Índice de curvas para campos vetoriais definidos no bordo ou suaves por partes

• Autor(a) principal: Furlan, Pablo Vandré Jacob
• Data de Publicação: 2017
• Tipo de documento: Tese
• Idioma: por
• Título da fonte: Repositório Institucional da UFG
• Texto Completo: http://repositorio.bc.ufg.br/tede/handle/tede/8081
• topic: Índice de Poincaré; Órbita periódica; Campo vetorial descontínuo; Poincaré index; Periodic orbit; Discontinuous vector fields; GEOMETRIA E TOPOLOGIA::SISTEMAS DINAMICOS

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.