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Center conditions II: Parametric and model center problems

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Abstract

We consider an Abel equation (*)y’=p(x)y 2 +q(x)y 3 withp(x), q(x) polynomials inx. A center condition for (*) (closely related to the classical center condition for polynomial vector fields on the plane) is thaty 0=y(0)≡y(1) for any solutiony(x) of (*).

We introduce a parametric version of this condition: an equation (**)y’=p(x)y 2 +εq(x)y 3 p, q as above, ℂ, is said to have a parametric center, if for any ε and for any solutiony(ε,x) of (**),y(ε,0)≡y(ε,1).

We show that the parametric center condition implies vanishing of all the momentsm k (1), wherem k (x)=∫ x0 pk(t)q(t)(dt),P(x)=∫ x0 p(t)dt. We investigate the structure of zeroes ofm k (x) and on this base prove in some special cases a composition conjecture, stated in [10], for a parametric center problem.

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Authors and Affiliations

  1. Jerusalem College of Engineering, Ramat Bet Hakerem, 91035, Jerusalem, Israel

    M. Briskin

  2. Département de Mathématiques, Université de Paris VI, U.F.R. 920, 46-56, 4 Place Jussieu, B.P. 172, 75252, Paris, France

    J. -P. Francoise

  3. Department of Theoretical Mathematics, The Weizmann Institute of Science, 76100, Rehovot, Israel

    Y. Yomdin

Authors
  1. M. Briskin
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  2. J. -P. Francoise
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  3. Y. Yomdin
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Correspondence to M. Briskin.

Additional information

The research of the first and the third author was supported by the Israel Science Foundation, Grant No. 101/95-1 and by the Minerva Foundation.

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Briskin, M., Francoise, J.P. & Yomdin, Y. Center conditions II: Parametric and model center problems. Isr. J. Math. 118, 61–82 (2000). https://doi.org/10.1007/BF02803516

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  • DOI: https://doi.org/10.1007/BF02803516

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