Electron. J. Diff. Equ.,
Vol. 2012 (2012), No. 165, pp. 1-15.
Stable piecewise polynomial vector fields
Claudio Pessoa, Jorge Sotomayor
Abstract:
Let
and
be the semi-planes 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 Sotomayor-Teixeira (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 Sotomayor-Teixeira and
Sotomayor-Machado (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, UNESP--IBILCE
Av. Cristovão Colombo, 2265
15.054--000, S. J. Rio Preto, SP, Brasil
email: pessoa@ibilce.unesp.br
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Jorge Sotomayor
Instituto de Matemática e Estatística,
Universidade de São Paulo
Rua do Matão 1010, Cidade Universitária
05.508-090, São Paulo, SP, Brasil
email: sotp@ime.usp.br
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