ON QUASIPERIODIC PERTURBATIONS OF ELLIPTIC EQUILIBRIUM POINTS
Angel Jorba(1) and Carles Simo(2)
(1) Dept. de Matematica Aplicada I (ETSEIB), Universitat Politecnica de
Catalunya, Diagonal 647, 08028 Barcelona (Spain).
jorba@ma1.upc.es
(2) Dept. de Matematica Aplicada i Analisi, Universitat de Barcelona,
Gran Via 585, 08007 Barcelona (Spain).
carles@maia.ub.es
Abstract
This work focusses on quasiperiodic time-dependent perturbations of
ordinary differential equations near elliptic equilibrium points.
This means studying
$$
\dot{x}=(A+\varepsilon Q(t,\varepsilon))x+\varepsilon g(t,\varepsilon)+
h(x,t,\varepsilon),
$$
where $A$ is elliptic and $h$ is ${\cal O}(x^2)$. It is shown 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.
Keywords: quasiperiodic perturbations, elliptic points, quasiperiodic
solutions, small divisors, quasiperiodic Floquet theorem, KAM theory.
AMS Subject Classifications: 34C27, 34C50, 58F27, 58F30