Electron. J. Diff. Equ., Vol. 2012 (2012), No. 165, pp. 1-15.

Stable piecewise polynomial vector fields

Claudio Pessoa, Jorge Sotomayor

Let $N=\{y>0\}$ and $S=\{y<0\}$ be the semi-planes of $\mathbb{R}^2$ having as common boundary the line $D=\{y=0\}$. Let X and Y be polynomial vector fields defined in N and $S$, 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 $Z_{\epsilon}$, 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 $\mathbb{R}^2$. 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
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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