On classical series expansions for quasi-periodic motions

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