 94310 Jorba A., Simo C.
 On Quasiperiodic Perturbations of Elliptic Equilibrium Points
(107K, LaTeX)
Oct 10, 94

Abstract ,
Paper (src),
View paper
(auto. generated ps),
Index
of related papers

Abstract. In this work we study quasiperiodic timedependent perturbations of
ordinary differential equations near elliptic equilibrium points, that
is, equations like
$$
\dot{x}=(A+\varepsilon Q(t,\varepsilon))x+\varepsilon g(t,\varepsilon)+
h(x,t,\varepsilon),
$$
where $A$ is elliptic and $h$ is ${\cal O}(x^2)$. Our results show
that, under suitable hypothesis of analyticity, nonresonance and
nondegeneracy with respect to $\varepsilon$, there exists a Cantorian
set ${\cal E}$ such that for all
$\varepsilon\in{\cal E}$ there exists a quasiperiodic solution such
that it goes to zero when $\varepsilon$ does. This quasiperiodic
solution has the same set of basic frequencies as the perturbation.
Moreover, the relative measure of the set
$[0,\,\varepsilon_0]\setminus{\cal E}$ in $[0,\,\varepsilon_0]$
is exponentially small in $\varepsilon_0$. The case $g\equiv 0$,
$h\equiv 0$ (quasiperiodic Floquet theorem) is also considered.
Finally, the Hamiltonian case is studied. In this situation, most
of the invariant tori that are near the equilibrium point are not
destroyed, but only slightly deformed and ``shaken" in a quasiperiodic
way. This quasiperiodic ``shaking" has the same basic frequencies
as the perturbation.
 Files:
94310.tex