# American Institute of Mathematical Sciences

August  2011, 30(3): 873-890. doi: 10.3934/dcds.2011.30.873

## The cyclicity of the period annulus of a quadratic reversible system with a hemicycle

 1 School of Mathematics and System Sciences, Beijing University of Aeronautics and Astronautics, LIMB of the Ministry of education, Beijing, 100191, China, China

Received  January 2010 Revised  January 2011 Published  March 2011

The cyclicity of the period annulus of a quadratic reversible and non-Hamiltonian system under quadratic perturbations is studied. The centroid curve method and other mathematical techniques are combined to prove that the related Abelian integral has at most two zeros. This gives a proof of Conjecture 1 in [8] for one case.
Citation: Linping Peng, Yazhi Lei. The cyclicity of the period annulus of a quadratic reversible system with a hemicycle. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 873-890. doi: 10.3934/dcds.2011.30.873
##### References:
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##### References:
 [1] C. Chicone and M. Jacobs, Bifurcation of limit cycles from quadratic isochrones,, J. Differential Equations, 91 (1991), 268. doi: 10.1016/0022-0396(91)90142-V. Google Scholar [2] S. N. Chow, C. Li and Y. Yi, The cyclicity of period annulus of degenerate quadratic Hamiltonian system with elliptic segment loops,, Ergodic Theory Dynam. Systems, 22 (2002), 349. doi: 10.1017/S0143385702000184. Google Scholar [3] F. Chen, C. Li, J. Llibre and Z. H. Zhang, A unified proof on the weak Hilbert 16th problem for $n=2$,, J. Differential Equations, 221 (2006), 309. doi: 10.1016/j.jde.2005.01.009. Google Scholar [4] G. Chen, C. Li, C. Liu and J. Llibre, The cyclicity of period annuli of some classes of reversible quadratic systems,, Discrete Contin. Dyn. Syst., 16 (2006), 157. doi: 10.3934/dcds.2006.16.157. Google Scholar [5] B. Coll, C. Li and R. Prohens, Quadratic perturbations of a class of quadratic reversible systems with two centers,, Discrete Contin. Dyn. Syst., 24 (2009), 699. doi: 10.3934/dcds.2009.24.699. Google Scholar [6] F. Dumortier, C. Li and Z. Zhang, Unfolding of a quadratic integrable system with two centers and two unbounded heteroclinic loops,, J. Differential Equations, 139 (1997), 146. doi: 10.1006/jdeq.1997.3285. Google Scholar [7] L. Gavrilov, The infinitesimal 16th Hilbert problem in the quadratic case,, Invent. Math., 143 (2001), 449. doi: 10.1007/PL00005798. Google Scholar [8] S. Gautier, L. Gavrilov and I. D. Iliev, Perturbations of quadratic center of genus one,, Discrete Contin. Dyn. Syst., 25 (2009), 511. doi: 10.3934/dcds.2009.25.511. Google Scholar [9] E. Horozov and I. D. Iliev, On the number of limit cycles in perturbations of quadratic Hamiltonian system,, Proc. London Math. Soc., 69 (1994), 198. doi: 10.1112/plms/s3-69.1.198. Google Scholar [10] I. D. Iliev, Perturbations of quadratic centers,, Bull. Sci. Math., 122 (1998), 107. doi: 10.1016/S0007-4497(98)80080-8. Google Scholar [11] I. D. Iliev, C. Li and J. Yu, Bifurcation of limit cycles from quadratic non-Hamiltonian systems with two centers and two heteroclinic loops,, Nonlinearity, 18 (2005), 305. doi: 10.1088/0951-7715/18/1/016. Google Scholar [12] C. Li and Z. Zhang, A criterion for determing the monotonicity of ratio of two Ablian integrals,, J. Differential Equations, 124 (1996), 407. doi: 10.1006/jdeq.1996.0017. Google Scholar [13] C. Li and Z. H. Zhang, Remarks on weak 16th problem for $n=2$,, Nonlinearity, 15 (2002), 1975. doi: 10.1088/0951-7715/15/6/310. Google Scholar [14] L. Peng, Unfolding of a quadratic integrable system with a homoclinic loop,, Acta Math. Sin.(Engl. ser.), 18 (2002), 737. doi: 10.1007/s10114-002-0196-4. Google Scholar [15] G. Swirszcz, Cyclicity of infinite contour around certain reversible quadratic center,, J. Differential Equations, 265 (1999), 239. Google Scholar [16] J. Yu and C. Li, Bifurcation of a class of planar non-Hamiltonian integrable systems with one center and one homoclinic loop,, J. Math. Anal. Appl., 269 (2002), 227. doi: 10.1016/S0022-247X(02)00018-5. Google Scholar [17] H. Zoladék, Quadratic systems with center and their perturbations,, J. Differential Equations, 109 (1994), 223. doi: 10.1006/jdeq.1994.1049. Google Scholar [18] Z. Zhang, T. Ding et al, "Qualitative Theory of Differential Equations,", Scientific press, (1985). Google Scholar [19] Z. Zhang and C. Li, On the number of limit cycles of a class of quadratic Hamiltonian systems under quadratic perturbations,, Adv. in Math. (China), 26 (1997), 445. Google Scholar

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