On the Number of Zeros of Abelian Integral for a Class of Cubic Hamilton Systems with the Phase Portrait “Butterfly”

Abstract

The present paper is devoted to study the number of zeros of Abelian integral for the near-Hamilton system $$\begin{aligned} {\left\{ \begin{array}{ll} \dot{x} = 2y(bx^2+2cy^2)+\varepsilon f(x,y),\\ \dot{y} = 2x(1-2ax^2-by^2)+\varepsilon g(x,y), \end{array}\right. } \end{aligned}$$ where \(a,b,c\in \mathbb {R}\), \(b 0\), \(b^2<4ac\), \(0<|\varepsilon |\ll 1\), f(x, y) and g(x, y) are polynomials in (x, y) of degree n. The generators of the corresponding Abelian integral satisfy three different Picard–Fuchs equations. We obtain an upper bound of the number of isolated zeros of the Abelian integral.