Bifurcations of Non Smooth Vector Fields on $\R^2$ by Geometric Singular Perturbations

TL;DR

This paper explores bifurcations in non-smooth vector fields on  using geometric singular perturbation techniques after regularization and blow-up, connecting non-smooth dynamics with singularly perturbed smooth systems.

Contribution

It introduces a novel approach applying geometric singular perturbation methods to non-smooth vector fields, bridging the gap between non-smooth and smooth singular systems.

Findings

Established new bifurcation results for non-smooth vector fields.

Demonstrated the effectiveness of regularization and blow-up techniques.

Connected non-smooth dynamics with classical singular perturbation theory.

Abstract

Our object of study is non smooth vector fields on $\R^2$. We apply the techniques of geometric singular perturbations in non smooth vector fields after regularization and a blow$-$up. In this way we are able to bring out some results that bridge the space between non$-$smooth dynamical systems presenting typical singularities and singularly perturbed smooth systems.