Conjuntos limite e transitividade de campos vetoriais suaves por partes em variedades Riemannianas bi-dimensionais
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Data
2020-12-14
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Universidade Federal de Goiás
Resumo
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 $S^2$.
Descrição
Palavras-chave
Sistemas de Filippov , Campos vetoriais suaves por partes , Conjuntos limite , Teorema de Poincaré-Bendixson , Minimalidade , Caos , Transitividade topológica , Filippov systems , Piecewise-smooth vector fields , Limit sets , Poincaré-Bendixson theorem , Minimality , Chaos , Topological transitivity
Citação
JUCÁ, J. S. Conjuntos limite e transitividade de campos vetoriais suaves por partes em variedades Riemannianas bi-dimensionais. 2020. 86 f. Tese (Doutorado em Matemática) - Universidade Federal de Goiás, Goiânia, 2020.