In this talk we briefly discuss some examples of chaotic behavior and we explain shortly what we mean for chaotic pattern. Then we will focus on chaotic phenomena associated to homoclinic trajectories (i.e. trajectories converging to a critical point as $t \to \pm \infty$).

It is well known that an autonomous system which has a homoclinic trajectory and subject to a small periodic forcing may exhibit a chaotic pattern. A motivating example in this context is given by a forced inverted pendulum.

Melnikov theory provides a computable sufficient condition for the existence of a transversal intersection between stable and unstable manifolds: in a smooth context this is enough to guarantee the persistence of the homoclinic and the insurgence of chaos.

In this talk we will briefly review these facts and we will show that, in a piecewise smooth context the situation is more complex: a geometrical obstruction may forbid chaotic phenomena which are replaced by new bifurcation scenarios. Further, if this obstruction is removed, chaos may arise again. Piecewise smooth systems are motivated by the study of dry friction, state dependent switches, or impacts.

In fact we will also show some results new in a smooth context, concerning multiplicity, position and size of the Cantor set $\Sigma$ of initial conditions from which chaos emanates.