Índice de curvas para campos vetoriais definidos no bordo ou suaves por partes

Detalhes bibliográficos
Ano de defesa: 2017
Autor(a) principal: Furlan, Pablo Vandré Jacob lattes
Orientador(a): Medrado, João Carlos da Rocha lattes
Banca de defesa: Medrado, João Carlos da Rocha, Euzébio, Rodrigo Donizete, Silva, Paulo Ricardo da, Lima, Maurício Firmino Silva, Buzzi, Claudio Aguinaldo
Tipo de documento: Tese
Tipo de acesso: Acesso aberto
Idioma: por
Instituição de defesa: Universidade Federal de Goiás
Programa de Pós-Graduação: Programa de Pós-graduação em Matemática (IME)
Departamento: Instituto de Matemática e Estatística - IME (RG)
País: Brasil
Palavras-chave em Português:
Palavras-chave em Inglês:
Área do conhecimento CNPq:
Link de acesso: http://repositorio.bc.ufg.br/tede/handle/tede/8081
Resumo: In this work, we establish a new method to calculate the index of curves in a neighborhood of a boundary and we show that the index of a trajectory of a vector field which intersects the boundary at two points is 1/2. Using this method we extended the index definition for discontinuous vector fields with a regular transition manifold and we calculate the index for closed curves that intersect the variety of transition = f−1(0), where f is a differentiable function, and is the union of the regions tangency, sewing, sliding and escaping. We also show that the index for solutions of the discontinuous vector field that are −closed of type 1 and intersect the boundary at 2-point is equal to 1. We also establish an index theory for discontinuous vector fields when the transition manifold is not regular in a point and we show that the index is given by the calculation in its regular regions and add ±1/2, depending on the dynamics at the non-regular point. We apply the theory of index developed in this work and we give quotas for the indices of continuous vector field and for polynomial vector fields on two zones. Finally, we demonstrate a version of the Poincaré-Hopf Theorem for discontinuous vector fields in compact manifolds.