Advanced Search
Article Contents
Article Contents

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

Abstract Related Papers Cited by
  • 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.
    Mathematics Subject Classification: Primary: 34C07, 34C08; Secondary: 37G15.


    \begin{equation} \\ \end{equation}
  • [1]

    C. Chicone and M. Jacobs, Bifurcation of limit cycles from quadratic isochrones, J. Differential Equations, 91 (1991), 268-326.doi: 10.1016/0022-0396(91)90142-V.


    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), 349-374.doi: 10.1017/S0143385702000184.


    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), 309-342.doi: 10.1016/j.jde.2005.01.009.


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


    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), 699-729.doi: 10.3934/dcds.2009.24.699.


    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), 146-193.doi: 10.1006/jdeq.1997.3285.


    L. Gavrilov, The infinitesimal 16th Hilbert problem in the quadratic case, Invent. Math., 143 (2001), 449-497.doi: 10.1007/PL00005798.


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


    E. Horozov and I. D. Iliev, On the number of limit cycles in perturbations of quadratic Hamiltonian system, Proc. London Math. Soc., 69 (1994), 198-224.doi: 10.1112/plms/s3-69.1.198.


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


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


    C. Li and Z. Zhang, A criterion for determing the monotonicity of ratio of two Ablian integrals, J. Differential Equations, 124 (1996), 407-424.doi: 10.1006/jdeq.1996.0017.


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


    L. Peng, Unfolding of a quadratic integrable system with a homoclinic loop, Acta Math. Sin.(Engl. ser.), 18 (2002), 737-754.doi: 10.1007/s10114-002-0196-4.


    G. Swirszcz, Cyclicity of infinite contour around certain reversible quadratic center, J. Differential Equations, 265 (1999), 239-266.


    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.


    H. Zoladék, Quadratic systems with center and their perturbations, J. Differential Equations, 109 (1994), 223-273.doi: 10.1006/jdeq.1994.1049.


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


    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), 445-460.

  • 加载中

Article Metrics

HTML views() PDF downloads(68) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint