Monodromic nilpotent singular points with odd Andreev number and the center problem

TL;DR

This paper investigates the center problem for nilpotent singular points with odd Andreev number in planar vector fields, providing new characterizations and solutions for specific cases.

Contribution

It characterizes systems with odd Andreev number where inverse integrating factors do not solve the center problem and solves the case for n=3.

Findings

Characterization of systems with odd Andreev number where inverse integrating factors fail

Necessary center conditions for all Andreev numbers

Complete solution of the center problem for n=3

Abstract

Given a nilpotent singular point of a planar vector field, its monodromy is associated with its Andreev number $n$. The parity of $n$ determines whether the existence of an inverse integrating factor implies that the singular point is a nilpotent center. For $n$ odd, this is not always true. We give a characterization for a family of systems having Andreev number $n$ such that the center problem cannot be solved by the inverse integrating factor method. Moreover, we study general properties of this family, determining necessary center conditions for every $n$ and solving the center problem in the case $n=3$.