Gautier, Sébastien Quadratic centers defining elliptic surfaces. (English) Zbl 1171.34034 J. Differ. Equations 245, No. 12, 3545-3569 (2008). The purpose of the paper is to classify all the first integrals of planar quadratic differential systems (not necessarily having a center) which define elliptic curves. In the first part, a full classification, up to an affine equivalence, is given of all elliptic foliations, that is those having a center. There are 18 reversible cases, 5 reversible Lotka-Volterra cases and 6 generic Lotka-Volterra cases in the classification.In the second part, all degenerate cases (i.e. without a center) are determined and the topology of their singular fibers is investigated.The results from the first part would be useful to add several new classes of quadratic centers whose limit cycles bifurcating under small perturbations are tractable by the technique (also known as the Abelian integrals method) previously used for the elliptic Hamiltonian cases. Reviewer: Iliya Iliev (Sofia) Cited in 12 Documents MSC: 34C99 Qualitative theory for ordinary differential equations 37C15 Topological and differentiable equivalence, conjugacy, moduli, classification of dynamical systems 34C08 Ordinary differential equations and connections with real algebraic geometry (fewnomials, desingularization, zeros of abelian integrals, etc.) 14J27 Elliptic surfaces, elliptic or Calabi-Yau fibrations 34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations 34A05 Explicit solutions, first integrals of ordinary differential equations Keywords:quadratic centers; elliptic phase curves; elliptic surface; singular surface; Kodaira’s classification; birational mapping blowing up PDFBibTeX XMLCite \textit{S. Gautier}, J. Differ. Equations 245, No. 12, 3545--3569 (2008; Zbl 1171.34034) Full Text: DOI arXiv References: [1] Barth, W.; Peters, C.; Van de Ven, A., Compact Complex Surfaces, Ergeb. Math. Grenzgeb. (1984), Springer-Verlag · Zbl 0718.14023 [2] Chen, G.; Li, C.; Liu, C.; Llibre, J., The cyclicity of period annuli of some classes of reversible quadratic systems, Discrete Contin. Dyn. Syst., 16, 1, 157-177 (2006) · Zbl 1119.34028 [3] S. Gautier, Feuilletages elliptiques quadratiques plans et leurs pertubations, PhD thesis, 7/12/2007; S. Gautier, Feuilletages elliptiques quadratiques plans et leurs pertubations, PhD thesis, 7/12/2007 [4] Gautier, S.; Gavrilov, L.; Iliev, I. D., Pertubations of quadratic centers of genus one · Zbl 1178.34037 [5] Gavrilov, L., The infinitesimal 16th Hilbert problem in the quadratic case, Invent. Math., 143, 449-497 (2001) · Zbl 0979.34024 [6] Griffiths, P.; Harris, J., Principles of Algebraic Geometry (1978), John Wiley and Sons: John Wiley and Sons New York · Zbl 0408.14001 [7] Iliev, I. D., Pertubations of quadratic centers, Bull. Sci. Math., 22, 107-161 (1998) · Zbl 0920.34037 [8] Iliev, I. D.; Li, C.; Yu, J., Bifurcations of limit cycles from quadratic non-Hamiltonian systems with two centres and two unbounded heteroclinic loops, Nonlinearity, 18, 1, 305-330 (2005) · Zbl 1077.34035 [9] Ilyashenko, Y., Centennial history of Hilbert’s 16th problem, Bull. Amer. Math. Soc., 39, 3, 301-354 (2002) · Zbl 1004.34017 [10] Jouanolou, J. C., Equations de Pfaff algébriques, Lecture Notes in Math., vol. 708 (1979) · Zbl 0477.58002 [11] Kas, K., Weierstrass normal forms and invariants of elliptic surfaces, Trans. Amer. Math. Soc., 225 (1977) · Zbl 0402.14014 [12] Kodaira, K., On compact analytic surfaces, II, Ann. of Math., 77, 563-626 (1963) · Zbl 0118.15802 [13] R. Miranda, The basic theory of elliptic surfaces, Dottorato di Ricerca in Matematica (Doctorate in Mathematical Research) ETS Editrice, Pisa, 1989; R. Miranda, The basic theory of elliptic surfaces, Dottorato di Ricerca in Matematica (Doctorate in Mathematical Research) ETS Editrice, Pisa, 1989 · Zbl 0744.14026 [14] Petrov, G. S., Nonoscillation of elliptic integrals, Funct. Anal. Appl., 3, 45-50 (1990) [15] Yu, J.; Li, C., Bifurcation of a class of planar non-Hamiltonian integrable systems with one center and one homoclinic loop, J. Math. Anal. Appl., 269, 1, 227-243 (2002) · Zbl 1019.34042 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.