Shilnikov problem in Filippov dynamical systems

In this paper we introduce the concept of sliding Shilnikov orbits for $3$D Filippov systems. In short, such an orbit is a piecewise smooth closed curve, composed by Filippov trajectories, which slides on the switching surface and connects a Filippov equilibrium to itself, namely a pseudo saddle-focus. A version of the Shilnikov's Theorem is provided for such systems. Particularly, we show that sliding Shilnikov orbits occur in generic one-parameter families of Filippov systems, and that arbitrarily close to a sliding Shilnikov orbit there exist countably infinitely many sliding periodic orbits. Here, no additional Shilnikov-like assumption is needed in order to get this last result. In addition, we show the existence of sliding Shilnikov orbits in discontinuous piecewise linear differential systems. As far as we know, the examples of Fillippov systems provided in this paper are the first exhibiting such a sliding phenomenon

On the periodic solutions of a generalized smooth or non-smooth perturbed planar double pendulum

We provide sufficient conditions for the existence of periodic solutions with small amplitude of the non--linear planar double pendulum perturbed by smooth or non--smooth functions.Comment: arXiv admin note: substantial text overlap with arXiv:1109.637

Generic bifurcation of reversible vector fields on a 2-dimensional manifold

In this paper we deal with reversible vector fields on a 2-dimensional manifold having a codimension one submanifold as its symmetry axis. We classify generically the one parameter families of such vector fields. As a matter of fact, aspects of structural stability and codimension one bifurcation are analysed

Foliations, solvability and global injectivity

Let F: R^n -> R^n be a C^\infty map such that DF(x) is invertible for all x in R^n. We know that F is a local diffeomorphism but, in general, it is not globally injective. We discuss the relations between some additional hypothesis that guarantee the global injectivity of F. Further, based on one of these hypotheses, we establish a necessary condition for the existence of F: R^n -> R^n such that det DF = h, where h: R^n -> [0,\infty) is a given C^\infty function