Via a blow up we establish a bridge between non-smooth dynamical systems and the geometric singular perturbation theory.
The paper is organized as follows: in Section 2 we give the basic theory about Non-Smooth Vector Fields on the plane, in Section 3 we give the theory about the regularization process, in Section 4 we present a few relevant methods of Geometric Singular Perturbation Theory, in Section 5 we present the singularities treated in Theorem 1, give its topological normal forms and prove Theorem 1, in Section 6 we present the singularities treated in Theorem 2, give its topological normal forms and prove Theorem 2.
The understanding of the phase portrait of the vector field associated to a SP-problem is the main goal of the geometric singular perturbation theory.
Nearly twenty years later, Szmolyan and Wechselberger  extended "Geometric Singular Perturbation Theory (see Fenichel [6, 7], O'Malley , Jones , and Kaper )" to canards problems in [R.
According to the Geometric Singular Perturbation Theory, the zero-order approximation in [epsilon] of the slow manifold associated with the Hodgkin-Huxley model (60a,60b), (60c), and (60d) is obtained by posing [epsilon] = 0 in (60c) and (60d).