Regularization of discontinuous foliations: Blowing up and sliding conditions via Fenichel theory.

We study the regularization of an oriented 1-foliation F on M ∖ Σ where M is a smooth manifold and Σ ⊂ M is a closed subset, which can be interpreted as the discontinuity locus of F . In the spirit of Filippov's work, we define a sliding and sewing dynamics on the discontinuity locus Σ as some sort of limit of the dynamics of a nearby smooth 1-foliation and obtain conditions to identify whether a point belongs to the sliding or sewing regions. [ABSTRACT FROM AUTHOR]