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On classical series expansions for quasi-periodic motions. (English) Zbl 0891.58016
Summary: We reconsider the problem of convergence of classical expansions in a parameter $$\varepsilon$$ 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 $$\varepsilon^s$$ 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 $$\varepsilon$$ expansions. We also discuss some relations between our expansions and Lindstedt’s ones.

##### MSC:
 37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) 37K35 Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems
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