Índice de curvas para campos vetoriais definidos no bordo ou suaves por partes
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2017-11-27
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Universidade Federal de Goiás
Resumo
In this work, we establish a new method to calculate the index of curves in a
neighborhood of a boundary and we show that the index of a trajectory of a vector
field which intersects the boundary at two points is 1/2.
Using this method we extended the index definition for discontinuous vector fields
with a regular transition manifold and we calculate the index for closed curves that
intersect the variety of transition = f−1(0), where f is a differentiable function,
and is the union of the regions tangency, sewing, sliding and escaping. We also
show that the index for solutions of the discontinuous vector field that are −closed
of type 1 and intersect the boundary at 2-point is equal to 1. We also establish
an index theory for discontinuous vector fields when the transition manifold is not
regular in a point and we show that the index is given by the calculation in its
regular regions and add ±1/2, depending on the dynamics at the non-regular point.
We apply the theory of index developed in this work and we give quotas for the
indices of continuous vector field and for polynomial vector fields on two zones.
Finally, we demonstrate a version of the Poincaré-Hopf Theorem for discontinuous
vector fields in compact manifolds.
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FURLAN, Pablo Vandré Jacob. Índice de curvas para campos vetoriais definidos no bordo ou suaves por partes. 2017. 112 f. Tese (Doutorado em Matemática) - Universidade Federal de Goiás, Goiânia, 2017.