American Institute of Mathematical Sciences

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May  2010, 9(3): 583-610. doi: 10.3934/cpaa.2010.9.583

Bifurcations of limit cycles in a reversible quadratic system with a center, a saddle and two nodes

Iliya D. Iliev 1, , Chengzhi Li 2, and  Jiang Yu 3,

1. 

Institute of Mathematics, Bulgarian Academy of Sciences, Bl. 8, 1113 Sofia, Bulgaria
2. 

LMAM and School of Mathematical Sciences, Peking University, Beijing 100871, China
3. 

Department of Mathematics, Shanghai Jiaotong University, Shanghai 200030, China

Received  May 2009 Revised  October 2009 Published  January 2010

We study the bifurcations of limit cycles in a class of planar reversible quadratic systems whose critical points are a center, a saddle and two nodes, under small quadratic perturbations. By using the properties of related complete elliptic integrals and the geometry of some planar curves defined by them, we prove that at most two limit cycles bifurcate from the period annulus around the center. This bound is exact.
Keywords: period annulus, Bifurcation of limit cycles, elliptic phase curves., small perturbations, reversible quadratic system.
Mathematics Subject Classification: Primary: 34C07, 34C08; Secondary: 37G1.
Citation: Iliya D. Iliev, Chengzhi Li, Jiang Yu. Bifurcations of limit cycles in a reversible quadratic system with a center, a saddle and two nodes. Communications on Pure & Applied Analysis, 2010, 9 (3) : 583-610. doi: 10.3934/cpaa.2010.9.583
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