Persistence of lower dimensional invariant tori for near-integrable Hamiltonians.

Speaker. Frank Trujillo
Date. 08.06.22 at 2:00 pm

Abstract. The classical KAM theory establishes the persistence, under sufficiently small perturbations, of most of the n-dimensional invariant tori for non-degenerate integrable Hamiltonians with n degrees of freedom. The surviving tori are those carrying a quasi-periodic motion by a Diophantine vector and, in particular, their restricted dynamics is minimal. On the other hand, such systems also admit n-dimensional invariant tori whose restricted dynamics is not minimal. These tori, which we call resonant, are foliated by lower dimensional invariant tori, that is, by invariant tori whose dimension is strictly smaller than the number of degrees of freedom of the system. The codimension of these invariant subtori (with respect to the resonant torus), is called the number of resonances of the resonant torus.
In this talk, I will present a criterion for the existence of at least one lower dimensional invariant torus, associated to a resonant invariant torus (with any number of resonances) in the unperturbed system, for a class of near-integrable non-convex Hamiltonians.