2026-09-19T17:49:57Z https://www.ideals.illinois.edu/oai-pmh

oai:www.ideals.illinois.edu:2142/107880 2023-07-11 col_2142_5131 col_2142_16340 com_2142_5130 com_2142_16339 com_2142_8903

Lerman, Eugene Loja Fernandes, Rui A Kerman, Ely Zharnitsky, Vadim Klajbor-Goderich, Stefanie 2020-08-26T21:54:18Z 2020-08-26T21:54:18Z 2020-04-22 2020-05 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). 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 application/pdf http://hdl.handle.net/2142/107880 en Copyright 2020 Stefanie Klajbor-Goderich equivariant dynamics relative equilibria bifurcation Equivariant dynamics and categories of equivariant vector fields text Thesis Mathematics Mathematics University of Illinois at Urbana-Champaign Dissertation Ph.D.