Regularization of sliding global bifurcations derived from the local fold singularity of Filippov systems

In this paper we study the Sotomayor-Teixeira regularization of a general visible fold singularity of a Filippov system. Extending Geometric Fenichel Theory beyond the fold with asymptotic methods, we determine there the deviation of the orbits of the regularized system from the generalized solutions of the Filippov one. This result is applied to the regularization of some global sliding bifurcations as the Grazing-Sliding of periodic orbits or the Sliding Homoclinic to a Saddle, as well as to some classical problems in dry friction. Roughly speaking, we see that locally, and also globally, the regularization of the bifurcations preserve the topological features of the sliding ones.