This thesis is devoted to the study of several phenomena in Bifurcation Theory in Dynamical Systems and it has two parts. The first one deals with the classical problem of exponentially small splitting of separatrices in analytic Hamiltonian Systems of one and a half degrees of freedom and the second one studies low codimension local and global bifurcations of Filippov Systems.
The problem of the exponentially small splitting of separatrices was already considered by Poincar,; the Fundamental problem of mechanics, since its understanding is crucial for the study of existence of instabilities in close to integrable Hamiltonian Systems. In this kind of problems, due to the existence of exponentially small phenomena, one cannot apply Melnikov Theory to compute the splitting, and then one has to consider different techniques, which consist in extending the invariant manifolds to the complex.
In the first chapter of the thesis, we have studied this phenomenon in the most paradigmatic model, that is, the pendulum with a fast and periodic perturbation. In the second chapter, we have proved an analogous result for general Hamiltonian Systems of one degree of freedom with a perturbation fast and periodic in time, such that the unperturbed Hamiltonian has a hyperbolic or parabolic critical point whose stable and unstable invariant manifolds coincide along a separatrix. We have given results also in the so-called singular case, in which the perturbation has the same size as the unperturbed system.
These systems are models of the dynamics in a simple resonance of close to integrable Hamiltonian Systems of one and a half degrees of freedom.
The second part of the thesis is devoted to the study of Filippov Systems, which are systems mode led by ordinary differential equations which are discontinuous along a (or several) hypersurfaces. These systems are broadly used for modeling several phenomena, such as mechanical systems with dry friction or power converters, but they lack of a well established mathematical theory. In the third chapter of the thesis we have focused on the study of low codimension local and global bifurcations of planar Filippov Systems. First we have obtained a classification of the codimension-1 local and global bifurcations and the codimension-2 local bifurcations, and we have unfolded those which had more interesting dynamics around them.
In the fourth chapter, we have considered one of the most paradigmatic models in 3-dimensional Filippov Systems: the oscillator periodically forced with dry friction . For certain range of parameters, this system has a symmetric periodic orbit which is persistent. We have studied the discontinuity induced bifurcations undergone by this periodic orbit and we have proved the existence of infinitely many codimension·2 bifurcations points in the parameter space, from which emanate several codimension-1 bifurcation curves. We have also studied the dynamics considering an arbitrarily small friction, namely considering the non-smoothness as a perturbation.