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

Iliya D. Iliev; Chengzhi Li; Jiang Yu

doi:10.3934/cpaa.2010.9.583

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2010, Volume 9, Issue 3: 583-610. Doi: 10.3934/cpaa.2010.9.583

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

1.
Institute of Mathematics, Bulgarian Academy of Sciences, Bl. 8, 1113 Sofia
2.
LMAM and School of Mathematical Sciences, Peking University, Beijing 100871
3.
Department of Mathematics, Shanghai Jiaotong University, Shanghai 200030

Received: April 30, 2009

Revised: September 30, 2009

Published: April 2010

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Abstract

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.

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Mathematics Subject Classification: Primary: 34C07, 34C08; Secondary: 37G15.

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