• Previous Article
    Entropy by unit length for the Ginzburg-Landau equation on the line. A Hilbert space framework
  • CPAA Home
  • This Issue
  • Next Article
    Convergence to the Poisson kernel for the Laplace equation with a nonlinear dynamical boundary condition
May  2012, 11(3): 1269-1283. doi: 10.3934/cpaa.2012.11.1269

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

Received  December 2010 Revised  March 2011 Published  December 2011

In this paper, we study the bifurcation of limit cycles of a class of planar quadratic reversible system $\dot{x}=y+4x^2$, $\dot{y}=-x+2xy$ under quadratic perturbations. It is proved that the cyclicity of the period annulus is equal to two.
Citation: Yi Shao, Yulin Zhao. The cyclicity of the period annulus of a class of quadratic reversible system. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1269-1283. doi: 10.3934/cpaa.2012.11.1269
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. 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. 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. doi: 10.3934/dcds.2006.16.157.

[4]

L. Gavrilov, The infinitesimal 16th Hilbert problem in the quadratic case,, Invent. Math., 143 (2001), 449. 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. 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. doi: 10.3934/dcds.2009.25.511.

[7]

S. Gautier, Quadratic centers defining elliptic surface,, J. Differential Equations, 245 (2008), 3545. 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. doi: 10.1112/plms/s3-69.1.198.

[9]

P. Hartman, "Ordinary Differential Equations,", $2^{nd}$ edition, (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. 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. 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. doi: 10.1016/j.na.2009.04.062.

[13]

I. D. Iliev, Perturbations of quadratic centers,, Bull. Sci. Math., 122 (1998), 107. 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. 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. 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. 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. 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. 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. 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. 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. doi: 10.3934/dcds.2006.16.157.

[4]

L. Gavrilov, The infinitesimal 16th Hilbert problem in the quadratic case,, Invent. Math., 143 (2001), 449. 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. 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. doi: 10.3934/dcds.2009.25.511.

[7]

S. Gautier, Quadratic centers defining elliptic surface,, J. Differential Equations, 245 (2008), 3545. 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. doi: 10.1112/plms/s3-69.1.198.

[9]

P. Hartman, "Ordinary Differential Equations,", $2^{nd}$ edition, (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. 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. 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. doi: 10.1016/j.na.2009.04.062.

[13]

I. D. Iliev, Perturbations of quadratic centers,, Bull. Sci. Math., 122 (1998), 107. 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. 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. 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. 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. 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. doi: 10.1006/jdeq.1999.3704.

[1]

Bourama Toni. Upper bounds for limit cycle bifurcation from an isochronous period annulus via a birational linearization. Conference Publications, 2005, 2005 (Special) : 846-853. doi: 10.3934/proc.2005.2005.846

[2]

Linping Peng, Yazhi Lei. The cyclicity of the period annulus of a quadratic reversible system with a hemicycle. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 873-890. doi: 10.3934/dcds.2011.30.873

[3]

Jaume Llibre, Dana Schlomiuk. On the limit cycles bifurcating from an ellipse of a quadratic center. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1091-1102. doi: 10.3934/dcds.2015.35.1091

[4]

Iliya D. Iliev, Chengzhi Li, Jiang Yu. Bifurcations of limit cycles in a reversible quadratic system with a center, a saddle and two nodes. Communications on Pure & Applied Analysis, 2010, 9 (3) : 583-610. doi: 10.3934/cpaa.2010.9.583

[5]

Jaume Llibre, Yilei Tang. Limit cycles of discontinuous piecewise quadratic and cubic polynomial perturbations of a linear center. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1769-1784. doi: 10.3934/dcdsb.2018236

[6]

Adriana Buică, Jaume Giné, Maite Grau. Essential perturbations of polynomial vector fields with a period annulus. Communications on Pure & Applied Analysis, 2015, 14 (3) : 1073-1095. doi: 10.3934/cpaa.2015.14.1073

[7]

Yulin Zhao. On the monotonicity of the period function of a quadratic system. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 795-810. doi: 10.3934/dcds.2005.13.795

[8]

Ben Niu, Weihua Jiang. Dynamics of a limit cycle oscillator with extended delay feedback. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1439-1458. doi: 10.3934/dcdsb.2013.18.1439

[9]

Valery A. Gaiko. The geometry of limit cycle bifurcations in polynomial dynamical systems. Conference Publications, 2011, 2011 (Special) : 447-456. doi: 10.3934/proc.2011.2011.447

[10]

Haihua Liang, Yulin Zhao. Quadratic perturbations of a class of quadratic reversible systems with one center. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 325-335. doi: 10.3934/dcds.2010.27.325

[11]

G. Chen, C. Li, C. Liu, Jaume Llibre. The cyclicity of period annuli of some classes of reversible quadratic systems. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 157-177. doi: 10.3934/dcds.2006.16.157

[12]

Magdalena Caubergh, Freddy Dumortier, Robert Roussarie. Alien limit cycles in rigid unfoldings of a Hamiltonian 2-saddle cycle. Communications on Pure & Applied Analysis, 2007, 6 (1) : 1-21. doi: 10.3934/cpaa.2007.6.1

[13]

Jihua Yang, Liqin Zhao. Limit cycle bifurcations for piecewise smooth integrable differential systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2417-2425. doi: 10.3934/dcdsb.2017123

[14]

Fangfang Jiang, Junping Shi, Qing-guo Wang, Jitao Sun. On the existence and uniqueness of a limit cycle for a Liénard system with a discontinuity line. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2509-2526. doi: 10.3934/cpaa.2016047

[15]

Stijn Luca, Freddy Dumortier, Magdalena Caubergh, Robert Roussarie. Detecting alien limit cycles near a Hamiltonian 2-saddle cycle. Discrete & Continuous Dynamical Systems - A, 2009, 25 (4) : 1081-1108. doi: 10.3934/dcds.2009.25.1081

[16]

Sze-Bi Hsu, Junping Shi. Relaxation oscillation profile of limit cycle in predator-prey system. Discrete & Continuous Dynamical Systems - B, 2009, 11 (4) : 893-911. doi: 10.3934/dcdsb.2009.11.893

[17]

Fabio Scalco Dias, Luis Fernando Mello. The center--focus problem and small amplitude limit cycles in rigid systems. Discrete & Continuous Dynamical Systems - A, 2012, 32 (5) : 1627-1637. doi: 10.3934/dcds.2012.32.1627

[18]

Jaume Llibre, Claudia Valls. Algebraic limit cycles for quadratic polynomial differential systems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2475-2485. doi: 10.3934/dcdsb.2018070

[19]

José Luis Bravo, Manuel Fernández, Ignacio Ojeda, Fernando Sánchez. Uniqueness of limit cycles for quadratic vector fields. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 483-502. doi: 10.3934/dcds.2019020

[20]

Lijun Wei, Xiang Zhang. Limit cycle bifurcations near generalized homoclinic loop in piecewise smooth differential systems. Discrete & Continuous Dynamical Systems - A, 2016, 36 (5) : 2803-2825. doi: 10.3934/dcds.2016.36.2803

2017 Impact Factor: 0.884

Metrics

  • PDF downloads (4)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]