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**Absolutely convergent series expansions for quasi periodic motions.**
*(English)*
Zbl 0896.34035

The problem of finding quasiperiodic series in a small parameter \(\varepsilon\) of solutions of certain analytic differential equations depending on \(\varepsilon\) occupied many astronomers and mathematicians during the last century (Linstedt, Gyldén, Poincaré). What they were looking for are formal series in \(\varepsilon\) which have coefficients that are analytic quasiperiodic functions and which satisfy the differential equation as if the series were convergent. One version of this problem is to find quasiperiodic series for a fixed frequency \(\omega\), in which case one requires that all coefficients are quasiperiodic with frequency \(\omega\).

In general such series exist if one supplies a certain number of constants to the equation and the problem then becomes one of “killing the constants” for the particular equation of the class of equations considered. Poincaré constructed in several ways quasiperiodic series expansions of the formal solution. It is important to note that a quasiperiodic series expansion of the formal solution is not unique. This in contrast to the formal solution itself, for which the term Linstedt series is reserved.

The Linstedt series are described together with a natural quasi-periodic series expansion and its absolute divergence is explained. Then a large class of compensations for the terms in the series are described. Taking into account these compensations new quasiperiodic absolutely convergent series are obtained. The proof of the absolute convergence generalizes Siegel’s method. Then it is proved that these new series are indeed an expansion of the formal solution.

Contents of the paper:

1. Introduction (The Hamiltonian problem).

2. The classical approach to the Hamiltonian problem – absolutely divergent series (The formal solution and its series expansion, “killing the constants” and the Linstedt series, convergence and divergence of the formal solution).

3. Siegel’s method (First, second and third lemma).

4. Generalization of Siegel’s first lemma (resonance on linear index set, generalization of Siegel’s first lemma, the basic compensations).

5. Generalization of Siegel’s third lemma.

6. Absolutely convergent series (interpretation of the series, “killing the constants”, theorem, other examples).

In general such series exist if one supplies a certain number of constants to the equation and the problem then becomes one of “killing the constants” for the particular equation of the class of equations considered. Poincaré constructed in several ways quasiperiodic series expansions of the formal solution. It is important to note that a quasiperiodic series expansion of the formal solution is not unique. This in contrast to the formal solution itself, for which the term Linstedt series is reserved.

The Linstedt series are described together with a natural quasi-periodic series expansion and its absolute divergence is explained. Then a large class of compensations for the terms in the series are described. Taking into account these compensations new quasiperiodic absolutely convergent series are obtained. The proof of the absolute convergence generalizes Siegel’s method. Then it is proved that these new series are indeed an expansion of the formal solution.

Contents of the paper:

1. Introduction (The Hamiltonian problem).

2. The classical approach to the Hamiltonian problem – absolutely divergent series (The formal solution and its series expansion, “killing the constants” and the Linstedt series, convergence and divergence of the formal solution).

3. Siegel’s method (First, second and third lemma).

4. Generalization of Siegel’s first lemma (resonance on linear index set, generalization of Siegel’s first lemma, the basic compensations).

5. Generalization of Siegel’s third lemma.

6. Absolutely convergent series (interpretation of the series, “killing the constants”, theorem, other examples).

Reviewer: I.Ginchev (Varna)

### MSC:

34C27 | Almost and pseudo-almost periodic solutions to ordinary differential equations |

34A25 | Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc. |

30B10 | Power series (including lacunary series) in one complex variable |