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On the number of zeros of Melnikov functions

Sergey Benditkis; Dmitry Novikov — 2011

Annales de la faculté des sciences de Toulouse Mathématiques

We provide an effective uniform upper bound 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 [BNY10]. The bound follows from an effective construction of the Gauss-Manin connection for iterated integrals.

Pseudo-abelian integrals on slow-fast Darboux systems

Marcin Bobieński; Pavao Mardešić; Dmitry Novikov — 2013

Annales de l’institut Fourier

We study pseudo-abelian integrals associated with polynomial deformations of slow-fast Darboux integrable systems. Under some assumptions we prove local boundedness of the number of their zeros.

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