Jorba A., Villanueva J.
On the Persistence of Lower Dimensional Invariant Tori under
Quasiperiodic Perturbations
(154K, LaTeX)
ABSTRACT. In this work we consider time dependent quasiperiodic perturbations
of autonomous Hamiltonian systems. We focus on the effect that this
kind of perturbations has on lower dimensional invariant tori. Our
results show that, under standard conditions of analyticity,
nondegeneracy and
nonresonance, most of these tori survive, adding the frequencies of
the perturbation to the ones they already have.
The paper also contains estimates on the amount of surviving tori. The
worst situation happens when the initial tori are normally elliptic.
In this case, a torus (identified by the vector of intrinsic
frequencies)
can be continued with respect to a perturbative
parameter $\epsilon\in[0,\epsilon_0]$, except for a set of
$\epsilon$ of measure exponentially small with $\epsilon_0$.
In case that $\epsilon$ is fixed (and sufficiently small), we prove the
existence of invariant tori for every vector of
frequencies close to the one of the initial torus, except for a set of
frequencies of measure exponentially small with the distance to the
unperturbed torus. As a particular case, if the perturbation is
autonomous, these results also give the same kind of estimates
on the measure of destroyed tori.
Finally, these results are applied to some problems of
celestial mechanics, in order to help in the description of the
phase space of some concrete models.