Attività del Dipartimento

Colloquium di Matematica

Small, smooth perturbations of integrable Hamiltonian systems with n degrees of freedom preserve invariant Lagrangian tori which have diophantine rotation vector; this is the content of classical KAM theory. Less is known about the fate of tori for which the unperturbed frequency vector is resonant. We will show that for an (n-r)-dimensional family of r-dimensional isotropic invariant tori for the unperturbed problem,  a generic perturbation will preserve 2^(n-r) many invariant r-tori. Furthermore, in a neighborhood of each preserved r-torus there are Cantor families of invariant d-tori for r < d leq m leq n, where m is the Morse index of a certain functional. 
These new higher dimensional tori have frequencies that decompose into an order-1 component and an order \epsilon^(1/3) component; they are lower dimensional KAM tori analogs of classical Birkhoff - Lewis orbits of long period that occur near elliptic periodic orbits.

 
org: PROCESI Michela