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This article is cited in 4 scientific papers (total in 4 papers)
Bounding the length of iterated integrals of the first nonzero Melnikov function
Pavao Mardešića, Dmitry Novikovb, Laura Ortiz-Bobadillac, Jessie Pontigo-Herrerab a Université de Bourgogne, Institute de Mathématiques de Bourgogne — UMR 5584 CNRS, Université de Bourgogne, 9 avenue Alain Savary, BP 47870, 21078 Dijon, FRANCE
b Faculty of Mathematics and Computer Science, Weizmann Institute of Science, Rehovot, 7610001, Israel
c Instituto de Matemáticas, Universidad Nacional Autónoma de México (UNAM), Área de la Investigación Científica, Circuito exterior, Ciudad Universitaria, 04510, Ciudad de México, México
Abstract:
We consider small polynomial deformations of integrable systems of the form $dF=0$, $F\in\mathbb C[x,y]$ and the first nonzero term $M_\mu$ of the displacement function $\Delta(t,\epsilon)=\sum_{i=\mu}M_i(t)\epsilon^i$ along a cycle $\gamma(t)\in F^{-1}(t)$. It is known that $M_\mu$ is an iterated integral of length at most $\mu$. The bound $\mu$ 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 $M_\mu$ 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 $M_\mu$ to be given by an abelian integral, i.e., by an iterated integral of length $1$. We conjecture that our bound is optimal.
Key words and phrases:
Hilbert 16th problem, center problem, Poincaré return map, abelian integrals, limit cycles, free group automorphism.
Citation:
Pavao Mardešić, Dmitry Novikov, Laura Ortiz-Bobadilla, Jessie Pontigo-Herrera, “Bounding the length of iterated integrals of the first nonzero Melnikov function”, Mosc. Math. J., 18:2 (2018), 367–386
Linking options:
https://www.mathnet.ru/eng/mmj676 https://www.mathnet.ru/eng/mmj/v18/i2/p367
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Abstract page: | 164 | References: | 31 |
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