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On the convergence of formal series containing small divisors. (English) Zbl 0957.37059

Simó, Carles (ed.), Hamiltonian systems with three or more degrees of freedom. Proceedings of the NATO Advanced Study Institute, S’Agaró, Spain, June 19-30, 1995. Dordrecht: Kluwer Academic Publishers. NATO ASI Ser., Ser. C, Math. Phys. Sci. 533, 345-349 (1999).
Quasi-periodic solutions of perturbed integrable Hamiltonian systems may be found either by KAM theory or by Poincaré-Lindstedt series. To show convergence of the latter it is necessary to compensate for small divisors by grouping terms in a useful way. The authors present a problem related to multiple resonances for which they expect the Poincaré-Lindstedt series to give stronger results than the KAM approach. They indicate how to group terms by using labelled, weighted, rooted trees.
For another problem the authors use a KAM-type theorem to prove convergence of the Poincaré-Lindstedt series for isoenergetic maximal quasi-periodic solutions.
For the entire collection see [Zbl 0942.00030].

MSC:

37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
70H08 Nearly integrable Hamiltonian systems, KAM theory
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