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Local and 'sigma'-semilocal structural stability structural of piecewise smooth vector fields in compact 3-dimensional manifolds

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Author(s):
Matheus Manzatto de Castro
Total Authors: 1
Document type: Master's Dissertation
Press: Campinas, SP.
Institution: Universidade Estadual de Campinas (UNICAMP). Instituto de Matemática, Estatística e Computação Científica
Defense date:
Examining board members:
Ricardo Miranda Martins; Marco Antonio Teixeira; Paulo Ricardo da Silva
Advisor: Ricardo Miranda Martins
Abstract

This dissertation is focused on the study of the structural stability of piecewise smooth vector fields on a three dimensional compact manifold $M$, where the sliding manifold is a two dimensional oriented manifold embedded in $M$. Therefore, we will, in a comprehensive way, look for conditions such that a piecewise smooth vector fields admit some type of structural stability. The first step to achieve this goal will be to define and understand the concept of tangential singularity and define what is understood by a generic tangential singularity. We will show the existence of a set, contained in the set of piecewise smooth vector fields, such that each of its elements has only generic singularities, moreover, it will be possible to prove that such a set is residual in the set of piecewise smooth vector fields. The second step of the process will be to prove the theorem known as the Vishik Normal Form, which will allow us to analyze local fields with only generic singularities. We will be able to find necessary and sufficient conditions for the existence of local structural stability. Finally, we will extend the concept of local structural stability to a neighborhood of the sliding manifold and then we will find conditions, based on Peixoto's theorems and the results referring to local structural stability, for the existence of this type of structural stability. Key-words: Filippov Systems, Discontinuous Vector Fields, Structural Stability (AU)

FAPESP's process: 17/23692-6 - Structural stability of piecewise smooth dynamical systems on torus and spheres
Grantee:Matheus Manzatto de Castro
Support Opportunities: Scholarships in Brazil - Master