Abelian Integrals from an Unfolding of Codimension-3 Singularities with Nilpotent Linear Part

Abstract

In this paper, we study the maximum number of limit cycles for the unfolding of codimension-3 planar singularities with nilpotent linear parts. In [J. Math. Anal. Appl. 499 (2017)], the authors proved that when parameter [Formula: see text] is rational, the corresponding problem could be transformed to solving semi-algebraic systems. At the same time, it is pointed out that when [Formula: see text], the logarithmic function will appear according to the method, which makes it impossible to solve the problem. In this paper, we use some techniques to avoid the occurrence of logarithmic function, and get the corresponding system to produce at most two limit cycles.