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The main purpose of this work is to provide a non-local approach to studyaspects of structural stability of 3D Filippov systems. We introduce a notionof semi-local structural stability which detects when a piecewise smooth vectorfield is robust around the whole switching manifold and we give a completecharacterization of such systems. In particular, we present some methods in thequalitative theory of piecewise smooth vector fields, emphasizing a geometricalanalysis of the foliations generated by their orbits. Such approach displayssurprisingly a rich dynamical behaviour that has been studied in details inthis work.  It is worth to say that this subject has not been treated recently from anon-local point of view, and we hope that the approach adopted in this workcontributes to the understanding of the structural stability forpiecewise-smooth vector fields in its most global sense.

On Structural Stability of 3D Filippov Systems: A Semi-Local Approach