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

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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 $\mu$ epends 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 $\mu$ 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.
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https://hal.archives-ouvertes.fr/hal-01900091
Contributor : Sébastien Mazzarese <>
Submitted on : Saturday, October 20, 2018 - 9:53:38 PM
Last modification on : Thursday, February 7, 2019 - 4:54:00 PM

### Identifiers

• HAL Id : hal-01900091, version 1
• ARXIV : 1703.03837

### Citation

Pavao Mardešić, Dmitry Novikov, Laura Ortiz-Bobadilla, Jessie Diana Pontigo-Herrera. Bounding the Length of Iterated Integrals of the First Nonzero Melnikov Function. Moscow mathematical journal, 2018, 18 (2), pp.367-386. ⟨hal-01900091⟩

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