|
|
|
Quadratic perturbations of a quadratic reversible center of genus one |
Linping PENG( ) |
| School of Mathematics and System Sciences, Beijing University of Aeronautics and Astronautics, LIMB of the Ministry of Education, Beijing 100191, China |
|
|
|
|
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
|
|
Corresponding Author(s):
PENG Linping,Email:penglp@buaa.edu.cn
|
|
Issue Date: 01 October 2011
|
|
| 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
doi: 10.1016/j.jde.2005.01.009
|
| 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
doi: 10.3934/dcds.2006.16.157
|
| 3 |
Chicone C, Jacobs M. Bifurcation of limit cycles from quadratic isochrones. J Differ Equations , 1991, 91: 268-326
doi: 10.1016/0022-0396(91)90142-V
|
| 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
doi: 10.1017/S0143385702000184
|
| 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
doi: 10.3934/dcds.2009.24.699
|
| 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
doi: 10.1006/jdeq.1997.3285
|
| 7 |
Gautier S, Gavrilov L, Iliev I-D. Perturbations of quadratic center of genus one. Discrete Cont Dyn S , 2009, 25: 511-535
doi: 10.3934/dcds.2009.25.511
|
| 8 |
Gavrilov L. The infinitesimal 16th Hilbert problem in the quadratic case. Invent Math , 2001, 143: 449-497
doi: 10.1007/PL00005798
|
| 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
doi: 10.1112/plms/s3-69.1.198
|
| 10 |
Iliev I-D. Perturbations of quadratic centers. Bull Sci Math , 1998, 122: 107-161
doi: 10.1016/S0007-4497(98)80080-8
|
| 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
doi: 10.1088/0951-7715/18/1/016
|
| 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
doi: 10.3934/cpaa.2010.9.583
|
| 13 |
Li C, Zhang Z. A criterion for determing the monotonicity of ratio of two Ablian integrals. J Differ Equations , 1996, 124: 407-424
doi: 10.1006/jdeq.1996.0017
|
| 14 |
Li C, Zhang Z-H. Remarks on weak 16th problem for n = 2. Nonlinearity , 2002, 15: 1975-1992
doi: 10.1088/0951-7715/15/6/310
|
| 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
doi: 10.3934/dcds.2010.27.325
|
| 16 |
Peng L. Unfolding of a quadratic integrable system with a homoclinic loop. Acta Math Sinica (English Ser) , 2002, 29: 737-754
doi: 10.1007/s10114-002-0196-4
|
| 17 |
Swirszcz G. Cyclicity of infinite contour around certain reversible quadratic center. J Differ Equations , 1999, 265: 239-266
doi: 10.1006/jdeq.1998.3538
|
| 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
doi: 10.1016/S0022-247X(02)00018-5
|
| 19 |
Zhang Z, Ding T, Huang W, Dong Z. Qualitative Theory of Differential Equations. Beijing: Science Press, 1985 (in Chinese)
|
| 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
doi: 10.1006/jdeq.1994.1049
|
|
Viewed |
|
|
|
Full text
|
|
|
|
|
Abstract
|
|
|
|
|
Cited |
|
|
|
|
| |
Shared |
|
|
|
|
| |
Discussed |
|
|
|
|
