On the center-focus problem for the equation $\protect \frac{\protect \mathrm{d}y}{\protect \mathrm{d}x} + \Sigma _{i=1}^{n} a_i(x) y^i = 0, 0\le x \le 1$ where $a_i$ are polynomials

Gavrilov, Lubomir

doi:10.5802/ahl.41

On the center-focus problem for the equation

\frac{d y}{d x} + Σ_{i = 1}^{n} a_{i} (x) y^{i} = 0, 0 \leq x \leq 1

where

a_{i}

are polynomials

Gavrilov, Lubomir ¹

¹ Institut de Mathématiques de Toulouse UMR5219 Université de Toulouse CNRS UPS IMT 31062 Toulouse Cedex 9 (France)

Annales Henri Lebesgue, Volume 3 (2020), pp. 615-648.

Abstract
Résumé

We study irreducible components of the set of polynomial plane differential systems with a center, which can be seen as a modern formulation of the classical center-focus problem. The emphasis is given on the interrelation between the geometry of the center set and the Picard–lefschetz theory of the bifurcation (or Poincaré–Pontryagin–Melnikov) functions. Our main illustrative example is the center-focus problem for the Abel equation on a segment, which is compared to the related polynomial Liénard equation.

Nous étudions les composantes irréductibles de l’ensemble des champs polynomiaux plans avec centre, ce qui peut être vu comme une formulation moderne du problème classique du centre-foyer de Poincaré. L’accent est mis sur l’interrelation entre la géométrie de l’ensemble des centres et la théorie de Picard–Lefschetz des fonctions de bifurcation (ou de Poincaré–Pontryagin–Melnikov). Notre exemple principal est le problème du centre-foyer pour l’équation d’Abel sur un segment, comparée à l’équation de Liénard associée.

Received: 2018-11-28
Revised: 2019-08-11
Accepted: 2019-11-06
Published online: 2020-07-10

DOI: 10.5802/ahl.41

Classification: 37F75, 34C14, 34C05
Keywords: center-focus problem, Abel equation, Liénard equation

Author's affiliations:

Gavrilov, Lubomir ¹

¹ Institut de Mathématiques de Toulouse UMR5219 Université de Toulouse CNRS UPS IMT 31062 Toulouse Cedex 9 (France)

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Gavrilov, Lubomir. On the center-focus problem for the equation $\protect \frac{\protect \mathrm{d}y}{\protect \mathrm{d}x} + \Sigma _{i=1}^{n} a_i(x) y^i = 0, 0\le x \le 1$ where $a_i$ are polynomials. Annales Henri Lebesgue, Volume 3 (2020), pp. 615-648. doi : 10.5802/ahl.41. http://archive.numdam.org/articles/10.5802/ahl.41/

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