Solution of the parametric center problem for the Abel differential equation

TL;DR

This paper characterizes when the Abel differential equation has a parametric center by linking it to polynomial compositions of its antiderivatives and the vanishing of certain generalized moments.

Contribution

It provides a complete algebraic characterization of the parametric center condition for Abel equations using polynomial composition and moment conditions.

Findings

The Abel equation has a parametric center iff its antiderivatives are composed with a common polynomial W.

The parametric center condition is equivalent to the vanishing of all generalized moments involving P and Q.

The paper establishes necessary and sufficient conditions connecting polynomial structure to the center property.

Abstract

The Abel differential equation $y'=p(x)y^2+q(x)y^3$ with $p,q\in \mathbb R[x]$ is said to have a center on a segment $[a,b]$ if all its solutions, with the initial value $y(a)$ small enough, satisfy the condition $y(b)=y(a)$. The problem of description of conditions implying that the Abel equation has a center may be interpreted as a simplified version of the classical Center-Focus problem of Poincar\'e. The Abel equation is said to have a "parametric center" if for each $\varepsilon \in \mathbb R$ the equation $y'=p(x)y^2+\varepsilon q(x)y^3$ has a center. In this paper we show that the Abel equation has a parametric center if and only if the antiderivatives $P=\int p(x) dx,$ $Q=\int q(x) dx$ satisfy the equalities $P=\widetilde P \circ W,\ $ $Q=\widetilde Q\circ W$ for some polynomials $\widetilde P,$ $\widetilde Q,$ and $W$ such that $W(a)=W(b)$. We also show that the last condition is necessary and sufficient for the "generalized moments" $\int_a^b P^id Q$ and $\int_a^b Q^id P$ to vanish for all $i\geq 0.$