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
[Sur le problème du centre-foyer pour l’équation

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

où les

a_{i}

sont des polynômes]

Gavrilov, Lubomir ¹

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

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

Résumé
Abstract

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.

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.

Reçu le : 2018-11-28
Révisé le : 2019-08-11
Accepté le : 2019-11-06
Publié le : 2020-07-10

DOI : 10.5802/ahl.41

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

Affiliations des auteurs :

Gavrilov, Lubomir ¹

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

@article{AHL_2020__3__615_0,
     author = {Gavrilov, Lubomir},
     title = {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},
     journal = {Annales Henri Lebesgue},
     pages = {615--648},
     publisher = {\'ENS Rennes},
     volume = {3},
     year = {2020},
     doi = {10.5802/ahl.41},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/ahl.41/}
}

TY  - JOUR
AU  - Gavrilov, Lubomir
TI  - 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
JO  - Annales Henri Lebesgue
PY  - 2020
SP  - 615
EP  - 648
VL  - 3
PB  - ÉNS Rennes
UR  - http://archive.numdam.org/articles/10.5802/ahl.41/
DO  - 10.5802/ahl.41
LA  - en
ID  - AHL_2020__3__615_0
ER  -

%0 Journal Article
%A Gavrilov, Lubomir
%T 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
%J Annales Henri Lebesgue
%D 2020
%P 615-648
%V 3
%I ÉNS Rennes
%U http://archive.numdam.org/articles/10.5802/ahl.41/
%R 10.5802/ahl.41
%G en
%F AHL_2020__3__615_0

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, Tome 3 (2020), pp. 615-648. doi : 10.5802/ahl.41. http://archive.numdam.org/articles/10.5802/ahl.41/

Bibliographie
Cité par

[BFY98] Briskin, Miriam; Françoise, Jean-Pierre; Yomdin, Yosef The Bautin ideal of the Abel equation, Nonlinearity, Volume 11 (1998) no. 3, pp. 431-443 | DOI | MR | Zbl

[BFY99] Briskin, Miriam; Françoise, Jean-Pierre; Yomdin, Yosef Center conditions, compositions of polynomials and moments on algebraic curves, Ergodic Theory Dyn. Syst., Volume 19 (1999) no. 5, pp. 1201-1220 | DOI | MR | Zbl

[BFY00] Briskin, Miriam; Françoise, Jean-Pierre; Yomdin, Yosef Center conditions. II. Parametric and model center problems, Isr. J. Math., Volume 118 (2000), pp. 61-82 | DOI | MR | Zbl

[Bru06] Brudnyi, Alexander On the center problem for ordinary differential equations, Am. J. Math., Volume 128 (2006) no. 2, pp. 419-451 | DOI | MR | Zbl

[BRY10] Briskin, Miriam; Roytvarf, Nina; Yomdin, Yosef Center conditions at infinity for Abel differential equations, Ann. Math., Volume 172 (2010) no. 1, pp. 437-483 | DOI | MR | Zbl

[Che72] Cherkas, Leonid A. On the conditions for the center for certain equations of the form $y y^{'} = P (x) + Q (x) y + R (x) y^{2}$ , Differ. Uravn, Volume 8 (1972), pp. 1435-1439 | MR

[Che77] Chen, Kuo-Tsai Iterated path integrals, Bull. Am. Math. Soc., Volume 83 (1977) no. 5, pp. 831-879 | DOI | MR | Zbl

[Chr99] Christopher, Colin An algebraic approach to the classification of centers in polynomial Liénard systems, J. Math. Anal. Appl., Volume 229 (1999) no. 1, pp. 319-329 | DOI | Zbl

[Chr00] Christopher, Colin Abel equations: composition conjectures and the model problem, Bull. Lond. Math. Soc., Volume 32 (2000) no. 3, pp. 332-338 | DOI | MR | Zbl

[CL07] Christopher, Colin; Li, Chengzhi Limit cycles of differential equations, Advanced Courses in Mathematics – CRM Barcelona, Birkhäuser, 2007 | Zbl

[CLN96] Cerveau, Dominique; Lins Neto, Alcides Irreducible components of the space of holomorphic foliations of degree two in $ℂ P (n)$ , $n \geq 3$ , Ann. Math., Volume 143 (1996) no. 3, pp. 577-612 | DOI | MR

[Dul08] Dulac, Henri Détermination et intégration d’une certaine classe d’équations différentielles ayant pour point singulier un centre, Bull. Sci. Math. II. Sér., Volume 32 (1908), pp. 230-252 | Zbl

[FGX16] Françoise, Jean-Pierre; Gavrilov, Lubomir; Xiao, Dongmei Hilbert’s 16th problem on a period annulus and Nash space of arcs (2016) (https://arxiv.org/abs/1610.07582, to appear in Math. Proc. Camb. Philos. Soc.)

[Fra96] Françoise, Jean-Pierre Successive derivatives of a first return map, application to the study of quadratic vector fields, Ergodic Theory Dyn. Syst., Volume 16 (1996) no. 1, pp. 87-96 | DOI | MR | Zbl

[Gav05] Gavrilov, Lubomir Higher order Poincaré-Pontryagin functions and iterated path integrals, Ann. Fac. Sci. Toulouse, Math., Volume 14 (2005) no. 4, pp. 663-682 | DOI | Numdam | MR | Zbl

[GGS19] Giné, Jaume; Grau, Maite; Santallusia, Xavier A counterexample to the composition condition conjecture for polynomial abel differential equations, Ergodic Theory Dyn. Syst., Volume 39 (2019) no. 12, pp. 3347-3352 | DOI | MR | Zbl

[GI09] Gavrilov, Lubomir; Iliev, Iliya D. Quadratic perturbations of quadratic codimension-four centers, J. Math. Anal. Appl., Volume 357 (2009) no. 1, pp. 69-76 | DOI | MR | Zbl

[GM07] Gavrilov, Lubomir; Movasati, Hossein The infinitesimal 16th Hilbert problem in dimension zero, Bull. Sci. Math., Volume 131 (2007) no. 3, pp. 242-257 | DOI | MR | Zbl

[GMN09] Gavrilov, Lubomir; Movasati, Hossein; Nakai, Isao On the non-persistence of Hamiltonian identity cycles, J. Differ. Equations, Volume 246 (2009) no. 7, pp. 2706-2723 | DOI | MR | Zbl

[Hai87] Hain, Richard M. The geometry of the mixed Hodge structure on the fundamental group, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985) (Bloch, Spencer J., ed.) (Proceedings of Symposia in Pure Mathematics), Volume 46, American Mathematical Society (1987) no. 2, pp. 247-282 | DOI | MR

[Har95] Harris, Joe Algebraic geometry. A first course, Graduate Texts in Mathematics, 133, Springer, 1995 (corrected reprint of the 1992 original) | Zbl

[Ili98] Iliev, Iliya D. On second order bifurcations of limit cycles, J. Lond. Math. Soc., Volume 58 (1998) no. 2, pp. 353-366 | DOI | MR

[LN80] Lins Neto, Alcides On the number of solutions of the equation $d x / d t = \sum_{j = 0}^{n} a_{j} (t) x^{j}$ , $0 \leq t \leq 1$ , for which $x (0) = x (1)$ , Invent. Math., Volume 59 (1980) no. 1, pp. 67-76 | MR | Zbl

[LN14] Lins Neto, Alcides Foliations with a Morse center, J. Singul., Volume 9 (2014), pp. 82-100 | MR | Zbl

[Mov04] Movasati, Hossein Center conditions: rigidity of logarithmic differential equations, J. Differ. Equations, Volume 197 (2004) no. 1, pp. 197-217 | DOI | MR | Zbl

[PM09] Pakovich, Fedor; Muzychuk, Mikhael Solution of the polynomial moment problem, Proc. Lond. Math. Soc., Volume 99 (2009) no. 3, pp. 633-657 | DOI | MR

[Rou98] Roussarie, Robert Bifurcation of planar vector fields and Hilbert’s sixteenth problem, Progress in Mathematics, 164, Birkhäuser, 1998 | MR | Zbl

[Yom03] Yomdin, Yosef The center problem for the Abel equation, compositions of functions, and moment conditions, Mosc. Math. J., Volume 3 (2003) no. 3, pp. 1167-1195 (dedicated to Vladimir Igorevich Arnold on the occasion of his 65th birthday) | DOI | MR | Zbl

[Zar19] Zare, Yadollah Center conditions: Pull-back of differential equations, Trans. Am. Math. Soc., Volume 372 (2019) no. 5, pp. 3167-3189 | DOI | MR | Zbl

[Żoł94] Żoładek, Henryk Quadratic systems with center and their perturbations, J. Differ. Equations, Volume 109 (1994) no. 2, pp. 223-273 | DOI | MR | Zbl

Cité par Sources :