Quadratic perturbations of a quadratic reversible center of genus one

Abstract

In this paper, we study a reversible and non-Hamitonian system with a period annulus bounded by a hemicycle in the Poincaré disk. It is proved that the cyclicity of the period annulus under quadratic perturbations is equal to two. This verifies some results of the conjecture given by Gautier et al.

Keywords

Quadratic reversible and non-Hamiltonian system / genus one / period annulus / limit cycle / cyclicity

References

[1] Chen F., Li C., Liu C., Llibre J. A unified proof on the weak Hilbert 16th problem for n = 2. J Differ Equations, 2006, 221, 309-342

[2] Chen G., Li C., Liu C., Llibre J. The cyclicity of period annuli of some classes of reversible quadratic systems. Discrete Cont Dyn S, 2006, 16, 157-177

[3] Chicone C., Jacobs M. Bifurcation of limit cycles from quadratic isochrones. J Differ Equations, 1991, 91, 268-326

[4] Chow S. -N., Li C., Yi Y. The cyclicity of period annulus of degenerate quadratic Hamiltonian system with elliptic segment loops. Ergod Theor Dyn Syst, 2002, 22, 349-374

[5] Coll B., Li C., Prohens R. Quadratic perturbations of a class of quadratic reversible systems with two centers. Discrete Cont Dyn S, 2009, 24, 699-729

[6] Dumortier F., Li C., Zhang Z. Unfolding of a quadratic integrable system with two centers and two unbounded heteroclinic loops. J Differ Equations, 1997, 139, 146-193

[7] Gautier S., Gavrilov L., Iliev I. -D. Perturbations of quadratic center of genus one. Discrete Cont Dyn S, 2009, 25, 511-535

[8] Gavrilov L. The infinitesimal 16th Hilbert problem in the quadratic case. Invent Math, 2001, 143, 449-497

[9] Horozov E., Iliev I. -D. On the number of limit cycles in perturbations of quadratic Hamiltonian system. Proc London Math Soc, 1994, 69, 198-224

[10] Iliev I. -D. Perturbations of quadratic centers. Bull Sci Math, 1998, 122, 107-161

[11] Iliev I. -D., Li C., Yu J. Bifurcation of limit cycles from quadratic non-Hamiltonian systems with two centers and two heteroclinic loops. Nonlinearity, 2005, 18 1 305-330

[12] Iliev I. -D., Li C., Yu J. Bifurcation of limit cycles in a reversible quadratic system with a center, a saddle and two nodes. Commun Pur Appl Anal, 2010, 9 3 583-610

[13] Li C., Zhang Z. A criterion for determing the monotonicity of ratio of two Ablian integrals. J Differ Equations, 1996, 124, 407-424

[14] Li C., Zhang Z. -H. Remarks on weak 16th problem for n = 2. Nonlinearity, 2002, 15, 1975-1992

[15] Liang H., Zhao Y. Quadratic perturbations of a class of quadratic reversible systems with one center. Discrete Cont Dyn S, 2010, 27 1 325-335

[16] Peng L. Unfolding of a quadratic integrable system with a homoclinic loop. Acta Math Sinica (English Ser), 2002, 29, 737-754

[17] Swirszcz G. Cyclicity of infinite contour around certain reversible quadratic center. J Differ Equations, 1999, 265, 239-266

[18] Yu J., Li C. Bifurcation of a class of planar non-Hamiltonian integrable systems with one center and one homoclinic loop. J Math Anal Appl, 2002, 269, 227-243

[19] Zhang Z., Ding T., Huang W., Dong Z. Qualitative Theory of Differential Equations, 1985, Beijing: Science Press.

[20] Zhang Z., Li C. On the number of limit cycles of a class of quadratic Hamiltonian systems under quadratic perturbations. Adv Math, 1997, 26, 445-460

[21] Zoladék H. Quadratic systems with center and their perturbations. J Differ Equations, 1994, 109, 223-273