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

Abstract

In this paper we consider some piecewise smooth 2-dimensional systems having a possibly non-smooth homoclinic γ→(t). We assume that the critical point 0→ lies on the discontinuity surface Ω0. We consider 4 scenarios which differ for the presence or not of sliding close to 0→ and for the possible presence of a transversal crossing between γ→(t) and Ω0. 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.