Stability in Hamiltonian Systems : KAM stability versus instability around an invariant torus

Sammanfattning: In his ICM-54 lecture, Kolmogorov introduced a now fundamental result regarding the persistence of a large (in the measure theoretic sense) set of invariant tori, in a certain category of almost-integrable Hamiltonian systems. 44 years later, in his ICM-98 talk, Herman conjectured that given any analytic Hamiltonian system with an invariant diophantine torus, this torus will always be accumulated by a positive measure set of invariant KAM tori, i.e. it will be KAM stable. In this thesis, we build upon recent results and provide a counterexample in three degrees of freedom to KAM stability around an invariant torus, in the category of smooth Hamiltonian systems. The thesis is self-contained in the sense that it also includes a brief introduction to Hamiltonian systems, as well as an exposition of Kolmogorov's classic result.