Zeros of Abelian integral for a class of quintic planar vector fields with a double homoclinic loop under non-symmetry perturbations

In this paper, we study the number of zeros of Abelian integral for a class of quintic planar vector fields with a double homoclinic loop under non-symmetry perturbations. We investigate the bounds of Abelian integral in bounded annuli and unbounded annuli by means of checking ECT-system and qualitative theory of planar vector fields.