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