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:
[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.

[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.

[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.

[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.

[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.

[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.

[7]

L. Gavrilov, The infinitesimal 16th Hilbert problem in the quadratic case,, Invent. Math., 143 (2001), 449. doi: 10.1007/PL00005798.

[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.

[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.

[10]

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

[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.

[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.

[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.

[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.

[15]

G. Swirszcz, Cyclicity of infinite contour around certain reversible quadratic center,, J. Differential Equations, 265 (1999), 239.

[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.

[17]

H. Zoladék, Quadratic systems with center and their perturbations,, J. Differential Equations, 109 (1994), 223. doi: 10.1006/jdeq.1994.1049.

[18]

Z. Zhang, T. Ding et al, "Qualitative Theory of Differential Equations,", Scientific press, (1985).

[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.

show all references

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.

[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.

[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.

[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.

[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.

[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.

[7]

L. Gavrilov, The infinitesimal 16th Hilbert problem in the quadratic case,, Invent. Math., 143 (2001), 449. doi: 10.1007/PL00005798.

[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.

[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.

[10]

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

[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.

[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.

[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.

[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.

[15]

G. Swirszcz, Cyclicity of infinite contour around certain reversible quadratic center,, J. Differential Equations, 265 (1999), 239.

[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.

[17]

H. Zoladék, Quadratic systems with center and their perturbations,, J. Differential Equations, 109 (1994), 223. doi: 10.1006/jdeq.1994.1049.

[18]

Z. Zhang, T. Ding et al, "Qualitative Theory of Differential Equations,", Scientific press, (1985).

[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.

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