On the center-focus problem for the equation dy dx+Σ i=1 n a i (x)y i =0,0x1 where a i are polynomials
Annales Henri Lebesgue, Volume 3 (2020), pp. 615-648.

Metadata

Keywords center-focus problem, Abel equation, Liénard equation

Abstract

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.


References

[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 yy =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 P(n), n3, 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) (Proceedings of Symposia in Pure Mathematics) Volume 46 (1987) no. 2, pp. 247-282 | DOI | MR

[Har95] Harris, Joe Algebraic geometry. A first course, Graduate Texts in Mathematics, Volume 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 dx/dt= j=0 n a j (t)x j , 0t1, 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, Volume 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