[{"ddc":["510"],"keyword":["Coupled cell systems","Network dynamics","Dimension reduction","Bifurcation theory","Symmetry","Monoid representation theory"],"language":[{"iso":"eng"}],"external_id":{"arxiv":["2510.06740"]},"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"}],"file":[{"file_name":"homogeneous-coupled-cell-systems-with-high-dimensional-internal-dynamics.pdf","access_level":"closed","file_id":"64980","file_size":1951746,"date_created":"2026-03-16T08:40:04Z","creator":"svdg","date_updated":"2026-03-16T08:40:04Z","relation":"main_file","success":1,"content_type":"application/pdf"}],"publication":"Chaos, Solitons & Fractals","title":"Homogeneous coupled cell systems with high-dimensional internal dynamics","publisher":"Elsevier BV","date_created":"2026-03-16T08:39:07Z","year":"2026","article_number":"118196","article_type":"original","file_date_updated":"2026-03-16T08:40:04Z","_id":"64979","user_id":"97359","department":[{"_id":"101"},{"_id":"841"}],"status":"public","type":"journal_article","doi":"10.1016/j.chaos.2026.118196","date_updated":"2026-03-16T08:42:56Z","author":[{"orcid":"0000-0002-8054-2058","last_name":"von der Gracht","full_name":"von der Gracht, Sören","id":"97359","first_name":"Sören"},{"full_name":"Nijholt, Eddie","last_name":"Nijholt","first_name":"Eddie"},{"last_name":"Rink","full_name":"Rink, Bob","first_name":"Bob"}],"volume":208,"citation":{"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>","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>.","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).","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} }","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>.","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>"},"intvolume":"       208","publication_status":"published","has_accepted_license":"1","publication_identifier":{"issn":["0960-0779"]}},{"date_created":"2025-03-27T10:15:06Z","publisher":"The Royal Society","title":"Higher-order interactions lead to ‘reluctant’ synchrony breaking","issue":"2301","year":"2024","language":[{"iso":"eng"}],"keyword":["higher-order interactions","synchrony breaking","network dynamics","coupled cell systems"],"ddc":["510"],"publication":"Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences","file":[{"content_type":"application/pdf","relation":"main_file","creator":"svdg","date_created":"2025-03-27T10:16:20Z","date_updated":"2025-03-27T10:19:48Z","file_name":"higher-order-interactions-lead-to-reluctant-synchrony-breaking.pdf","access_level":"open_access","file_id":"59172","file_size":820435}],"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"}],"volume":480,"author":[{"last_name":"von der Gracht","orcid":"0000-0002-8054-2058","full_name":"von der Gracht, Sören","id":"97359","first_name":"Sören"},{"first_name":"Eddie","full_name":"Nijholt, Eddie","last_name":"Nijholt"},{"last_name":"Rink","full_name":"Rink, Bob","first_name":"Bob"}],"date_updated":"2025-03-27T10:19:56Z","oa":"1","doi":"10.1098/rspa.2023.0945","has_accepted_license":"1","publication_identifier":{"issn":["1364-5021","1471-2946"]},"publication_status":"published","intvolume":"       480","citation":{"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} }","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>.","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>.","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>"},"department":[{"_id":"101"}],"user_id":"97359","_id":"59171","file_date_updated":"2025-03-27T10:19:48Z","type":"journal_article","status":"public"}]