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

摘要

The present paper is devoted to study the number of zeros of Abelian integral for the near-Hamilton system x=2y(bx2+2cy2)+epsilon f(x,y),y=2x(1-2ax2-by2)+epsilon g(x,y),where a,b,c is an element of R, b<0, c>0, b2<4ac, 0<|epsilon | 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.