Integrable 2D and 3D piecewise smooth vector fields with chaotic behavior and preserving energy or not

Abstract

In this paper we exhibit 2D and 3D piecewise smooth vector fields that have, at the same time, first integrals and chaotic behavior. The fact of existing first integrals means that the trajectories are tangent to foliations. Also, in some of these examples the trajectories live in an unique level of the first integrals, and so the system preserves energy, and in other cases the trajectories visit distinct levels of the first integrals and the system does not preserves the energy.