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Quadratic perturbations of a quadratic reversible Lotka-Volterra system with two centers
1. | School of Mathematics and System Sciences, Beijing University of Aeronautics and Astronautics, LIMB of the Ministry of education, Beijing, 100191 |
2. | Department of Mathematics, University of Texas-Pan American, Edinburg, TX 78539 |
3. | School of Mathematics, Soochow University, Suzhou, Jiangsu 215006, China |
References:
[1] |
C. Chicone and M. Jacobs, Bifurcation of limit cycles from quadratic isochronous, J. Differential Equations, 91 (1991), 268-326.
doi: 10.1016/0022-0396(91)90142-V. |
[2] |
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. |
[3] |
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. |
[4] |
F. Dumortier, R. Roussarie and C. Rousseau, Hilbert's 16th problem for quadratic vector fields, J. Differential Equations, 110 (1994), 86-133.
doi: 10.1006/jdeq.1994.1061. |
[5] |
J. P. Francoise, Successive derivatives of a first return map, application to the study of quadratic vector fields, Ergodic Theory Dynam. Syst., 16 (1996), 87-96.
doi: 10.1017/S0143385700008725. |
[6] |
L. Gavrilov, The infinitesimal 16th Hilbert problem in the quadratic case, Invent. Math., 143 (2001), 449-497.
doi: 10.1007/PL00005798. |
[7] |
L. Gavrilov and I. D. Iliev, Quadratic perturbations of quadratic codimension-four centers, J. Differential Equations, 357 (2009), 69-76.
doi: 10.1016/j.jmaa.2009.04.004. |
[8] |
L. Gavrilov and I. D. Iliev, Bifurcations of limit cycles from infinity in quadratic systems, Canad. J. Math., 54 (2002), 1038-1064.
doi: 10.4153/CJM-2002-038-6. |
[9] |
S. Gautier, L. Gavrilov and I. D. Iliev, Perturbations of quadratic center of genus one, Disc. Contin. Dyn. Sys., 25 (2009), 511-535.
doi: 10.3934/dcds.2009.25.511. |
[10] |
M. Grau, F. Manosas and J. Villadelprat, A chebyshev criterion for Abelian integral, Trans. Amer. Math. Soc., 363 (2011), 109-129.
doi: 10.1090/S0002-9947-2010-05007-X. |
[11] |
D. Hilbert, Mathematical problems, Bull. Amer. Math. Soc., 8 (1902), 437-479.
doi: 10.1090/S0002-9904-1902-00923-3. |
[12] |
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. |
[13] |
I. D. Iliev, The cyclicity of the period annulus of the quadratic Hamiltonian thiangle, J. Differential Equations, 128 (1996), 309-326.
doi: 10.1006/jdeq.1996.0097. |
[14] |
I. D. Iliev, Perturbations of quadratic centers, Bull. Sci. Math., 122 (1998), 107-161.
doi: 10.1016/S0007-4497(98)80080-8. |
[15] |
Y. Ilyashenko and J. Llibre, A restricted version of Hilbert's 16th problem for quadratic vector fields, Mosc. Math. J., 10 (2010), 317-335. |
[16] |
C. Li and J. Llibre, The cyclicity of period annulus of a quadratic reversible Lotka-Voterra system, Nonlinearity, 22 (2009), 2971-2979.
doi: 10.1088/0951-7715/22/12/009. |
[17] |
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. |
[18] |
I. Petrovskii and E. Landis, On the number of limit cycles of the equation $dy/dx= \frac{P(x,y)}{Q(x,y)}$, where $P$ and $Q$ are polynomials of degree 2, Matem. Sb., 37 (1955), 209-250 (in Russian). |
[19] |
D. Schlomiuk, Algebric particular integrals, integrability and the problem of the center, Trans. Amer. Math, Soc., 338 (1993), 799-841.
doi: 10.1090/S0002-9947-1993-1106193-6. |
[20] |
Y. Shao and Y. Zhao, The cyclicity and period annulus of a class of quadratic reversible Lotka-Volterra system of genus one, J. Math. Anal. Appl., 377 (2011), 817-827.
doi: 10.1016/j.jmaa.2010.11.048. |
[21] |
H. Zoladek, Quadratic systems with center and perturbations, J. Differential Equations, 109 (1994), 223-273.
doi: 10.1006/jdeq.1994.1049. |
show all references
References:
[1] |
C. Chicone and M. Jacobs, Bifurcation of limit cycles from quadratic isochronous, J. Differential Equations, 91 (1991), 268-326.
doi: 10.1016/0022-0396(91)90142-V. |
[2] |
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. |
[3] |
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. |
[4] |
F. Dumortier, R. Roussarie and C. Rousseau, Hilbert's 16th problem for quadratic vector fields, J. Differential Equations, 110 (1994), 86-133.
doi: 10.1006/jdeq.1994.1061. |
[5] |
J. P. Francoise, Successive derivatives of a first return map, application to the study of quadratic vector fields, Ergodic Theory Dynam. Syst., 16 (1996), 87-96.
doi: 10.1017/S0143385700008725. |
[6] |
L. Gavrilov, The infinitesimal 16th Hilbert problem in the quadratic case, Invent. Math., 143 (2001), 449-497.
doi: 10.1007/PL00005798. |
[7] |
L. Gavrilov and I. D. Iliev, Quadratic perturbations of quadratic codimension-four centers, J. Differential Equations, 357 (2009), 69-76.
doi: 10.1016/j.jmaa.2009.04.004. |
[8] |
L. Gavrilov and I. D. Iliev, Bifurcations of limit cycles from infinity in quadratic systems, Canad. J. Math., 54 (2002), 1038-1064.
doi: 10.4153/CJM-2002-038-6. |
[9] |
S. Gautier, L. Gavrilov and I. D. Iliev, Perturbations of quadratic center of genus one, Disc. Contin. Dyn. Sys., 25 (2009), 511-535.
doi: 10.3934/dcds.2009.25.511. |
[10] |
M. Grau, F. Manosas and J. Villadelprat, A chebyshev criterion for Abelian integral, Trans. Amer. Math. Soc., 363 (2011), 109-129.
doi: 10.1090/S0002-9947-2010-05007-X. |
[11] |
D. Hilbert, Mathematical problems, Bull. Amer. Math. Soc., 8 (1902), 437-479.
doi: 10.1090/S0002-9904-1902-00923-3. |
[12] |
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. |
[13] |
I. D. Iliev, The cyclicity of the period annulus of the quadratic Hamiltonian thiangle, J. Differential Equations, 128 (1996), 309-326.
doi: 10.1006/jdeq.1996.0097. |
[14] |
I. D. Iliev, Perturbations of quadratic centers, Bull. Sci. Math., 122 (1998), 107-161.
doi: 10.1016/S0007-4497(98)80080-8. |
[15] |
Y. Ilyashenko and J. Llibre, A restricted version of Hilbert's 16th problem for quadratic vector fields, Mosc. Math. J., 10 (2010), 317-335. |
[16] |
C. Li and J. Llibre, The cyclicity of period annulus of a quadratic reversible Lotka-Voterra system, Nonlinearity, 22 (2009), 2971-2979.
doi: 10.1088/0951-7715/22/12/009. |
[17] |
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. |
[18] |
I. Petrovskii and E. Landis, On the number of limit cycles of the equation $dy/dx= \frac{P(x,y)}{Q(x,y)}$, where $P$ and $Q$ are polynomials of degree 2, Matem. Sb., 37 (1955), 209-250 (in Russian). |
[19] |
D. Schlomiuk, Algebric particular integrals, integrability and the problem of the center, Trans. Amer. Math, Soc., 338 (1993), 799-841.
doi: 10.1090/S0002-9947-1993-1106193-6. |
[20] |
Y. Shao and Y. Zhao, The cyclicity and period annulus of a class of quadratic reversible Lotka-Volterra system of genus one, J. Math. Anal. Appl., 377 (2011), 817-827.
doi: 10.1016/j.jmaa.2010.11.048. |
[21] |
H. Zoladek, Quadratic systems with center and perturbations, J. Differential Equations, 109 (1994), 223-273.
doi: 10.1006/jdeq.1994.1049. |
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