Quadratic perturbations of quadratic codimension-four centers |
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Authors: | Lubomir Gavrilov Iliya D Iliev |
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Institution: | a Institut de Mathématiques de Toulouse, UMR 5219, Université Paul Sabatier (Toulouse III), 31062 Toulouse, Cedex 9, France b Institute of Mathematics, Bulgarian Academy of Sciences, Bl. 8, 1113 Sofia, Bulgaria |
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Abstract: | We study the stratum in the set of all quadratic differential systems , with a center, known as the codimension-four case Q4. It has a center and a node and a rational first integral. The limit cycles under small quadratic perturbations in the system are determined by the zeros of the first Poincaré-Pontryagin-Melnikov integral I. We show that the orbits of the unperturbed system are elliptic curves, and I is a complete elliptic integral. Then using Picard-Fuchs equations and the Petrov's method (based on the argument principle), we set an upper bound of eight for the number of limit cycles produced from the period annulus around the center. |
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Keywords: | Quadratic codimension-four centers Limit cycles Zeros of Abelian integrals |
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