Estimate of the number of zeros of Abelian integrals for a perturbation of hyperelliptic Hamiltonian system with nilpotent center

Abstract In this paper, we present a complete study of the zeros of Abelian integrals obtained by integrating the 1-form ( α  +  βx  +  γ x 2 ) ydx over the compact level curves of the hyperelliptic Hamiltonian of degree five H ( x , y ) = y 2 2 + 1 4 x 4 - 1 5 x 5 . Such a family of compact level curves surround a nilpotent center. It is proved that the lowest upper bound of the number of the isolated zeros of Abelian integral is two in any compact period annulus, and there exists some α , β and γ such that system could appear at least two limit cycles bifurcating from the nilpotent center. The proof relies on the Chebyshev criterion for Abelian integrals (Grau et al, Trans Amer Math Soc 2011) and some techniques in polynomial algebra.