Estimate of the number of zeros of Abelian integrals for a kind of quartic Hamiltonians with two centers

Abstract

In this paper, we give the upper bound of the number of zeros of Abelian integral I ( h ) = ∮ Γ h P ( x , y ) d x - Q ( x , y ) d y , where Γ h is the closed orbit defined by H ( x , y ) = - x 2 + λ x 4 + y 4 = h , λ > 0 , h ∈ Σ ; Σ is the maximal open interval on which the ovals { Γ h } exist; P ( x , y ) and Q ( x , y ) are real polynomials in x and y of degree at most n .