Solution of the parametric center problem for the Abel differential equation

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8 Scopus citations


The Abel differential equation y' = p(x)y2 + q(x)y3 with p, q ∈ ℝ[x] is said to have a center on an interval [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" ∫ba Pi dQ and ∫ba QidP to vanish for all i ≥ 0.

Original languageEnglish
Pages (from-to)2343-2369
Number of pages27
JournalJournal of the European Mathematical Society
Issue number8
StatePublished - 1 Jan 2017


  • Abel equation
  • Centers
  • Composition conjecture
  • Moment problem
  • Periodic orbits

ASJC Scopus subject areas

  • General Mathematics
  • Applied Mathematics


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