Singular perturbation of a saddle-node boundary equilibrium

Abstract

In this work, we analyze piecewise smooth planar vector fields defined in two regions, where the switching manifold is a curve passing through the origin. In one of these regions, a smooth vector field acts with a saddle-node equilibrium at the origin, while in the other, the vector field is transversal to the switching manifold. It is well known that this piecewise smooth vector field has codimension two in the set of all piecewise smooth vector fields, and a singular perturbation in this scenario is new in the literature. Also, since the equilibrium considered is over the switching manifold, it will be called a boundary equilibrium. We regularized the piecewise smooth vector field using the method known as Sotomayor-Teixeira. After this step, we adopt a singular perturbation approach. In this process, we identify a non-normally hyperbolic equilibrium point. For a complete understanding of the dynamics around this point, we apply a sequence of transformations to it, known as “blow-ups”.