Stable piecewise polynomial vector fields

TL;DR

This paper extends stability analysis and regularization methods for discontinuous piecewise polynomial vector fields in R^2, providing new conditions for structural stability, genericity results, and a procedure for studying behavior at infinity.

Contribution

It generalizes existing stability conditions and regularization techniques to piecewise polynomial vector fields, including at infinity, with new genericity results.

Findings

Extended stability conditions for piecewise polynomial vector fields.

Provided a method for analyzing behavior at infinity via compactification.

Established genericity results for the stability conditions.

Abstract

Consider in R^2 the semi-planes N={y>0} and S={y<0}$ having as common boundary the straight line D={y=0}$. In N and S are defined polynomial vector fields X and Y, respectively, leading to a discontinuous piecewise polynomial vector field Z=(X,Y). This work pursues the stability and the transition analysis of solutions of Z between N and S, started by Filippov (1988) and Kozlova (1984) and reformulated by Sotomayor-Teixeira (1995) in terms of the regularization method. This method consists in analyzing a one parameter family of continuous vector fields Z_{\epsilon}$, 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 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.