PERIODIC AND QUASI-PERIODIC ORBITS IN NEARLY INTEGRABLE HAMILTONIAN SYSTEMS

The study of periodic and quasi-periodic orbits in nearly integrable Hamiltonian systems is a long standing and challenging problem, that dates back to Poincaré. Quoting Poincaré, they represent "the only opening through which we can try to enter a place which, up to now, was deemed inaccessible". The aim of this thesis is to find effective and constructive algorithms for constructing both periodic and quasi-periodic solutions via a modification of the normal form methods related to Kolmogorov's theorem. The thesis is divided in two parts. The first part concerns the classical problem of the continuation of periodic orbits surviving to the breaking of invariant maximal or lower dimensional completely resonant tori in nearly integrable Hamiltonian systems: we here propose a new scheme which allows to deal with the problem of degeneracy at any order of perturbation. The second part regards the development of a variation of the Kolmogorov's normalization algorithm, by avoiding the so-called translation step at the price of fixing only the final frequency, while the initial one can only be determined a posteriori.