The perturbations of a class of hyper-elliptic Hamilton systems with a double homoclinic loop through a nilpotent saddle

In this paper, we consider the upper bounds of a number of isolated zeros of Abelian integrals associated to system ẋ=y, ẏ=−x3(x2−1) under the perturbations of ϵ(α0+α1x+α2x2+α3x3+α4x4)y∂∂y, where 0<|ϵ|≪10<|ϵ|≪1 and αi∈Rαi∈R, i=0,…,4i=0,…,4. The unperturbed system has a double homoclinic loop with a nilpotent saddle of order 1. The sharp upper bounds are obtained for each of the cases of α1=α4=0α1=α4=0, α1=α3=0α1=α3=0, α2=α3=0α2=α3=0 and α3=α4=0α3=α4=0 when Abelian integrals are defined in the bounded period annuli.