  The main purpose of this work is to provide a non-local approach to study
aspects of structural stability of 3D Filippov systems. We introduce a notion
of semi-local structural stability which detects when a piecewise smooth vector
field is robust around the whole switching manifold and we give a complete
characterization of such systems. In particular, we present some methods in the
qualitative theory of piecewise smooth vector fields, emphasizing a geometrical
analysis of the foliations generated by their orbits. Such approach displays
surprisingly a rich dynamical behaviour that has been studied in details in
this work.
  It is worth to say that this subject has not been treated recently from a
non-local point of view, and we hope that the approach adopted in this work
contributes to the understanding of the structural stability for
piecewise-smooth vector fields in its most global sense.
