Electron. J. Diff. Equ.,
Vol. 2012 (2012), No. 165, pp. 115.
Stable piecewise polynomial vector fields
Claudio Pessoa, Jorge Sotomayor
Abstract:
Let
and
be the semiplanes of
having as common boundary the line
.
Let X and Y be
polynomial vector fields defined in N and
,
respectively,
leading to a discontinuous piecewise
polynomial vector field Z=(X,Y). This work pursues the
stability and the transition analysis of solutions of Z between
N and S, started by Filippov (1988) and Kozlova (1984) and
reformulated by SotomayorTeixeira (1995) in terms of the
regularization method. This method consists in analyzing a one
parameter family of continuous vector fields
,
defined by averaging X and Y. This family approaches Z when
the parameter goes to zero. The results of SotomayorTeixeira and
SotomayorMachado (2002) providing conditions on (X,Y) for the
regularized vector fields to be structurally stable on planar
compact connected regions are extended to discontinuous piecewise
polynomial vector fields on
.
Pertinent genericity
results for vector fields satisfying the above stability
conditions are also extended to the present case. A procedure for
the study of discontinuous piecewise vector fields at infinity
through a compactification is proposed here.
Submitted February 28, 2012. Published September 22, 2012.
Math Subject Classifications: 34C35, 58F09, 34D30.
Key Words: Structural stability; piecewise vector fields;
compactification
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Claudio Pessoa
Universidade Estadual Paulista, UNESPIBILCE
Av. Cristovão Colombo, 2265
15.054000, S. J. Rio Preto, SP, Brasil
email: pessoa@ibilce.unesp.br


Jorge Sotomayor
Instituto de Matemática e Estatística,
Universidade de São Paulo
Rua do Matão 1010, Cidade Universitária
05.508090, São Paulo, SP, Brasil
email: sotp@ime.usp.br

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