Persistence of lower-dimensional tori under the first Melnikov's non-resonance condition
The dynamics of integrable Hamiltonian systems is simple, since all the compact energy surfaces are foliated by invariant tori which carry quasiperiodic motions. There are not many examples of this kind of systems (this is an active field of research); however, the KAM theory proves that for all Hamiltonian systems near a nondegenerate integrable Hamiltonian system the quasiperiodic motions in invariant tori are typical. In the middle of the 1960's, Melnikov founded a KAM persistence result for lower-dimensional tori, satisfying the now called first and second Melnikov's non-resonance conditions. In the present paper, the authors prove the persistence of lower-dimensional tori without assuming the second Melnikov's non-resonance condition. They also prove that this result is optimal in the sense that they are able to construct an example showing that the first Melnikov's non-resonance condition is necessary.
scientific article; zbMATH DE number 1770825