Geodesic
Solution of the parametric center problem for the Abel differential equation
Journal of the European Mathematical Society, Tome 19 (2017) no. 8, pp. 2343-2369

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DOI

The Abel differential equation y′=p(x)y2+q(x)y3 with p,q∈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∈R the equation y′=p(x)y2+εq(x)y3 has a center. In this paper we show that the Abel equation has a parametric center if and only if the antiderivatives P=∫p(x)dx, Q=∫q(x)dx satisfy the equalities P=P∘W, Q=Q​∘W for some polynomials P, Q​, and W such that W(a)=W(b). We also show that the last condition is necessary and sufficient for the “generalized moments” ∫ab​PidQ and ∫ab​QidP to vanish for all i geq0.
DOI : 10.4171/jems/719
Classification : 34-XX, 37-XX
Keywords: Periodic orbits, centers, Abel equation, moment problem, composition conjecture 
Fedor Pakovich. Solution of the parametric center problem for the Abel differential equation. Journal of the European Mathematical Society, Tome 19 (2017) no. 8, pp. 2343-2369. doi: 10.4171/jems/719
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     author = {Fedor Pakovich},
     title = {Solution of the parametric center problem for the {Abel} differential equation},
     journal = {Journal of the European Mathematical Society},
     pages = {2343--2369},
     year = {2017},
     volume = {19},
     number = {8},
     doi = {10.4171/jems/719},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/jems/719/}
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