# American Institute of Mathematical Sciences

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

## Perturbations of quadratic centers of genus one

 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.
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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