Abelian integrals in unfoldings of codimension 3 singularities with nilpotent linear parts

This paper is concerned with the upper bound of the number of limit cycles in unfolding of codimension 3 planar singularities with nilpotent linear parts. After making a central rescaling, the problem reduces to a perturbation problem of a one-parameter family of quadratic reversible systems. As the parameter a∈(−1,1)∖{0} is rational, except the case a=−23, based on the Chebyshev criterion for Abelian integrals and a rationalizing transformation, the problem could be solved theoretically. To illustrate our approaches, two particular cases (corresponding to nilpotent codimension 3 saddle and elliptic case respectively) are proved where the upper bound of the number of limit cycles is two.