Lindstedt series for lower dimensional tori
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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.