LINDSTEDT SERIES FOR LOWER DIMENSIONAL TORI
Angel Jorba(1), Rafael de la Llave(2) and Maorong Zou(2)
(1) Dept. de Matematica Aplicada I (ETSEIB), Universitat Politecnica de
Catalunya, Diagonal 647, 08028 Barcelona (Spain).
E-mail: jorba@ma1.upc.es
(2) Dept. of Mathematics, University of Texas at Austin,
Austin, TX 78712 (USA).
E-mails: llave@math.utexas.edu, mzou@math.utexas.edu
Abstract
We consider the existence and effective computation of low-dimensional
(less independent frequencies than degrees of freedom) invariant tori
of a near-integrable system. Lindstedt method is a systematic
procedure to compute formal power series expansions of quasi-periodic
solutions. This procedure is very suitable for numerical
computations. Under some non-degeneracy assumptions it is possible to
show that a finite number of this low dimensional tori persist in the
sense of formal power series expansions of the perturbation parameter
($\varepsilon$).
Contrary to the series for full dimensional tori, whose convergence is
established by KAM theory, the convergence of the expansions for
low-dimensional tori is not settled -- even if its reasonable to
suspect they diverge for typical systems --. Nevertheless, we show
that these tori are analytic functions in $\varepsilon$ in a complex
disk minus a thin wedge ending at the origin. The formal power series
obtained in the Lindstedt method are an asymptotic expansion to them
on this set.
The main technical tool is a KAM theorem that shows that near a torus
which is approximately invariant and approximately reducible (the
variation equations can be reduced to constant up to some small error)
there is a truly invariant torus. We point out that this KAM theorem
presents small divisors involving the normal and intrinsic frequencies
of the torus whereas the Linsdedt procedure only presents small
divisors coming from the intrinsic frequencies.
Note also that the quasi-invariant, quasi reducible tori that are the
input for the KAM procedure may have been produced by other methods
than Lindstedt series, notably numerical computations or other
perturbative expansions.
Keywords: Low-dimensional invariant tori, KAM theory, Lindstedt series.