Moscow Mathematical Journal

Volume 18, Issue 2, April–June 2018 pp. 367–386.

Bounding the Length of Iterated Integrals of the First Nonzero Melnikov Function

Authors: Pavao Mardešić (1), Dmitry Novikov (2), Laura Ortiz-Bobadilla (3), and Jessie Pontigo-Herrera (4)
Author institution:(1) Université de Bourgogne, Institute de Mathématiques de Bourgogne - UMR 5584 CNRS, Université de Bourgogne, 9 avenue Alain Savary, BP 47870, 21078 Dijon, FRANCE
(2) Faculty of Mathematics and Computer Science, Weizmann Institute of Science, Rehovot, 7610001, Israel
(3) Instituto de Matemáticas, Universidad Nacional Autónoma de México (UNAM), Área de la Investigación Científica, Circuito exterior, Ciudad Universitaria, 04510, Ciudad de México, México
(4) Faculty of Mathematics and Computer Science, Weizmann Institute of Science, Rehovot, 7610001 Israel

Summary:

We consider small polynomial deformations of integrable systems of the form dF = 0, F∈ℂ[x,y] and the first nonzero term M_μ of the displacement function Δ(t, ε) = ∑_i=μ M_i (t)εⁱ along a cycle γ(t)∈F⁻¹(t). It is known that Mμ is an iterated integral of length at most μ. The bound μ depends on the deformation of dF.

In this paper we give a universal bound for the length of the iterated integral expressing the first nonzero term M_μ depending only on the geometry of the unperturbed system dF=0. The result generalizes the result of Gavrilov and Iliev providing a sufficient condition for M_μ to be given by an abelian integral, i.e., by an iterated integral of length 1. We conjecture that our bound is optimal.

2010 Math. Subj. Class. Primary: 34C07; Secondary: 34C05, 34C08.

Keywords: Hilbert 16th problem, center problem, Poincaré return map, abelian integrals, limit cycles, free group automorphism.

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