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arXiv · 1007.0672

On the number of zeros of Melnikov functions

Dmitry Novikov·Sergey Benditkis
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Abstract

We provide an effective uniform upper bond for the number of zeros of the first non-vanishing Melnikov function of a polynomial perturbations of a planar polynomial Hamiltonian vector field. The bound depends on degrees of the field and of the perturbation, and on the order $k$ of the Melnikov function. The generic case $k=1$ was considered by Binyamini, Novikov and Yakovenko (\cite{BNY-Inf16}). The bound follows from an effective construction of the Gauss-Manin connection for iterated integrals.

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Dmitry Novikov, Sergey Benditkis. 2010-07-05. On the number of zeros of Melnikov functions. https://arxiv.org/abs/1007.0672

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