doi: 10.3934/dcdsb.2021299
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The cyclicity of a class of quadratic reversible centers defining elliptic curves

School of Mathematics (Zhuhai), Sun Yat-sen University, Zhuhai, 519082, China

*Corresponding author: Changjian Liu

Received  August 2021 Revised  November 2021 Early access December 2021

Fund Project: The work was supported by the NSF of China (No.11771315)

In this paper, the cyclicity of period annulus of an one-parameter family quadratic reversible system under quadratic perturbations is studied which is equivalent to the number of zeros of any nontrivial linear combination of three Abelian integrals. By the criteria established in [28] and the asymptotic expansions of Abelian integrals, we obtain that the cyclicity is two when the parameter in $ (-\infty,-2)\cup[-\frac{8}{5},+\infty) $. Moreover, we develop new criteria which combined with the asymptotic expansions of Abelian integrals show that the cyclicity is three when the parameter belongs to $ (-2,-\frac{8}{5}) $.

Citation: Guilin Ji, Changjian Liu. The cyclicity of a class of quadratic reversible centers defining elliptic curves. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021299
References:
[1] A.A. AndronovA.A. Vitt and S.E. Khaikin, Theory of Oscillators, Pergamon Press, Oxford, New York, Toronto, 1966. 
[2]

G. ChenC. LiC. 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]

L. ChenX. MaG. Zhang and C. Li, Cyclicity of several quadratic reversible systems with center of genus one, J. Appl. Anal. Comput., 1 (2011), 439-447.  doi: 10.11948/2011030.

[4]

S. N. ChowC. Li and Y. Yi, The cyclicity of period annulus of degenerate quadratic Hamiltonian system with elliptic segment loop, Ergodic Theory Dynam. Systems, 22 (2002), 349-374.  doi: 10.1017/S0143385702000184.

[5]

B. CollC. Li and R. Prohens, Quadratic perturbations of a class quadratic reversible systems with two centers, Discrete Contin. Dyn. Syst., 24 (2009), 699-729.  doi: 10.3934/dcds.2009.24.699.

[6]

F. DumortierC. Li and Z. Zhang, Unfolding of a quadratic integrable system with two centers and two unbounded heteroclinic loops, J. Differ. Equations, 139 (1997), 146-193.  doi: 10.1006/jdeq.1997.3285.

[7]

J.-P. FrançoiseL. Gavrilov and D. Xiao, Hilbert's 16th problem on a period annulus and Nash space of arcs, Math. Proc. Camb. Philos. Soc., 169 (2020), 377-409.  doi: 10.1017/S0305004119000239.

[8]

S. GautierL. Gavrilov and I. D. Iliev, Perturbation of quadratic centers of genus one, Discrete Contin. Dyn. Syst., 25 (2009), 511-535.  doi: 10.3934/dcds.2009.25.511.

[9]

M. GrauF. Mañosas and J. Villadelprat, A Chebyshev criterion for abelian integrals, Trans. Am. Math. Soc., 363 (2011), 109-129.  doi: 10.1090/S0002-9947-2010-05007-X.

[10]

M. Han, Asymptotic expansions of Melnikov functions and limit cycle bifurcations, Int. J. Bifur. Chaos, 22 (2012), 1250296.  doi: 10.1142/S0218127412502963.

[11]

M. HanJ. YangA. Tarta and Y. Gao, Limit cycles near homoclinic and heteroclinic loops, J. Dynam. Differ. Equ., 20 (2008), 923-944.  doi: 10.1007/s10884-008-9108-3.

[12]

I. D. Iliev, The cyclicity of the period annulus of the quadratic Hamiltonian triangle, J. Differ. Equations, 128 (1996), 309-326.  doi: 10.1006/jdeq.1996.0097.

[13]

I. D. Iliev, High-order Melnikov functions for degenerate cubic Hamiltonians, Adv. Differential Equations, 1 (1996), 689-708. 

[14]

I. D. Iliev, Perturbations of quadratic centers, Bull. Sci. Math., 122 (1998), 107-161.  doi: 10.1016/S0007-4497(98)80080-8.

[15]

I. D. IlievC. Li and J. Yu, Bifurcation of limit cycles from quadratic non-Hamiltonian systems with two centers and two unbounded heteroclinic loops, Nonlinearity, 18 (2005), 305-330.  doi: 10.1088/0951-7715/18/1/016.

[16]

I. D. IlievC. Li and J. Yu, Bifurcation of limit cycles in a reversible quadratic system with a center, a saddle and two nodes, Commun. Pure Appl. Anal., 9 (2010), 583-610.  doi: 10.3934/cpaa.2010.9.583.

[17]

C. Li, Abelian integrals and limit cycles, Qual. Theory Dyn. Syst., 11 (2012), 111-128.  doi: 10.1007/s12346-011-0051-z.

[18]

C. Li and J. Llibre, A unified study on the cyclicity of period annulus of the reversible quadratic Hamiltonian systems, J. Dynam. Differ. Equations, 16 (2004), 271-295.  doi: 10.1007/s10884-004-2781-y.

[19]

C. Li and J. Llibre, The cyclicity of period annulus of a quadratic reversible Lotka-Volterra system, Nonlinearity, 22 (2009), 2971-2979.  doi: 10.1088/0951-7715/22/12/009.

[20]

C. Li and Z. Zhang, A criterion for determining the monotonocity of the ratio of two Abelian integrals, J. Differ. Equations, 124 (1996), 407-424.  doi: 10.1006/jdeq.1996.0017.

[21]

J. Li, Limit cycles bifurcated from a reversible quadratic center, Qual. Theory Dyn. Syst., 6 (2005), 205-215.  doi: 10.1007/BF02972673.

[22]

H. LiangK. Wu and Y. Zhao, Quadratic perturbations of a class of quadratic reversible center of genus one, Sci. China Math., 56 (2013), 577-596.  doi: 10.1007/s11425-012-4488-6.

[23]

H. Liang and Y. Zhao, Quadratic perturbations of a class of quadratic reversible systems with one center, Discrete Contin. Dyn. Syst., 27 (2010), 325-335.  doi: 10.3934/dcds.2010.27.325.

[24]

H. Liang and Y. Zhao, Limit cycles bifurcated from a class of quadratic reversible center of genus one, J. Math. Anal. Appl., 391 (2012), 240-254.  doi: 10.1016/j.jmaa.2012.02.014.

[25]

C. Liu, The cyclicity of period annuli of a class of quadratic reversible systems with two centers, J. Differ. Equations, 252 (2012), 5260-5273.  doi: 10.1016/j.jde.2012.02.005.

[26]

C. Liu, Limit cycles bifurcated from some reversible quadratic centres with a non-algebraic first integral, Nonlinearity, 25 (2012), 1653-1660.  doi: 10.1088/0951-7715/25/6/1653.

[27]

C. Liu and D. Xiao, The monotonicity of the ratio of two Abelian integrals, Trans. Amer. Math. Soc., 365 (2013), 5525-5544.  doi: 10.1090/S0002-9947-2013-05934-X.

[28]

C. Liu and D. Xiao, The smallest upper bound on the number of zeros of Abelian integrals, J. Differ. Equations, 269 (2020), 3816-3852.  doi: 10.1016/j.jde.2020.03.016.

[29]

F. Mañosas and J. Villadelprat, Bounding the number of zeros of certain Abelian integrals, J. Differ. Equations, 251 (2011), 1656-1669.  doi: 10.1016/j.jde.2011.05.026.

[30]

L. Peng, Unfolding of a quadratic integrable system with a homoclinic loop, Acta Math. Sin. (Engl. Ser.), 18 (2002), 737-754.  doi: 10.1007/s10114-002-0196-4.

[31]

Y. Shao and Y. Zhao, The cyclicity of a class of quadratic reversible system of genus one, Chaos Solitons Fractals, 44 (2011), 827-835.  doi: 10.1016/j.chaos.2011.06.015.

[32]

Y. Shao and Y. Zhao, The cyclicity of the period annulus of a class of quadratic reverdible system, Commun. Pure Appl. Anal., 11 (2012), 1269-1283.  doi: 10.3934/cpaa.2012.11.1269.

[33]

Y. Sun and C. Liu, The poincaré bifurcation of a SD oscillator, Discrete Contin. Dyn. Syst. B, 26 (2021), 1565-1577.  doi: 10.3934/dcdsb.2020173.

[34]

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.

[35]

Y. ZhaoZ. Liang and G. Lu, The cyclicity of period annulus of the quadratic Hamiltonian systems with non-Morsean point, J. Differ. Equations, 162 (2000), 199-223.  doi: 10.1006/jdeq.1999.3704.

[36]

Y. Zhao and H. Zhu, Bifurcation of limit cycles from a non-Hamiltonian quadratic integrable system with homoclinic loop, Infinite Dimensional Dynamical Systems, 64 (2013), 445-479.  doi: 10.1007/978-1-4614-4523-4.

[37]

Y. Zhao and S. Zhu, Perturbations of the non-generic quadratic Hamiltonian vector fields with hyperbolic segment, Bull. Sci. Math., 125 (2001), 109-138.  doi: 10.1016/S0007-4497(00)01069-1.

[38]

H. Żołądek, Quadratic systems with center and their perturbations, J. Differ. Equations, 109 (1994), 223-273.  doi: 10.1006/jdeq.1994.1049.

show all references

References:
[1] A.A. AndronovA.A. Vitt and S.E. Khaikin, Theory of Oscillators, Pergamon Press, Oxford, New York, Toronto, 1966. 
[2]

G. ChenC. LiC. 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]

L. ChenX. MaG. Zhang and C. Li, Cyclicity of several quadratic reversible systems with center of genus one, J. Appl. Anal. Comput., 1 (2011), 439-447.  doi: 10.11948/2011030.

[4]

S. N. ChowC. Li and Y. Yi, The cyclicity of period annulus of degenerate quadratic Hamiltonian system with elliptic segment loop, Ergodic Theory Dynam. Systems, 22 (2002), 349-374.  doi: 10.1017/S0143385702000184.

[5]

B. CollC. Li and R. Prohens, Quadratic perturbations of a class quadratic reversible systems with two centers, Discrete Contin. Dyn. Syst., 24 (2009), 699-729.  doi: 10.3934/dcds.2009.24.699.

[6]

F. DumortierC. Li and Z. Zhang, Unfolding of a quadratic integrable system with two centers and two unbounded heteroclinic loops, J. Differ. Equations, 139 (1997), 146-193.  doi: 10.1006/jdeq.1997.3285.

[7]

J.-P. FrançoiseL. Gavrilov and D. Xiao, Hilbert's 16th problem on a period annulus and Nash space of arcs, Math. Proc. Camb. Philos. Soc., 169 (2020), 377-409.  doi: 10.1017/S0305004119000239.

[8]

S. GautierL. Gavrilov and I. D. Iliev, Perturbation of quadratic centers of genus one, Discrete Contin. Dyn. Syst., 25 (2009), 511-535.  doi: 10.3934/dcds.2009.25.511.

[9]

M. GrauF. Mañosas and J. Villadelprat, A Chebyshev criterion for abelian integrals, Trans. Am. Math. Soc., 363 (2011), 109-129.  doi: 10.1090/S0002-9947-2010-05007-X.

[10]

M. Han, Asymptotic expansions of Melnikov functions and limit cycle bifurcations, Int. J. Bifur. Chaos, 22 (2012), 1250296.  doi: 10.1142/S0218127412502963.

[11]

M. HanJ. YangA. Tarta and Y. Gao, Limit cycles near homoclinic and heteroclinic loops, J. Dynam. Differ. Equ., 20 (2008), 923-944.  doi: 10.1007/s10884-008-9108-3.

[12]

I. D. Iliev, The cyclicity of the period annulus of the quadratic Hamiltonian triangle, J. Differ. Equations, 128 (1996), 309-326.  doi: 10.1006/jdeq.1996.0097.

[13]

I. D. Iliev, High-order Melnikov functions for degenerate cubic Hamiltonians, Adv. Differential Equations, 1 (1996), 689-708. 

[14]

I. D. Iliev, Perturbations of quadratic centers, Bull. Sci. Math., 122 (1998), 107-161.  doi: 10.1016/S0007-4497(98)80080-8.

[15]

I. D. IlievC. Li and J. Yu, Bifurcation of limit cycles from quadratic non-Hamiltonian systems with two centers and two unbounded heteroclinic loops, Nonlinearity, 18 (2005), 305-330.  doi: 10.1088/0951-7715/18/1/016.

[16]

I. D. IlievC. Li and J. Yu, Bifurcation of limit cycles in a reversible quadratic system with a center, a saddle and two nodes, Commun. Pure Appl. Anal., 9 (2010), 583-610.  doi: 10.3934/cpaa.2010.9.583.

[17]

C. Li, Abelian integrals and limit cycles, Qual. Theory Dyn. Syst., 11 (2012), 111-128.  doi: 10.1007/s12346-011-0051-z.

[18]

C. Li and J. Llibre, A unified study on the cyclicity of period annulus of the reversible quadratic Hamiltonian systems, J. Dynam. Differ. Equations, 16 (2004), 271-295.  doi: 10.1007/s10884-004-2781-y.

[19]

C. Li and J. Llibre, The cyclicity of period annulus of a quadratic reversible Lotka-Volterra system, Nonlinearity, 22 (2009), 2971-2979.  doi: 10.1088/0951-7715/22/12/009.

[20]

C. Li and Z. Zhang, A criterion for determining the monotonocity of the ratio of two Abelian integrals, J. Differ. Equations, 124 (1996), 407-424.  doi: 10.1006/jdeq.1996.0017.

[21]

J. Li, Limit cycles bifurcated from a reversible quadratic center, Qual. Theory Dyn. Syst., 6 (2005), 205-215.  doi: 10.1007/BF02972673.

[22]

H. LiangK. Wu and Y. Zhao, Quadratic perturbations of a class of quadratic reversible center of genus one, Sci. China Math., 56 (2013), 577-596.  doi: 10.1007/s11425-012-4488-6.

[23]

H. Liang and Y. Zhao, Quadratic perturbations of a class of quadratic reversible systems with one center, Discrete Contin. Dyn. Syst., 27 (2010), 325-335.  doi: 10.3934/dcds.2010.27.325.

[24]

H. Liang and Y. Zhao, Limit cycles bifurcated from a class of quadratic reversible center of genus one, J. Math. Anal. Appl., 391 (2012), 240-254.  doi: 10.1016/j.jmaa.2012.02.014.

[25]

C. Liu, The cyclicity of period annuli of a class of quadratic reversible systems with two centers, J. Differ. Equations, 252 (2012), 5260-5273.  doi: 10.1016/j.jde.2012.02.005.

[26]

C. Liu, Limit cycles bifurcated from some reversible quadratic centres with a non-algebraic first integral, Nonlinearity, 25 (2012), 1653-1660.  doi: 10.1088/0951-7715/25/6/1653.

[27]

C. Liu and D. Xiao, The monotonicity of the ratio of two Abelian integrals, Trans. Amer. Math. Soc., 365 (2013), 5525-5544.  doi: 10.1090/S0002-9947-2013-05934-X.

[28]

C. Liu and D. Xiao, The smallest upper bound on the number of zeros of Abelian integrals, J. Differ. Equations, 269 (2020), 3816-3852.  doi: 10.1016/j.jde.2020.03.016.

[29]

F. Mañosas and J. Villadelprat, Bounding the number of zeros of certain Abelian integrals, J. Differ. Equations, 251 (2011), 1656-1669.  doi: 10.1016/j.jde.2011.05.026.

[30]

L. Peng, Unfolding of a quadratic integrable system with a homoclinic loop, Acta Math. Sin. (Engl. Ser.), 18 (2002), 737-754.  doi: 10.1007/s10114-002-0196-4.

[31]

Y. Shao and Y. Zhao, The cyclicity of a class of quadratic reversible system of genus one, Chaos Solitons Fractals, 44 (2011), 827-835.  doi: 10.1016/j.chaos.2011.06.015.

[32]

Y. Shao and Y. Zhao, The cyclicity of the period annulus of a class of quadratic reverdible system, Commun. Pure Appl. Anal., 11 (2012), 1269-1283.  doi: 10.3934/cpaa.2012.11.1269.

[33]

Y. Sun and C. Liu, The poincaré bifurcation of a SD oscillator, Discrete Contin. Dyn. Syst. B, 26 (2021), 1565-1577.  doi: 10.3934/dcdsb.2020173.

[34]

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.

[35]

Y. ZhaoZ. Liang and G. Lu, The cyclicity of period annulus of the quadratic Hamiltonian systems with non-Morsean point, J. Differ. Equations, 162 (2000), 199-223.  doi: 10.1006/jdeq.1999.3704.

[36]

Y. Zhao and H. Zhu, Bifurcation of limit cycles from a non-Hamiltonian quadratic integrable system with homoclinic loop, Infinite Dimensional Dynamical Systems, 64 (2013), 445-479.  doi: 10.1007/978-1-4614-4523-4.

[37]

Y. Zhao and S. Zhu, Perturbations of the non-generic quadratic Hamiltonian vector fields with hyperbolic segment, Bull. Sci. Math., 125 (2001), 109-138.  doi: 10.1016/S0007-4497(00)01069-1.

[38]

H. Żołądek, Quadratic systems with center and their perturbations, J. Differ. Equations, 109 (1994), 223-273.  doi: 10.1006/jdeq.1994.1049.

Figure 1.  The phase portrait of system (2.1)
Figure 2.  The real constant $ \alpha_2 $ and $ \frac{W[\bar{F}_1,\bar{F}_3](x)}{W[\bar{F}_1,\bar{F}_2](x)} $ have at most two intersections
Figure 3.  The phase portrait of system (1.5) with $ b\in(-\infty,-2)\cup(-2,0]\cup[2,+\infty) $
Figure 4.  The range of $ (u,b). $
Figure 5.  The value of $ N_3(u,b) $
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