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 $\gamma(t)$. We assume that the critical point $\vec{0}$ lies on the discontinuity surface $\Omega^0$. We consider $4$ scenarios which differ for the presence or not of sliding close to $\vec{0}$ and for the possible presence of a transversal crossing between $\gamma(t)$ and $\Omega^0$. We assume that the systems are subject to a small non-autonomous perturbation, 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 homoclinics.