On classical series expansions for quasi-periodic motions

by Antonio Giorgilli and Ugo Locatelli

**Abstract:**
We reconsider the problem of convergence of classical
expansions in a parameter $\epsilon$ for quasiperiodic motions on
invariant tori in nearly integrable Hamiltonian systems. Using a
reformulation of the algorithm proposed by Kolmogorov, we show that if
the frequencies satisfy the nonresonance condition proposed by Bruno,
then one can construct a normal form such that the coefficient of
$\epsilon^s$ is a sum of $O(C^s)$ terms each of which is bounded by
$O(C^s)$. This allows us to produce a direct proof of the classical
$\epsilon$ expansions. We also discuss some relations between our
expansions and the Lindstedt's ones.

MPEJ **3** N. 5 (1997)

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