Solution of the parametric center problem for the Abel differential equation
Solution of the parametric center problem for the Abel differential equation
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DOI：
10.4171/jems/719
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发表时间：
2014-07
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通讯作者：
F. Pakovich
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作者：
F. Pakovich
阿贝尔微分方程$y '= p（x）y^2+q（x）y^3$（p，q\in \mathbb R[x]$）称其中心在线段$[a，B]$上，如果它的所有解，在初始值$y（a）$足够小的情况下，满足条件$y（B）=y（a）$。描述Abel方程有中心的条件的问题可以解释为经典的Poincar\'e中心焦点问题的简化版本。阿贝尔方程被称为有一个“参数中心”，如果对于每个\mathbb R$中的$\vareps\，方程$y '= p（x）y^2+\vareps q（x）y^3$有一个中心。本文证明了Abel方程有参数中心当且仅当对某些多项式$\widetilde P，$ $\widetilde Q，$和$W$，反导数$P =\int p（x）dx，$ $Q=\int q（x）dx$满足等式$P=\widetilde P \circ W，\ $ $Q=\widetilde Q\circ W$使得$W（a）=W（B）$.我们还证明了最后一个条件是“广义矩”$\int_a^B P^id Q$和$\int_a^B Q^id P$对所有i\geq 0.$为零的充要条件。
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 $i\geq 0.$