Perturbative series expansions: Theoretical aspects and numerical investigations

Perturbation theory is introduced by means of models borrowed from Celestial Mechanics, namely the two-body and three-body problems. Such models allow one to introduce in a simple way the concepts of integrable and nearly-integrable systems, which can be conveniently investigated using Hamiltonian formalism. After discussing the problem of the convergence of perturbative series expansions, we introduce the basic notions of KAM theory, which allows (under quite general assumptions) to state the persistence of invariant tori. The value at which such surfaces break-down can be determined by means of numerical algorithms. Among the others, we review three methods to which we refer as Greene, Pade and Lyapunov. We present some concrete applications to discrete models of the three different techniques, in order to provide complementary information about the break-down of invariant tori.