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Iterated integrals, Gelfand-Leray residue, and first return mapping. (English) Zbl 1108.37043

Summary: Recently, one of the authors gave an algorithm for calculating the first nonzero Poincaré-Pontryagin function of a small polynomial perturbation of a polynomial Hamiltonian, under a generic hypothesis. We generalize this algorithm and show that any Poincaré-Pontryagin function of order \(\ell\), denoted by \(M_\ell\), can be written as a sum of an iterated integral of length at most \(\ell\) and of a combination of all previous Poincaré-Pontryagin functions, \(M_1,M_2, \dots,M_{\ell-1}\), and their derivatives. This extends some results obtained recently and allows us to identify the Bautin ideal with the ideal generated by iterated integrals.

MSC:

37J40 Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion
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