On Moment Condition and Center Condition for Abel Equation

TL;DR

This paper investigates conditions under which Abel equations have centers, establishing that certain moment conditions must hold and partially confirming a longstanding conjecture in the field.

Contribution

The paper proves that moment conditions are necessary for the Abel equation to have a center and provides a partial positive answer to a 2001 conjecture.

Findings

Moment conditions are necessary for the existence of a center.

Established a partial confirmation of a longstanding conjecture.

Provided analytical conditions linking the functions in Abel equations.

Abstract

In this paper we consider Abel equation $x' = g(t)x^2+f(t)x^3$, where $f$ and $g$ are analytical functions. We proved that if the equation has a center at $x=0$, then the Moment Conditions, i. e., $m_k=\int_{-1}^1f(t)(G(t))^kdt=0,~~k=0,1,2$, is satisfied where $G(t)=\int_{-1}^tg(s)ds$. Besides, we give partial a positive answer to a conjecture proposed by Y. Lijun and T. Yun in 2001.