Conjuntos limite e transitividade de campos vetoriais suaves por partes em variedades Riemannianas bi-dimensionais

• Ano de defesa: 2020
• Autor(a) principal: Jucá, Joaby de Souza
• Orientador(a): Euzébio, Rodrigo Donizete
• Banca de defesa: Euzébio, Rodrigo Donizete, Varão Filho, José Régis Azevedo, Oliveira, Regilene Delazari dos Santos, Buzzi, Claudio Aguinaldo, Tonon, Durval José
• 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: Sistemas de Filippov; Campos vetoriais suaves por partes; Conjuntos limite; Teorema de Poincaré-Bendixson; Minimalidade; Caos; Transitividade topológica
• Palavras-chave em Inglês: Filippov systems; Piecewise-smooth vector fields; Limit sets; Poincaré-Bendixson theorem; Minimality; Chaos; Topological transitivity
• Área do conhecimento CNPq: CIENCIAS EXATAS E DA TERRA::MATEMATICA
• Link de acesso: http://repositorio.bc.ufg.br/tede/handle/tede/11039

In this work we study piecewise-smooth vector fields defined on a two-di\-men\-sio\-nal differential manifold M, according to the Filippov convention. In the first part, M is considered as being any Riemannian manifold and we present a classification of the possible limit sets for a maximal trajectory whose its positive branch is contained on a compact subset $K\subset M$ (Theorem 3.1). We consider the occurrence of sliding motion and we verify the presence of limit sets with non-empty interior that can present a non-deterministic chaotic behavior. Moreover, we provide some examples and classes of systems satisfying the hypotheses of the main results. In the second part, we study the topological transitivity of piecewise-smooth vector fields defined on the two-dimensional sphere $S^2$. We guarantee the existence of an one-parameter family of topologically transitive piecewise-smooth vector fields on $S^2$ (Theorem 4.1), which does not happen for continuous vector fields on $S^2$. We prove that the occurrence of transitivity on $S^2$ implies the existence of escaping and sliding regions. We also prove they connect to each other through infinitely many Filippov trajectories. Moreover, we prove that there exist no robustly transitive piecewise-smooth vector fields on $S^2$.