Regularization of discontinuous vector fields in dimension three

Jaume Llibre; Marco Antonio Teixeira

doi:10.3934/dcds.1997.3.235

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1997, Volume 3, Issue 2: 235-241. Doi: 10.3934/dcds.1997.3.235

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Regularization of discontinuous vector fields in dimension three

Jaume Llibre^1, and
Marco Antonio Teixeira^2,

1.
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia
2.
Departamento de Matemática, Universidade Estadual de Campinas, Caixa Postal 6065, 13083-970, Campinas, S.P.

Received: August 31, 1996

Published: March 1997

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Abstract

In this paper vector fields around the origin in dimension three which are approximations of discontinuous ones are studied. In a former work of Sotomayor and Teixeira [6] it is shown, via regularization, that Filippov's conditions are the natural ones to extend the orbit solutions through the discontinuity set for vector fields in dimension two. In this paper we show that this is also the case for discontinuous vector fields in dimension three. Moreover, we analyse the qualitative dynamics of the local flow in a neighborhood of the codimension zero regular and singular points of the discontinuity surface.

Keywords:

Regularization and discontinuous vector fields.

Mathematics Subject Classification: 58F09, 34C35.

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