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

摘要

In this paper, we consider the upper bounds of a number of isolated zeros of Abelian integrals associated to system x = y, y = ?x~3(x~2 ? 1) under the perturbations of ? (α_0 + α_1x + α_2x~2 + α_3x~3 + α_4x~4)y ?/(?y), where 0 < |?| ? 1 and α_i ∈ R, i = 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 α_1 = α_4 = 0, α_1 = α_3 = 0, α_2 = α_3 = 0 and α_3 = α_4 = 0 when Abelian integrals are defined in the bounded period annuli.