{"title":"On the Dynamical Hierarchy in Gathering Protocols with Circulant  Topologies","date_updated":"2024-01-03T17:00:48Z","type":"preprint","user_id":"32655","page":"38","author":[{"id":"32655","last_name":"Gerlach","first_name":"Raphael","full_name":"Gerlach, Raphael"},{"id":"97359","first_name":"Sören","last_name":"von der Gracht","full_name":"von der Gracht, Sören","orcid":"0000-0002-8054-2058"},{"full_name":"Dellnitz, Michael","first_name":"Michael","last_name":"Dellnitz"}],"status":"public","date_created":"2023-05-12T10:16:06Z","publication":"arXiv:2305.06632","year":"2023","_id":"44840","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."}],"language":[{"iso":"eng"}],"citation":{"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} }","ama":"Gerlach R, von der Gracht S, Dellnitz M. On the Dynamical Hierarchy in Gathering Protocols with Circulant  Topologies. <i>arXiv:230506632</i>.","apa":"Gerlach, R., von der Gracht, S., &#38; Dellnitz, M. (n.d.). On the Dynamical Hierarchy in Gathering Protocols with Circulant  Topologies. In <i>arXiv:2305.06632</i>.","ieee":"R. Gerlach, S. von der Gracht, and M. Dellnitz, “On the Dynamical Hierarchy in Gathering Protocols with Circulant  Topologies,” <i>arXiv:2305.06632</i>. .","short":"R. Gerlach, S. von der Gracht, M. Dellnitz, ArXiv:2305.06632 (n.d.).","mla":"Gerlach, Raphael, et al. “On the Dynamical Hierarchy in Gathering Protocols with Circulant  Topologies.” <i>ArXiv:2305.06632</i>.","chicago":"Gerlach, Raphael, Sören von der Gracht, and Michael Dellnitz. “On the Dynamical Hierarchy in Gathering Protocols with Circulant  Topologies.” <i>ArXiv:2305.06632</i>, n.d."},"keyword":["Dynamical Systems","Coupled Systems","Distributed Computing","Robot Swarms","Autonomous Mobile Robots","Gathering"],"department":[{"_id":"101"}],"main_file_link":[{"url":"https://arxiv.org/abs/2305.06632"}],"external_id":{"arxiv":["2305.06632"]},"publication_status":"submitted","related_material":{"link":[{"url":"https://math.uni-paderborn.de/en/ag/chair-of-applied-mathematics/research/research-projects/swarmdynamics","relation":"supplementary_material"}]},"project":[{"_id":"106","grant_number":"453112019","name":"Algorithmen für Schwarmrobotik: Verteiltes Rechnen trifft Dynamische Systeme"}]}