We consider an Abel equation (*)y’=p(x)y2 +q(x)y3 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 thaty0=y(0)≡y(1) for any solutiony(x) of (*).
Folowing , we consider a parametric version of this condition: an equation (**)y’=p(x)y2 +εq(x)y3p, q as above, ε ∈ ℂ, is said to have a parametric center, if for any ɛ and for any solutiony(ɛ,x) of (**)y(ɛ, 0)≡y(ɛ, 1)..
We give another proof of the fact, shown in , that the parametric center condition implies vanishing of all the momentsmk (1), wheremk(x)=∫0xpk(t)q(t)(dt),P(x)=∫0xp(t)dt. We investigate the structure of zeroes ofmk(x) and generalize a “canonical representation” ofmk(x) given in . On this base we prove in some additional cases a composition conjecture, stated in [6, 7] for a parametric center problem.
M. Briskin, J.-P. Francoise and Y. Yomdin,Center conditions, composition of polynomials and moments on algebraic curves, Ergodic Theory and Dynamical Systems19 (1999), 1201–1220.MATHCrossRefMathSciNetGoogle Scholar
M. Briskin, J.-P. Francoise and Y. Yomdin,Center conditions II: Parametric and model center problems, Israel Journal of Mathematics, this volume.Google Scholar
C. J. Christopher,Small-amplitude limit cycles in polynomial Liénard systems, NoDEA, Nonlinear Differential Equations and Applications3 (1996), 183–190.MATHCrossRefMathSciNetGoogle Scholar
J. Devlin,Word problem related to periodic solutions of a non-autonomous system, Mathematical Proceedings of the Cambridge Philosophical Society108 (1990), 127–151.MATHMathSciNetGoogle Scholar
J. Devlin,Word problems related to derivatives of the displacement map, Mathematical Proceedings of the Cambridge Philosophical Society110 (1991), 569–579.MATHMathSciNetCrossRefGoogle Scholar