i>, vol. 83, no. 6, pp. 2329–2353, 2023, doi: <a href=\"https://doi.org/10.1137/23m1561075\">10.1137/23m1561075</a>.","apa":"von der Gracht, S., Nijholt, E., &#38; Rink, B. (2023). Hypernetworks: Cluster Synchronization Is a Higher-Order Effect. <i>SIAM Journal on Applied Mathematics</i>, <i>83</i>(6), 2329–2353. <a href=\"https://doi.org/10.1137/23m1561075\">https://doi.org/10.1137/23m1561075</a>"},"external_id":{"arxiv":["2302.08974"]},"status":"public","publisher":"Society for Industrial & Applied Mathematics (SIAM)","_id":"49326","page":"2329-2353","volume":83,"user_id":"97359","issue":"6","publication":"SIAM Journal on Applied Mathematics","abstract":[{"text":"Many networked systems are governed by non-pairwise interactions between nodes. The resulting higher-order interaction structure can then be encoded by means of a hypernetwork. In this paper we consider dynamical systems on hypernetworks by defining a class of admissible maps for every such hypernetwork. We explain how to classify robust cluster synchronization patterns on hypernetworks by finding balanced partitions, and we generalize the concept of a graph fibration to the hypernetwork context. We also show that robust synchronization patterns are only fully determined by polynomial admissible maps of high order. This means that, unlike in dyadic networks, cluster synchronization on hypernetworks is a higher-order, i.e., nonlinear, effect. We give a formula, in terms of the order of the hypernetwork, for the degree of the polynomial admissible maps that determine robust synchronization patterns. We also demonstrate that this degree is optimal by investigating a class of examples. We conclude by demonstrating how this effect may cause remarkable synchrony breaking bifurcations that occur at high polynomial degree.","lang":"eng"}],"date_created":"2023-11-29T10:50:05Z","department":[{"_id":"101"}],"type":"journal_article","keyword":["Applied Mathematics"],"author":[{"full_name":"von der Gracht, Sören","orcid":"0000-0002-8054-2058","last_name":"von der Gracht","first_name":"Sören","id":"97359"},{"full_name":"Nijholt, Eddie","first_name":"Eddie","last_name":"Nijholt"},{"first_name":"Bob","last_name":"Rink","full_name":"Rink, Bob"}],"publication_identifier":{"issn":["0036-1399","1095-712X"]},"year":"2023","title":"Hypernetworks: Cluster Synchronization Is a Higher-Order Effect","intvolume":"        83","publication_status":"published","date_updated":"2023-11-29T10:52:23Z","language":[{"iso":"eng"}],"doi":"10.1137/23m1561075"},{"user_id":"100441","doi":"10.1093/imanum/drad037","_id":"45971","publisher":"Oxford University Press (OUP)","language":[{"iso":"eng"}],"publication_status":"published","date_updated":"2024-04-03T09:15:27Z","title":"Error analysis for the numerical approximation of the harmonic map heat flow with nodal constraints","year":"2023","status":"public","author":[{"first_name":"Sören","last_name":"Bartels","full_name":"Bartels, Sören"},{"id":"100441","first_name":"Balázs","last_name":"Kovács","orcid":"0000-0001-9872-3474","full_name":"Kovács, Balázs"},{"first_name":"Zhangxian","last_name":"Wang","full_name":"Wang, Zhangxian"}],"publication_identifier":{"issn":["0272-4979","1464-3642"]},"type":"journal_article","keyword":["Applied Mathematics","Computational Mathematics","General Mathematics"],"department":[{"_id":"841"}],"date_created":"2023-07-10T12:32:10Z","abstract":[{"lang":"eng","text":"<jats:title>Abstract</jats:title>\r\n               <jats:p>An error estimate for a canonical discretization of the harmonic map heat flow into spheres is derived. The numerical scheme uses standard finite elements with a nodal treatment of linearized unit-length constraints. The analysis is based on elementary approximation results and only uses the discrete weak formulation.</jats:p>"}],"publication":"IMA Journal of Numerical Analysis","citation":{"mla":"Bartels, Sören, et al. “Error Analysis for the Numerical Approximation of the Harmonic Map Heat Flow with Nodal Constraints.” <i>IMA Journal of Numerical Analysis</i>, Oxford University Press (OUP), 2023, doi:<a href=\"https://doi.org/10.1093/imanum/drad037\">10.1093/imanum/drad037</a>.","ama":"Bartels S, Kovács B, Wang Z. Error analysis for the numerical approximation of the harmonic map heat flow with nodal constraints. <i>IMA Journal of Numerical Analysis</i>. Published online 2023. doi:<a href=\"https://doi.org/10.1093/imanum/drad037\">10.1093/imanum/drad037</a>","bibtex":"@article{Bartels_Kovács_Wang_2023, title={Error analysis for the numerical approximation of the harmonic map heat flow with nodal constraints}, DOI={<a href=\"https://doi.org/10.1093/imanum/drad037\">10.1093/imanum/drad037</a>}, journal={IMA Journal of Numerical Analysis}, publisher={Oxford University Press (OUP)}, author={Bartels, Sören and Kovács, Balázs and Wang, Zhangxian}, year={2023} }","apa":"Bartels, S., Kovács, B., &#38; Wang, Z. (2023). Error analysis for the numerical approximation of the harmonic map heat flow with nodal constraints. <i>IMA Journal of Numerical Analysis</i>. <a href=\"https://doi.org/10.1093/imanum/drad037\">https://doi.org/10.1093/imanum/drad037</a>","ieee":"S. Bartels, B. Kovács, and Z. Wang, “Error analysis for the numerical approximation of the harmonic map heat flow with nodal constraints,” <i>IMA Journal of Numerical Analysis</i>, 2023, doi: <a href=\"https://doi.org/10.1093/imanum/drad037\">10.1093/imanum/drad037</a>.","chicago":"Bartels, Sören, Balázs Kovács, and Zhangxian Wang. “Error Analysis for the Numerical Approximation of the Harmonic Map Heat Flow with Nodal Constraints.” <i>IMA Journal of Numerical Analysis</i>, 2023. <a href=\"https://doi.org/10.1093/imanum/drad037\">https://doi.org/10.1093/imanum/drad037</a>.","short":"S. Bartels, B. Kovács, Z. Wang, IMA Journal of Numerical Analysis (2023)."}},{"language":[{"iso":"eng"}],"_id":"53140","user_id":"100441","doi":"10.48550/ARXIV.2312.05511","author":[{"full_name":"Contri, Alessandro","first_name":"Alessandro","last_name":"Contri"},{"id":"100441","last_name":"Kovács","first_name":"Balázs","orcid":"0000-0001-9872-3474","full_name":"Kovács, Balázs"},{"full_name":"Massing, André","first_name":"André","last_name":"Massing"}],"year":"2023","status":"public","title":"Error analysis of BDF 1-6 time-stepping methods for the transient Stokes problem: velocity and pressure estimates","date_updated":"2024-04-03T09:12:47Z","date_created":"2024-04-03T09:08:38Z","department":[{"_id":"841"}],"type":"journal_article","citation":{"bibtex":"@article{Contri_Kovács_Massing_2023, title={Error analysis of BDF 1-6 time-stepping methods for the transient Stokes problem: velocity and pressure estimates}, DOI={<a href=\"https://doi.org/10.48550/ARXIV.2312.05511\">10.48550/ARXIV.2312.05511</a>}, journal={arXiv}, author={Contri, Alessandro and Kovács, Balázs and Massing, André}, year={2023} }","ama":"Contri A, Kovács B, Massing A. Error analysis of BDF 1-6 time-stepping methods for the transient Stokes problem: velocity and pressure estimates. <i>arXiv</i>. Published online 2023. doi:<a href=\"https://doi.org/10.48550/ARXIV.2312.05511\">10.48550/ARXIV.2312.05511</a>","mla":"Contri, Alessandro, et al. “Error Analysis of BDF 1-6 Time-Stepping Methods for the Transient Stokes Problem: Velocity and Pressure Estimates.” <i>ArXiv</i>, 2023, doi:<a href=\"https://doi.org/10.48550/ARXIV.2312.05511\">10.48550/ARXIV.2312.05511</a>.","chicago":"Contri, Alessandro, Balázs Kovács, and André Massing. “Error Analysis of BDF 1-6 Time-Stepping Methods for the Transient Stokes Problem: Velocity and Pressure Estimates.” <i>ArXiv</i>, 2023. <a href=\"https://doi.org/10.48550/ARXIV.2312.05511\">https://doi.org/10.48550/ARXIV.2312.05511</a>.","short":"A. Contri, B. Kovács, A. Massing, ArXiv (2023).","ieee":"A. Contri, B. Kovács, and A. Massing, “Error analysis of BDF 1-6 time-stepping methods for the transient Stokes problem: velocity and pressure estimates,” <i>arXiv</i>, 2023, doi: <a href=\"https://doi.org/10.48550/ARXIV.2312.05511\">10.48550/ARXIV.2312.05511</a>.","apa":"Contri, A., Kovács, B., &#38; Massing, A. (2023). Error analysis of BDF 1-6 time-stepping methods for the transient Stokes problem: velocity and pressure estimates. <i>ArXiv</i>. <a href=\"https://doi.org/10.48550/ARXIV.2312.05511\">https://doi.org/10.48550/ARXIV.2312.05511</a>"},"publication":"arXiv","abstract":[{"lang":"eng","text":"We present a new stability and error analysis of fully discrete approximation schemes for the transient Stokes equation. For the spatial discretization, we consider a wide class of Galerkin finite element methods which includes both inf-sup stable spaces and symmetric pressure stabilized formulations. We extend the results from Burman and Fernández [\\textit{SIAM J. Numer. Anal.}, 47 (2009), pp. 409-439] and provide a unified theoretical analysis of backward difference formulae (BDF methods) of order 1 to 6. The main novelty of our approach lies in the use of Dahlquist's G-stability concept together with multiplier techniques introduced by Nevannlina-Odeh and recently by Akrivis et al. [\\textit{SIAM J. Numer. Anal.}, 59 (2021), pp. 2449-2472] to derive optimal stability and error estimates for both the velocity and the pressure. When combined with a method dependent Ritz projection for the initial data, unconditional stability can be shown while for arbitrary interpolation, pressure stability is subordinate to the fulfillment of a mild inverse CFL-type condition between space and time discretizations."}]},{"abstract":[{"lang":"eng","text":"We present a novel method for high-order phase reduction in networks of\r\nweakly coupled oscillators and, more generally, perturbations of reducible\r\nnormally hyperbolic (quasi-)periodic tori. Our method works by computing an\r\nasymptotic expansion for an embedding of the perturbed invariant torus, as well\r\nas for the reduced phase dynamics in local coordinates. Both can be determined\r\nto arbitrary degrees of accuracy, and we show that the phase dynamics may\r\ndirectly be obtained in normal form. We apply the method to predict remote\r\nsynchronisation in a chain of coupled Stuart-Landau oscillators."}],"publication":"arXiv:2306.03320","citation":{"short":"S. von der Gracht, E. Nijholt, B. Rink, ArXiv:2306.03320 (n.d.).","chicago":"Gracht, Sören von der, Eddie Nijholt, and Bob Rink. “A Parametrisation Method for High-Order Phase Reduction in Coupled  Oscillator Networks.” <i>ArXiv:2306.03320</i>, n.d.","apa":"von der Gracht, S., Nijholt, E., &#38; Rink, B. (n.d.). A parametrisation method for high-order phase reduction in coupled  oscillator networks. In <i>arXiv:2306.03320</i>.","ieee":"S. von der Gracht, E. Nijholt, and B. Rink, “A parametrisation method for high-order phase reduction in coupled  oscillator networks,” <i>arXiv:2306.03320</i>. .","ama":"von der Gracht S, Nijholt E, Rink B. A parametrisation method for high-order phase reduction in coupled  oscillator networks. <i>arXiv:230603320</i>.","bibtex":"@article{von der Gracht_Nijholt_Rink, title={A parametrisation method for high-order phase reduction in coupled  oscillator networks}, journal={arXiv:2306.03320}, author={von der Gracht, Sören and Nijholt, Eddie and Rink, Bob} }","mla":"von der Gracht, Sören, et al. “A Parametrisation Method for High-Order Phase Reduction in Coupled  Oscillator Networks.” <i>ArXiv:2306.03320</i>."},"type":"preprint","department":[{"_id":"101"}],"external_id":{"arxiv":["2306.03320"]},"date_created":"2023-06-07T07:57:28Z","publication_status":"submitted","date_updated":"2023-06-07T07:59:06Z","year":"2023","status":"public","title":"A parametrisation method for high-order phase reduction in coupled  oscillator networks","author":[{"orcid":"0000-0002-8054-2058","last_name":"von der Gracht","first_name":"Sören","full_name":"von der Gracht, Sören","id":"97359"},{"first_name":"Eddie","last_name":"Nijholt","full_name":"Nijholt, Eddie"},{"full_name":"Rink, Bob","first_name":"Bob","last_name":"Rink"}],"user_id":"97359","main_file_link":[{"url":"https://arxiv.org/pdf/2306.03320"}],"page":"29","language":[{"iso":"eng"}],"_id":"45498"},{"date_created":"2022-09-06T11:38:15Z","keyword":["Applied Mathematics","General Physics and Astronomy","Mathematical Physics","Statistical and Nonlinear Physics"],"type":"journal_article","publication":"Nonlinearity","issue":"4","abstract":[{"lang":"eng","text":"We investigate bifurcations in feedforward coupled cell networks. Feedforward structure (the absence of feedback) can be defined by a partial order on the cells. We use this property to study generic one-parameter steady state bifurcations for such networks. Branching solutions and their asymptotics are described in terms of Taylor coefficients of the internal dynamics. They can be determined via an algorithm that only exploits the network structure. Similar to previous results on feedforward chains, we observe amplifications of the growth rates of steady state branches induced by the feedforward structure. However, contrary to these earlier results, as the interaction scenarios can be more complicated in general feedforward networks, different branching patterns and different amplifications can occur for different regions in the space of Taylor coefficients."}],"extern":"1","language":[{"iso":"eng"}],"doi":"10.1088/1361-6544/ac5463","year":"2022","title":"Amplified steady state bifurcations in feedforward networks","publication_identifier":{"issn":["0951-7715","1361-6544"]},"author":[{"orcid":"0000-0002-8054-2058","last_name":"von der Gracht","first_name":"Sören","full_name":"von der Gracht, Sören","id":"97359"},{"full_name":"Nijholt, Eddie","last_name":"Nijholt","first_name":"Eddie"},{"first_name":"Bob","last_name":"Rink","full_name":"Rink, Bob"}],"date_updated":"2022-09-07T08:36:46Z","publication_status":"published","intvolume":"        35","external_id":{"arxiv":["2105.02547"]},"citation":{"ieee":"S. von der Gracht, E. Nijholt, and B. Rink, “Amplified steady state bifurcations in feedforward networks,” <i>Nonlinearity</i>, vol. 35, no. 4, pp. 2073–2120, 2022, doi: <a href=\"https://doi.org/10.1088/1361-6544/ac5463\">10.1088/1361-6544/ac5463</a>.","apa":"von der Gracht, S., Nijholt, E., &#38; Rink, B. (2022). Amplified steady state bifurcations in feedforward networks. <i>Nonlinearity</i>, <i>35</i>(4), 2073–2120. <a href=\"https://doi.org/10.1088/1361-6544/ac5463\">https://doi.org/10.1088/1361-6544/ac5463</a>","chicago":"Gracht, Sören von der, Eddie Nijholt, and Bob Rink. “Amplified Steady State Bifurcations in Feedforward Networks.” <i>Nonlinearity</i> 35, no. 4 (2022): 2073–2120. <a href=\"https://doi.org/10.1088/1361-6544/ac5463\">https://doi.org/10.1088/1361-6544/ac5463</a>.","short":"S. von der Gracht, E. Nijholt, B. Rink, Nonlinearity 35 (2022) 2073–2120.","mla":"von der Gracht, Sören, et al. “Amplified Steady State Bifurcations in Feedforward Networks.” <i>Nonlinearity</i>, vol. 35, no. 4, IOP Publishing, 2022, pp. 2073–120, doi:<a href=\"https://doi.org/10.1088/1361-6544/ac5463\">10.1088/1361-6544/ac5463</a>.","bibtex":"@article{von der Gracht_Nijholt_Rink_2022, title={Amplified steady state bifurcations in feedforward networks}, volume={35}, DOI={<a href=\"https://doi.org/10.1088/1361-6544/ac5463\">10.1088/1361-6544/ac5463</a>}, number={4}, journal={Nonlinearity}, publisher={IOP Publishing}, author={von der Gracht, Sören and Nijholt, Eddie and Rink, Bob}, year={2022}, pages={2073–2120} }","ama":"von der Gracht S, Nijholt E, Rink B. Amplified steady state bifurcations in feedforward networks. <i>Nonlinearity</i>. 2022;35(4):2073-2120. doi:<a href=\"https://doi.org/10.1088/1361-6544/ac5463\">10.1088/1361-6544/ac5463</a>"},"page":"2073-2120","publisher":"IOP Publishing","_id":"33264","user_id":"97359","volume":35,"status":"public"},{"status":"public","page":"2673-2758","_id":"45970","publisher":"World Scientific Pub Co Pte Ltd","user_id":"100441","volume":32,"citation":{"ieee":"H. Garcke, B. Kovács, and D. Trautwein, “Viscoelastic Cahn–Hilliard models for tumor growth,” <i>Mathematical Models and Methods in Applied Sciences</i>, vol. 32, no. 13, pp. 2673–2758, 2022, doi: <a href=\"https://doi.org/10.1142/s0218202522500634\">10.1142/s0218202522500634</a>.","apa":"Garcke, H., Kovács, B., &#38; Trautwein, D. (2022). Viscoelastic Cahn–Hilliard models for tumor growth. <i>Mathematical Models and Methods in Applied Sciences</i>, <i>32</i>(13), 2673–2758. <a href=\"https://doi.org/10.1142/s0218202522500634\">https://doi.org/10.1142/s0218202522500634</a>","short":"H. Garcke, B. Kovács, D. Trautwein, Mathematical Models and Methods in Applied Sciences 32 (2022) 2673–2758.","chicago":"Garcke, Harald, Balázs Kovács, and Dennis Trautwein. “Viscoelastic Cahn–Hilliard Models for Tumor Growth.” <i>Mathematical Models and Methods in Applied Sciences</i> 32, no. 13 (2022): 2673–2758. <a href=\"https://doi.org/10.1142/s0218202522500634\">https://doi.org/10.1142/s0218202522500634</a>.","mla":"Garcke, Harald, et al. “Viscoelastic Cahn–Hilliard Models for Tumor Growth.” <i>Mathematical Models and Methods in Applied Sciences</i>, vol. 32, no. 13, World Scientific Pub Co Pte Ltd, 2022, pp. 2673–758, doi:<a href=\"https://doi.org/10.1142/s0218202522500634\">10.1142/s0218202522500634</a>.","bibtex":"@article{Garcke_Kovács_Trautwein_2022, title={Viscoelastic Cahn–Hilliard models for tumor growth}, volume={32}, DOI={<a href=\"https://doi.org/10.1142/s0218202522500634\">10.1142/s0218202522500634</a>}, number={13}, journal={Mathematical Models and Methods in Applied Sciences}, publisher={World Scientific Pub Co Pte Ltd}, author={Garcke, Harald and Kovács, Balázs and Trautwein, Dennis}, year={2022}, pages={2673–2758} }","ama":"Garcke H, Kovács B, Trautwein D. Viscoelastic Cahn–Hilliard models for tumor growth. <i>Mathematical Models and Methods in Applied Sciences</i>. 2022;32(13):2673-2758. doi:<a href=\"https://doi.org/10.1142/s0218202522500634\">10.1142/s0218202522500634</a>"},"year":"2022","title":"Viscoelastic Cahn–Hilliard models for tumor growth","publication_identifier":{"issn":["0218-2025","1793-6314"]},"author":[{"first_name":"Harald","last_name":"Garcke","full_name":"Garcke, Harald"},{"id":"100441","last_name":"Kovács","orcid":"0000-0001-9872-3474","first_name":"Balázs","full_name":"Kovács, Balázs"},{"full_name":"Trautwein, Dennis","last_name":"Trautwein","first_name":"Dennis"}],"date_updated":"2024-04-03T09:15:35Z","publication_status":"published","intvolume":"        32","language":[{"iso":"eng"}],"doi":"10.1142/s0218202522500634","issue":"13","publication":"Mathematical Models and Methods in Applied Sciences","abstract":[{"text":"<jats:p> We introduce a new phase field model for tumor growth where viscoelastic effects are taken into account. The model is derived from basic thermodynamical principles and consists of a convected Cahn–Hilliard equation with source terms for the tumor cells and a convected reaction–diffusion equation with boundary supply for the nutrient. Chemotactic terms, which are essential for the invasive behavior of tumors, are taken into account. The model is completed by a viscoelastic system consisting of the Navier–Stokes equation for the hydrodynamic quantities, and a general constitutive equation with stress relaxation for the left Cauchy–Green tensor associated with the elastic part of the total mechanical response of the viscoelastic material. For a specific choice of the elastic energy density and with an additional dissipative term accounting for stress diffusion, we prove existence of global-in-time weak solutions of the viscoelastic model for tumor growth in two space dimensions [Formula: see text] by the passage to the limit in a fully-discrete finite element scheme where a CFL condition, i.e. [Formula: see text], is required. </jats:p><jats:p> Moreover, in arbitrary dimensions [Formula: see text], we show stability and existence of solutions for the fully-discrete finite element scheme, where positive definiteness of the discrete Cauchy–Green tensor is proved with a regularization technique that was first introduced by Barrett and Boyaval [Existence and approximation of a (regularized) Oldroyd-B model, Math. Models Methods Appl. Sci. 21 (2011) 1783–1837]. After that, we improve the regularity results in arbitrary dimensions [Formula: see text] and in two dimensions [Formula: see text], where a CFL condition is required. Then, in two dimensions [Formula: see text], we pass to the limit in the discretization parameters and show that subsequences of discrete solutions converge to a global-in-time weak solution. Finally, we present numerical results in two dimensions [Formula: see text]. </jats:p>","lang":"eng"}],"date_created":"2023-07-10T11:47:27Z","type":"journal_article","keyword":["Applied Mathematics","Modeling and Simulation"],"department":[{"_id":"841"}]}]