## Abstract

The Abel differential equation y' = p(x)y^{2} + q(x)y^{3} 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)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 = ∫ 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" ∫^{b}_{a} P^{i} dQ and ∫^{b}_{a} Q^{i}dP to vanish for all i ≥ 0.

Original language | English |
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Pages (from-to) | 2343-2369 |

Number of pages | 27 |

Journal | Journal of the European Mathematical Society |

Volume | 19 |

Issue number | 8 |

DOIs | |

State | Published - 1 Jan 2017 |

## Keywords

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

## ASJC Scopus subject areas

- General Mathematics
- Applied Mathematics