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.

Voir la notice de l'article provenant de la source EMS Press

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 
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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},
     publisher = {mathdoc},
     volume = {19},
     number = {8},
     year = {2017},
     doi = {10.4171/jems/719},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/jems/719/}
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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. http://geodesic.mathdoc.fr/articles/10.4171/jems/719/

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