Zeros of Abelian integrals for a quartic Hamiltonian with figure-of-eight loop through a nilpotent saddle

Abstract

In this paper, we give the upper bound of the number of zeros of Abelian integral $I\left(h\right)={\oint }_{{\Gamma }_h}g(x,y)dy-f(x,y)dx$, where ${\Gamma }_h$ is the closed orbit defined by $H(x,y)=-x^2+x^4+y^4+r{x}^2{y}^2=h$, $r\ge 0$, $r\ne 2$, $h\in \Sigma$, $\Sigma$ is the maximal open interval on which the ovals $\left\{{\Gamma }_h\right\}$ exist, $f(x,y)$ and $g(x,y)$ are real polynomials in $x$ and $y$ of degree at most $n$.