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July  2009, 25(2): 511-535. doi: 10.3934/dcds.2009.25.511

Perturbations of quadratic centers of genus one

Sébastien Gautier 1, , Lubomir Gavrilov 1, and  Iliya D. Iliev 2,

1. 

Université de Toulouse, 31062 Toulouse cedex 9, France, France
2. 

Institute of Mathematics, Bulgarian Academy of Sciences, Bl. 8, 1113 Sofia, Bulgaria

Received  June 2008 Revised  December 2008 Published  June 2009

We propose a program for finding the cyclicity of period annuli of quadratic systems with centers of genus one. As a first step, we classify all such systems and determine the essential one-parameter quadratic perturbations which produce the maximal number of limit cycles. We compute the associated Poincaré-Pontryagin-Melnikov functions whose zeros control the number of limit cycles. To illustrate our approach, we determine the cyclicity of the annuli of two particular reversible systems.
Keywords: Limit cycles, Small perturbations, Quadratic centers, Genus-one curves., Elliptic integrals.
Mathematics Subject Classification: Primary: 34C07, 34C08; Secondary: 37G1.
Citation: Sébastien Gautier, Lubomir Gavrilov, Iliya D. Iliev. Perturbations of quadratic centers of genus one. Discrete and Continuous Dynamical Systems, 2009, 25 (2) : 511-535. doi: 10.3934/dcds.2009.25.511
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