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Quadratic centers defining elliptic surfaces. (English) Zbl 1171.34034

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.

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
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References:

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