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May  2012, 11(3): 1269-1283. doi: 10.3934/cpaa.2012.11.1269

## The cyclicity of the period annulus of a class of quadratic reversible system

 1 Department of Mathematics, Sun Yat-sen University, Guangzhou, 510275, China 2 Department of Mathematics, Sun Yat-Sen University, Guangzhou, 510275

Received  December 2010 Revised  March 2011 Published  December 2011

In this paper, we study the bifurcation of limit cycles of a class of planar quadratic reversible system $\dot{x}=y+4x^2$, $\dot{y}=-x+2xy$ under quadratic perturbations. It is proved that the cyclicity of the period annulus is equal to two.
Citation: Yi Shao, Yulin Zhao. The cyclicity of the period annulus of a class of quadratic reversible system. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1269-1283. doi: 10.3934/cpaa.2012.11.1269
##### References:
 [1] J. C. Artés, J. Llibre and D. Schlomiuk, The geometry of quadratic differential systems with a weak focus of second order,, Inter. J. Bifur. & Chaos, 16 (2006), 3127. doi: 10.1142/s9218127406016720. Google Scholar [2] F. Chen, C. Li, J. Llibre and Z. Zhang, A unified proof on the weak Hilbert 16th problem for n = 2,, J. Differential Equations, 221 (2006), 309. doi: 10.1007/978-3-7643-8410-4_14. Google Scholar [3] G. Chen, C. Li, C. Liu and J. Llibre, The cyclicity of the period annulus of some classes of reversible quadratic systems,, Discrete Contin. Dyn. Syst.-A, 16 (2006), 157. doi: 10.3934/dcds.2006.16.157. Google Scholar [4] L. Gavrilov, The infinitesimal 16th Hilbert problem in the quadratic case,, Invent. Math., 143 (2001), 449. doi: 10.1007/s002220000112. Google Scholar [5] M. Grau, F. Mañosas and J. Villadelprat, A Chebyshev criterion for Abelian integrals,, Trans. Amer. Math. Soc., 363 (2011), 109. doi: 10.1090/S0002-9947-2010-05007-X. Google Scholar [6] S. Gautier, L. Gavrilov and I. D. Iliev, Perturbations of quadratic centers of genus one,, Discrete Contin. Dyn. Syst., 25 (2009), 511. doi: 10.3934/dcds.2009.25.511. Google Scholar [7] S. Gautier, Quadratic centers defining elliptic surface,, J. Differential Equations, 245 (2008), 3545. doi: 10.1016/j.jde.2008.06.033. Google Scholar [8] E. Horozov and I. D. Iliev, On the number of limit cycles in perturbations of quadratic Hamiltonian systems,, Proc. London Math. Soc., 69 (1994), 198. doi: 10.1112/plms/s3-69.1.198. Google Scholar [9] P. Hartman, "Ordinary Differential Equations,", $2^{nd}$ edition, (1982). doi: 10.2307/2283267. Google Scholar [10] C. Li and Z.-H. Zhang, Remarks on 16th weak Hilbert problem for $n = 2$,, Nonlinearity, 15 (2002), 1975. doi: 10.1088/0951-7715/15/6/310. Google Scholar [11] H. Liang and Y. Zhao, Quadratic perturbations of a class of quadratic reversible systems with one center,, Disc. Contin. Dyn. Syst., 27 (2010), 325. doi: 10.3934/dcds.2010.27.325. Google Scholar [12] H. Liang and Y. Zhao, On the period function of reversible quadratic centers with their orbits inside quartics,, Nonlinear Anal., 71 (2009), 5655. doi: 10.1016/j.na.2009.04.062. Google Scholar [13] I. D. Iliev, Perturbations of quadratic centers,, Bull. Sci. Math., 122 (1998), 107. doi: 10.1016/S0007-4497(98)80080-8. Google Scholar [14] I. D. Iliev, C. Li and J. Yu, Bifurcations of limit cycles from quadratic non-Hamiltonian systems with two centres and two unbounded heteroclinic loops,, Nonlinearity, 18 (2005), 305. doi: 10.1088/0951-7715/18/1/016. Google Scholar [15] I. D. Iliev, C. Li and J. Yu, Bifurcations of limit cycles in a reversible quadratic system with a centre, a saddle and two nodes,, Comm. Pure. Anal. Appl., 9 (2010), 583. doi: 10.3934/cpaa.2010.9.583. 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] Y. Zhao, On the momotonicity of the period function of a quadratic system,, Disc. Contin. Dyn. Syst., 13 (2005), 795. doi: 10.3934/dcds.2005.13.795. Google Scholar [18] Y. Zhao, Z. Liang and G. Lu, The cyclicity of the period annulus of the quadratic Hamiltonian systems with non-morsean point,, J. Differential Equations, 162 (2000), 199. doi: 10.1006/jdeq.1999.3704. Google Scholar

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##### References:
 [1] J. C. Artés, J. Llibre and D. Schlomiuk, The geometry of quadratic differential systems with a weak focus of second order,, Inter. J. Bifur. & Chaos, 16 (2006), 3127. doi: 10.1142/s9218127406016720. Google Scholar [2] F. Chen, C. Li, J. Llibre and Z. Zhang, A unified proof on the weak Hilbert 16th problem for n = 2,, J. Differential Equations, 221 (2006), 309. doi: 10.1007/978-3-7643-8410-4_14. Google Scholar [3] G. Chen, C. Li, C. Liu and J. Llibre, The cyclicity of the period annulus of some classes of reversible quadratic systems,, Discrete Contin. Dyn. Syst.-A, 16 (2006), 157. doi: 10.3934/dcds.2006.16.157. Google Scholar [4] L. Gavrilov, The infinitesimal 16th Hilbert problem in the quadratic case,, Invent. Math., 143 (2001), 449. doi: 10.1007/s002220000112. Google Scholar [5] M. Grau, F. Mañosas and J. Villadelprat, A Chebyshev criterion for Abelian integrals,, Trans. Amer. Math. Soc., 363 (2011), 109. doi: 10.1090/S0002-9947-2010-05007-X. Google Scholar [6] S. Gautier, L. Gavrilov and I. D. Iliev, Perturbations of quadratic centers of genus one,, Discrete Contin. Dyn. Syst., 25 (2009), 511. doi: 10.3934/dcds.2009.25.511. Google Scholar [7] S. Gautier, Quadratic centers defining elliptic surface,, J. Differential Equations, 245 (2008), 3545. doi: 10.1016/j.jde.2008.06.033. Google Scholar [8] E. Horozov and I. D. Iliev, On the number of limit cycles in perturbations of quadratic Hamiltonian systems,, Proc. London Math. Soc., 69 (1994), 198. doi: 10.1112/plms/s3-69.1.198. Google Scholar [9] P. Hartman, "Ordinary Differential Equations,", $2^{nd}$ edition, (1982). doi: 10.2307/2283267. Google Scholar [10] C. Li and Z.-H. Zhang, Remarks on 16th weak Hilbert problem for $n = 2$,, Nonlinearity, 15 (2002), 1975. doi: 10.1088/0951-7715/15/6/310. Google Scholar [11] H. Liang and Y. Zhao, Quadratic perturbations of a class of quadratic reversible systems with one center,, Disc. Contin. Dyn. Syst., 27 (2010), 325. doi: 10.3934/dcds.2010.27.325. Google Scholar [12] H. Liang and Y. Zhao, On the period function of reversible quadratic centers with their orbits inside quartics,, Nonlinear Anal., 71 (2009), 5655. doi: 10.1016/j.na.2009.04.062. Google Scholar [13] I. D. Iliev, Perturbations of quadratic centers,, Bull. Sci. Math., 122 (1998), 107. doi: 10.1016/S0007-4497(98)80080-8. Google Scholar [14] I. D. Iliev, C. Li and J. Yu, Bifurcations of limit cycles from quadratic non-Hamiltonian systems with two centres and two unbounded heteroclinic loops,, Nonlinearity, 18 (2005), 305. doi: 10.1088/0951-7715/18/1/016. Google Scholar [15] I. D. Iliev, C. Li and J. Yu, Bifurcations of limit cycles in a reversible quadratic system with a centre, a saddle and two nodes,, Comm. Pure. Anal. Appl., 9 (2010), 583. doi: 10.3934/cpaa.2010.9.583. 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] Y. Zhao, On the momotonicity of the period function of a quadratic system,, Disc. Contin. Dyn. Syst., 13 (2005), 795. doi: 10.3934/dcds.2005.13.795. Google Scholar [18] Y. Zhao, Z. Liang and G. Lu, The cyclicity of the period annulus of the quadratic Hamiltonian systems with non-morsean point,, J. Differential Equations, 162 (2000), 199. doi: 10.1006/jdeq.1999.3704. Google Scholar
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