Solution of the parametric center problem for the Abel differential equation

Abstract

The Abel differential equation $y^{\prime }=p(x)y^2+q(x)y^3$ 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 Poincaré. The Abel equation is said to have a “parametric center” if for each $ϵ\in \mathbb{R}$ the equation $y^{\prime }=p(x)y^2+ϵq(x)y^3$ 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 $P=\tilde{P}\circ W$, $Q=\tilde{Q}\circ W$ for some polynomials $\tilde{P},$ $\tilde{Q},$ and $W$ such that $W(a)=W(b)$. We also show that the last condition is necessary and sufficient for the “generalized moments” ${\int }_a^b{P}^{i}dQ$ and ${\int }_a^b{Q}^{i}dP$ to vanish for all $igeq0.$