{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/107880"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/107880","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Equivariant dynamics and categories of equivariant vector fields","abstract":"We present a framework for studying the dynamics of equivariant vector fields near relative equilibria. To overcome the lack of linearization at a relative equilibrium or the possible non-smoothness of the orbit space, we categorify the space of equivariant vector fields. A category where the objects are equivariant vector fields was first introduced by Hepworth in the context of smooth stacks. Central to our approach is the ensuing notion of isomorphic equivariant vector fields. The idea is that considering equivariant vector fields, and their corresponding dynamics, up to isomorphism is a way to take into account the symmetries of the group action without passing to the orbit space. In particular, the category of equivariant vector fields near a relative equilibrium is equivalent to the category of equivariant vector fields on the slice representation of the relative equilibrium. We apply this to the stability and motion of relative equilibria, bifurcations to and from relative equilibria, and to the genericity of conditions for equivariant bifurcation from relative equilibria.","abstract_html":"We present a framework for studying the dynamics of equivariant vector fields near relative equilibria. To overcome the lack of linearization at a relative equilibrium or the possible non-smoothness of the orbit space, we categorify the space of equivariant vector fields. A category where the objects are equivariant vector fields was first introduced by Hepworth in the context of smooth stacks. Central to our approach is the ensuing notion of isomorphic equivariant vector fields. The idea is that considering equivariant vector fields, and their corresponding dynamics, up to isomorphism is a way to take into account the symmetries of the group action without passing to the orbit space. In particular, the category of equivariant vector fields near a relative equilibrium is equivalent to the category of equivariant vector fields on the slice representation of the relative equilibrium. We apply this to the stability and motion of relative equilibria, bifurcations to and from relative equilibria, and to the genericity of conditions for equivariant bifurcation from relative equilibria.","abstract_has_math":false,"creators":["Klajbor-Goderich, Stefanie"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Lerman, Eugene","Loja Fernandes, Rui A","Kerman, Ely","Zharnitsky, Vadim"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2020,"date_issued":"2020-08-26T21:54:18Z","date_published":"2020-08-26T21:54:18Z","updated_at":"2026-07-22T22:24:47Z","subjects":["equivariant dynamics","relative equilibria","bifurcation"],"languages":["en"],"rights":["Copyright 2020 Stefanie Klajbor-Goderich"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/2142/107880","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Lerman, Eugene","Loja Fernandes, Rui A","Kerman, Ely","Zharnitsky, Vadim"]},{"key":"dc:creator","label":"Author","values":["Klajbor-Goderich, Stefanie"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2020-08-26T21:54:18Z","2020-04-22","2020-05"]},{"key":"dc:type","label":"Dc Type","values":["text","Thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["equivariant dynamics","relative equilibria","bifurcation"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["en"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 2020 Stefanie Klajbor-Goderich"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["http://hdl.handle.net/2142/107880"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["We present a framework for studying the dynamics of equivariant vector fields near relative equilibria. To overcome the lack of linearization at a relative equilibrium or the possible non-smoothness of the orbit space, we categorify the space of equivariant vector fields. A category where the objects are equivariant vector fields was first introduced by Hepworth in the context of smooth stacks. Central to our approach is the ensuing notion of isomorphic equivariant vector fields. The idea is that considering equivariant vector fields, and their corresponding dynamics, up to isomorphism is a way to take into account the symmetries of the group action without passing to the orbit space. In particular, the category of equivariant vector fields near a relative equilibrium is equivalent to the category of equivariant vector fields on the slice representation of the relative equilibrium. We apply this to the stability and motion of relative equilibria, bifurcations to and from relative equilibria, and to the genericity of conditions for equivariant bifurcation from relative equilibria.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2020-08-25 without embargo terms","The student, Stefanie Klajbor-Goderich, accepted the attached license on 2020-04-13 at 15:35.","The student, Stefanie Klajbor-Goderich, submitted this Dissertation for approval on 2020-04-13 at 15:49.","This Dissertation was approved for publication on 2020-04-22 at 08:31.","DSpace SAF Submission Ingestion Package generated from Vireo submission #14971 on 2020-08-25 at 17:06:50","Made available in DSpace on 2020-08-26T21:54:18Z (GMT). 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To overcome the lack of linearization at a relative equilibrium or the possible non-smoothness of the orbit space, we categorify the space of equivariant vector fields. A category where the objects are equivariant vector fields was first introduced by Hepworth in the context of smooth stacks. Central to our approach is the ensuing notion of isomorphic equivariant vector fields. The idea is that considering equivariant vector fields, and their corresponding dynamics, up to isomorphism is a way to take into account the symmetries of the group action without passing to the orbit space. In particular, the category of equivariant vector fields near a relative equilibrium is equivalent to the category of equivariant vector fields on the slice representation of the relative equilibrium. We apply this to the stability and motion of relative equilibria, bifurcations to and from relative equilibria, and to the genericity of conditions for equivariant bifurcation from relative equilibria.","Submission original under an indefinite embargo labeled 'Open Access'. The submission was exported from vireo on 2020-08-25 without embargo terms","The student, Stefanie Klajbor-Goderich, accepted the attached license on 2020-04-13 at 15:35.","The student, Stefanie Klajbor-Goderich, submitted this Dissertation for approval on 2020-04-13 at 15:49.","This Dissertation was approved for publication on 2020-04-22 at 08:31.","DSpace SAF Submission Ingestion Package generated from Vireo submission #14971 on 2020-08-25 at 17:06:50","Made available in DSpace on 2020-08-26T21:54:18Z (GMT). No. of bitstreams: 3 KLAJBOR-GODERICH-DISSERTATION-2020.pdf: 642052 bytes, checksum: f4bfa739b2dc04f12a77ce4d8cadded2 (MD5) LICENSE.txt: 4222 bytes, checksum: 7e61c53c68f22e7eccd164c451e7f89d (MD5) PROQUEST_LICENSE.txt: 4568 bytes, checksum: 967ef136c19fa9ce3bc4f5bb82f12b1e (MD5) Previous issue date: 2020-04-22"],"dc:format":["application/pdf"],"dc:identifier":["http://hdl.handle.net/2142/107880"],"dc:language":["en"],"dc:rights":["Copyright 2020 Stefanie Klajbor-Goderich"],"dc:subject":["equivariant dynamics","relative equilibria","bifurcation"],"dc:title":["Equivariant dynamics and categories of equivariant vector fields"],"dc:type":["text","Thesis"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:24:47Z"}