In this paper we introduce the concept of sliding Shilnikov orbits for 3D
Filippov systems. In short, such an orbit is a piecewise smooth closed curve,
composed by Filippov trajectories, which slides on the switching surface and
connects a Filippov equilibrium to itself, namely a pseudo saddle-focus. A
version of the Shilnikov's Theorem is provided for such systems. Particularly,
we show that sliding Shilnikov orbits occur in generic one-parameter families
of Filippov systems, and that arbitrarily close to a sliding Shilnikov orbit
there exist countably infinitely many sliding periodic orbits. Here, no
additional Shilnikov-like assumption is needed in order to get this last
result. In addition, we show the existence of sliding Shilnikov orbits in
discontinuous piecewise linear differential systems. As far as we know, the
examples of Fillippov systems provided in this paper are the first exhibiting
such a sliding phenomenon
