 96500 Jorba A., Villanueva J.
 On the Normal Behaviour of Partially Elliptic Lower Dimensional Tori
of Hamiltonian Systems
(160K, LaTeX)
Oct 21, 96

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Abstract. The purpose of this paper is to study the dynamics near a reducible
lower dimensional invariant tori of a finitedimensional autonomous
Hamiltonian system with $\ell$ degrees of freedom. We will focus in
the case in which the torus has (some) elliptic directions.
First, let us assume that the torus is totally elliptic. In this
case, it is shown that the diffusion time (the time to move away from
the torus) is exponentially big with the initial distance to the
torus. The result is valid, in particular, when the torus is of
maximal dimension and when it is of dimension 0 (elliptic point). In
the maximal dimension case, our results coincide with previous
ones. In the zero dimension case, our results improve the existing
bounds in the literature.
Let us assume now that the torus (of dimension $r$, $0\le r<\ell$) is
partially elliptic (let us call $m_e$ to the number of these
directions). In this case we show that, given a fixed number of
elliptic directions (let us call $m_1\le m_e$ to this number), there
exist a Cantor family of invariant tori of dimension $r+m_1$, that
generalize the linear oscillations corresponding to these elliptic
directions. Moreover, the Lebesgue measure of the complementary of
this Cantor set (in the frequency space $\RR^{r+m_1}$) is proven to
be exponentially small with the distance to the initial torus. This
is a sort of ``Cantorian central manifold'' theorem, in which the
central manifold is completely filled up by invariant tori and it is
uniquely defined.
The proof of these results is based on the construction of suitable
normal forms around the initial torus.
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