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Entropy by unit length for the Ginzburg-Landau equation on the line. A Hilbert space framework
The cyclicity of the period annulus of a class of quadratic reversible system
1. | Department of Mathematics, Sun Yat-sen University, Guangzhou, 510275, China |
2. | Department of Mathematics, Sun Yat-Sen University, Guangzhou, 510275 |
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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