On the number of zeros of Melnikov functions

Details On the number of zeros of Melnikov functions On the number of zeros of Melnikov functions

2010-07-05

arxiv.org/abs/1007.0672v1

Mathematics

Dynamical Systems

Scientific paper

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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