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Campos vetoriais suaves por partes: índice de singularidades e alguns resultados sobre a existência de ciclos limites

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Author(s):
Joyce Aparecida Casimiro
Total Authors: 1
Document type: Doctoral Thesis
Press: Campinas, SP.
Institution: Universidade Estadual de Campinas (UNICAMP). Instituto de Matemática, Estatística e Computação Científica
Defense date:
Examining board members:
Ricardo Miranda Martins; Luis Fernando de Osório Mello; José Régis Azevedo Varão Filho; Ketty Abaroa de Rezende; Luci Any Francisco Roberto
Advisor: Ricardo Miranda Martins
Abstract

Filippov vector fields are the subject of highly relevant study, both in terms of their theoretical and applied aspects. Additionally, the analysis of minimal sets plays a fundamental role in understanding the global qualitative behavior of dynamical systems. Therefore, determining the existence or non-existence of these sets is a crucial and extensively explored topic in this research area. In this thesis, we investigate whether certain classes of Filippov vector fields exhibit limit cycles after small perturbations. The Euler characteristic of a compact two-dimensional manifold and the local behavior of smooth vector fields defined on it are interconnected through the Poincaré-Hopf Theorem. Until now, such a result had not been established for Filippov vector fields, and we demonstrate its validity in this context. While, in smooth cases, singularities consist of points where the vector field vanishes, in the context of Filippov vector fields, the notion of singularity also includes new types of points, namely, pseudo-equilibrium points and tangency points. In this context, the classical definition of the index for singularities in smooth vector fields is extended to encompass the singularities of Filippov vector fields (AU)

FAPESP's process: 18/25575-0 - Piecewise smooth differential equations in dimension 3
Grantee:Joyce Aparecida Casimiro
Support Opportunities: Scholarships in Brazil - Doctorate