New global bifurcation diagrams for piecewise smooth systems: Transversality of homoclinic points does not imply chaos

In this paper we consider some piecewise smooth 2-dimensional systems having a possibly non-smooth homoclinic Eγ (t). We assume that the critical point E0 lies on the discontinuity surface Omega(0). We consider 4 scenarios which differ for the presence or not of sliding close to E0 and for the possible presence of a transversal crossing between Eγ (t) and 0. We assume that the systems are subject to a small non-autonomous pertur- bation, and we obtain 4 new bifurcation diagrams. In particular we show that, in one of these scenarios, the existence of a transversal homoclinic point guarantees the persistence of the homoclinic trajectory but chaos cannot occur. Further we illustrate the presence of new phenomena involving an uncountable number of sliding homoclinic