# American Institute of Mathematical Sciences

1997, 3(2): 235-241. doi: 10.3934/dcds.1997.3.235

## Regularization of discontinuous vector fields in dimension three

 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., Brazil

Received  September 1996 Published  January 1997

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.
Citation: Jaume Llibre, Marco Antonio Teixeira. Regularization of discontinuous vector fields in dimension three. Discrete & Continuous Dynamical Systems - A, 1997, 3 (2) : 235-241. doi: 10.3934/dcds.1997.3.235
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