On Elliptic Lower-Dimensional Invariant Tori with Prescribed Frequencies in Hamiltonian Systems with Small Parameters

In this paper we consider the persistence of elliptic lower-dimensional invariant tori with prescribed frequencies in Hamiltonian systems with small parameters. Under the Brjuno nondegeneracy condition, if the prescribed frequencies satisfy a Diophantine condition, by the KAM technique we prove that for most of small parameters in the sense of Lebesgue measure, the Hamiltonian systems admit a lower-dimensional invariant torus whose frequency vector is a dilation of the prescribed frequencies.