Smoothing of nonsmooth differential systems near regular-tangential singularities and boundary limit cycles

Understanding how tangential singularities evolve under smoothing processes was one of the first problem concerning regularization of Filippov systems. In this paper, we are interested in $C^n$-regularizations of Filippov systems around visible regular-tangential singularities of even multiplicity. More specifically, using Fenichel Theory and Blow-up Methods, we aim to understand how the trajectories of the regularized system transits through the region of regularization. We apply our results to investigate $C^n$-regularizations of boundary limit cycles with even multiplicity contact with the switching manifold.