# American Institute of Mathematical Sciences

November  2014, 34(11): 4807-4826. doi: 10.3934/dcds.2014.34.4807

## 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

Received  December 2013 Revised  February 2014 Published  May 2014

This paper is concerned with the bifurcation of limit cycles from a quadratic reversible Lotka-Volterra system with two centers of genus one under small quadratic perturbations. It shows that the cyclicities of each period annulus and two period annuli of the considered system under small quadratic perturbations are two, respectively. This not only gives at least partially a positive answer to an open conjecture, but also improves the corresponding results in the literature. In addition, we present the configurations of limit cycles of the perturbed system as (2, 0), (1, 1), (1, 0), (0, 2), (0, 1) and (0, 0), where $(i,\, j)$ indicates that the perturbed system has $i$ limit cycles surrounding the positive singularity while it has $j$ limit cycles surrounding the negative one.
Citation: Linping Peng, Zhaosheng Feng, Changjian Liu. Quadratic perturbations of a quadratic reversible Lotka-Volterra system with two centers. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4807-4826. doi: 10.3934/dcds.2014.34.4807
##### 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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##### 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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