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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

Yi Shao 1, and  Yulin Zhao 2,

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.
Keywords: period annulus, limit cycle., Quadratic center.
Mathematics Subject Classification: Primary: 34C07, 34C10; Secondary: 37G1.
Citation: Yi Shao, Yulin Zhao. The cyclicity of the period annulus of a class of quadratic reversible system. Communications on Pure and 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), 3127-3194. doi: 10.1142/s9218127406016720.

[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), 309-342. doi: 10.1007/978-3-7643-8410-4_14.

[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), 157-177. doi: 10.3934/dcds.2006.16.157.

[4]

L. Gavrilov, The infinitesimal 16th Hilbert problem in the quadratic case, Invent. Math., 143 (2001), 449-497. doi: 10.1007/s002220000112.

[5]

M. Grau, F. Mañosas and J. Villadelprat, A Chebyshev criterion for Abelian integrals, Trans. Amer. Math. Soc., 363 (2011), 109-129. doi: 10.1090/S0002-9947-2010-05007-X.

[6]

S. Gautier, L. Gavrilov and I. D. Iliev, Perturbations of quadratic centers of genus one, Discrete Contin. Dyn. Syst., 25 (2009), 511-535. doi: 10.3934/dcds.2009.25.511.

[7]

S. Gautier, Quadratic centers defining elliptic surface, J. Differential Equations, 245 (2008), 3545-3569. doi: 10.1016/j.jde.2008.06.033.

[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), 198-224. doi: 10.1112/plms/s3-69.1.198.

[9]

P. Hartman, "Ordinary Differential Equations," $2^{nd}$ edition, Birkhäuser, 1982. doi: 10.2307/2283267.

[10]

C. Li and Z.-H. Zhang, Remarks on 16th weak Hilbert problem for $n = 2$, Nonlinearity, 15 (2002), 1975-1992. doi: 10.1088/0951-7715/15/6/310.

[11]

H. Liang and Y. Zhao, Quadratic perturbations of a class of quadratic reversible systems with one center, Disc. Contin. Dyn. Syst., 27 (2010), 325-335. doi: 10.3934/dcds.2010.27.325.

[12]

H. Liang and Y. Zhao, On the period function of reversible quadratic centers with their orbits inside quartics, Nonlinear Anal., 71 (2009), 5655-5671. doi: 10.1016/j.na.2009.04.062.

[13]

I. D. Iliev, Perturbations of quadratic centers, Bull. Sci. Math., 122 (1998), 107-161. doi: 10.1016/S0007-4497(98)80080-8.

[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), 305-330. doi: 10.1088/0951-7715/18/1/016.

[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), 583-610. doi: 10.3934/cpaa.2010.9.583.

[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), 227-243. doi: 10.1016/S0022-247X(02)00018-5.

[17]

Y. Zhao, On the momotonicity of the period function of a quadratic system, Disc. Contin. Dyn. Syst., 13 (2005), 795-810. doi: 10.3934/dcds.2005.13.795.

[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), 199-223. doi: 10.1006/jdeq.1999.3704.

show all references

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), 3127-3194. doi: 10.1142/s9218127406016720.

[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), 309-342. doi: 10.1007/978-3-7643-8410-4_14.

[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), 157-177. doi: 10.3934/dcds.2006.16.157.

[4]

L. Gavrilov, The infinitesimal 16th Hilbert problem in the quadratic case, Invent. Math., 143 (2001), 449-497. doi: 10.1007/s002220000112.

[5]

M. Grau, F. Mañosas and J. Villadelprat, A Chebyshev criterion for Abelian integrals, Trans. Amer. Math. Soc., 363 (2011), 109-129. doi: 10.1090/S0002-9947-2010-05007-X.

[6]

S. Gautier, L. Gavrilov and I. D. Iliev, Perturbations of quadratic centers of genus one, Discrete Contin. Dyn. Syst., 25 (2009), 511-535. doi: 10.3934/dcds.2009.25.511.

[7]

S. Gautier, Quadratic centers defining elliptic surface, J. Differential Equations, 245 (2008), 3545-3569. doi: 10.1016/j.jde.2008.06.033.

[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), 198-224. doi: 10.1112/plms/s3-69.1.198.

[9]

P. Hartman, "Ordinary Differential Equations," $2^{nd}$ edition, Birkhäuser, 1982. doi: 10.2307/2283267.

[10]

C. Li and Z.-H. Zhang, Remarks on 16th weak Hilbert problem for $n = 2$, Nonlinearity, 15 (2002), 1975-1992. doi: 10.1088/0951-7715/15/6/310.

[11]

H. Liang and Y. Zhao, Quadratic perturbations of a class of quadratic reversible systems with one center, Disc. Contin. Dyn. Syst., 27 (2010), 325-335. doi: 10.3934/dcds.2010.27.325.

[12]

H. Liang and Y. Zhao, On the period function of reversible quadratic centers with their orbits inside quartics, Nonlinear Anal., 71 (2009), 5655-5671. doi: 10.1016/j.na.2009.04.062.

[13]

I. D. Iliev, Perturbations of quadratic centers, Bull. Sci. Math., 122 (1998), 107-161. doi: 10.1016/S0007-4497(98)80080-8.

[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), 305-330. doi: 10.1088/0951-7715/18/1/016.

[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), 583-610. doi: 10.3934/cpaa.2010.9.583.

[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), 227-243. doi: 10.1016/S0022-247X(02)00018-5.

[17]

Y. Zhao, On the momotonicity of the period function of a quadratic system, Disc. Contin. Dyn. Syst., 13 (2005), 795-810. doi: 10.3934/dcds.2005.13.795.

[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), 199-223. doi: 10.1006/jdeq.1999.3704.

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