Electronic Journal of Differential Equations,
Vol. 2012 (2012), No. 165, pp. 1-15.

Title: Stable piecewise polynomial vector fields

Authors: Claudio Pessoa  (Univ. Estadual Paulista, Brasil)
         Jorge Sotomayor (Univ. de Sao Paulo,  Brasil)

Abstract:
 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