Limit cycles in planar piecewise smooth systems having non-regular switches, time scales or rotated properties

• Ano de defesa: 2022
• Autor(a) principal: Silva, Ana Maria Alves da
• Orientador(a): Euzébio, Rodrigo Donizete
• Banca de defesa: Euzébio, Rodrigo Donizete, Roberto, Luci Any Francisco, Martins, Ricardo Miranda, Andrade, Kamila da Silva, Oliveira, Regilene Delazari dos Santos
• Tipo de documento: Tese
• Tipo de acesso: Acesso aberto
• Idioma: eng
• 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: Campos de vetores suaves por partes; Trajetórias periódicas; Sistemas rápido-lento; Campos de vetores rodados por partes; Teoria de bifurcação
• Palavras-chave em Inglês: Piecewise smooth vector fields; Periodic trajectories; Fast-slow systems; Piecewise rotated vector fields; Bifurcation theory
• Área do conhecimento CNPq: CIENCIAS EXATAS E DA TERRA::MATEMATICA
• Link de acesso: http://repositorio.bc.ufg.br/tede/handle/tede/12415

In this thesis, periodic trajectories in planar discontinuous piecewise linear systems with a nonregular switching line are studied. We provide sharp upper bounds of one or two limit cycles for certain classes of the model considered. We also establish the stability and hyperbolicity of these limit cycles. In addition, we provide examples reaching one and two limit cycles for these classes. We perform the global analysis of a representative model through bifurcation theory to analyze the birth of limit cycles, sliding periodic trajectories, and tangential ones. We also provide some results addressing the coexistence of periodic trajectories. We studied Fast-Slow systems with nonregular switching line with a new approach. This study allows proving that a specific sliding periodic trajectory is in fact a homoclinic trajectory. This homoclinic trajectory arises from a bifurcation of sliding limit cycles that are not topologically equivalents. We propose the theory of piecewise rotated vector fields with the goal of understanding how the trajectories of two families of rotated vector fields behave as the same parameter is varied. In this context, we prove the non-intersection theorem for closed periodic trajectories for piecewise rotated vector fields.