Zeros of Abelian integrals for a quartic Hamiltonian with figure-of-eight loop through a nilpotent saddle. (February 2016) - Yang, Jihua; Zhao, Liqin
Abstract: In this paper, we give the upper bound of the number of zeros of Abelian integral I ( h ) = ∮ Γ h g ( x, y ) d y − f ( x, y ) d x, where Γ h is the closed orbit defined by H ( x, y ) = − x 2 + x 4 + y 4 + r x 2 y 2 = h, r ≥ 0, r ≠ 2, h ∈ Σ, Σ is the maximal open interval on which the ovals Γ h exist, f ( x, y ) and g ( x, y ) are real polynomials in x and y of degree at most n .