Estimating the Number of Zeros of Abelian Integrals for the Perturbed Cubic Z4-Equivariant Planar Hamiltonian System

Abstract

We analyze the dynamics of a class of [Formula: see text]-equivariant Hamiltonian systems of the form [Formula: see text], where [Formula: see text] is complex, the time [Formula: see text] is real, while [Formula: see text] and [Formula: see text] are real parameters. The topological phase portraits with at least one center are given. The finite generators of Abelian integral [Formula: see text] are obtained, where [Formula: see text] is a family of closed ovals defined by [Formula: see text] [Formula: see text], [Formula: see text] is the open interval on which [Formula: see text] is defined, [Formula: see text], [Formula: see text] are real polynomials in [Formula: see text] and [Formula: see text] with degree [Formula: see text]. We give an estimation of the number of isolated zeros of the corresponding Abelian integral by using its algebraic structure. We show that for the given polynomials [Formula: see text] and [Formula: see text] in [Formula: see text] and [Formula: see text] with degree [Formula: see text], the number of the limit cycles of the perturbed [Formula: see text]-equivariant Hamiltonian system does not exceed [Formula: see text] (taking into account the multiplicity).