International audienceWe provide an algorithm to compute the first non-zero derivative of the return map r ↦ L(r, ε) of a planar vector field which is a polynomial perturbation of a linear center. It yields a new method for finding the centre conditions in the case of a homogeneous perturbative part
We consider an Abel equation $(*)$ $y^{\prime}=p(x)y^2+q(x)y^3$ with $p(x)$, $q(x)$ polynomials in $x$. A center condition for ($*$) (closely related to the classical center condition for polynomial vector fields on the plane) is that $y_0=y(0)\equiv y(1)$ for any solution $y(x)$ of ($*$). This condition is given by the vanishing of all the Taylor coefficients $v_k(1)$ in the development $y(x)=y_0+\sum^{\infty}_{k=2}v_k(x)y^k_0$. A new basis for the ideals $I_k=\{v_2,\dots,v_k\}$ has recently been produced, defined by a linear recurrence relation. Studying this recurrence relation, we connect center conditions with a representability of $P=\int p$ and $Q=\int q$ in a certain composition form (developing further some results of Alwash and Lloyd), and with a behavior of the moments $\int P^kq$. On this base, explicit center equations are obtained for small degrees of $p$ and $q$.
International audienceWe 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 (*).We introduce a parametric version of this condition: an equation (**)y’=p(x)y2 +εq(x)y3p, 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 momentsmk(1), wheremk(x)=∫0xpk(t)q(t)(dt),P(x)=∫0xp(t)dt. We investigate the structure of zeroes ofmk(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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