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January  2020, 25(1): 401-414. doi: 10.3934/dcdsb.2019187

Bi-center problem and Hopf cyclicity of a Cubic Liénard system

Min Hu , Tao Li and  Xingwu Chen ,

Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China

* Corresponding author: Xingwu Chen(xingwu.chen@hotmail.com)

Received  June 2018 Published  January 2020 Early access  September 2019

Fund Project: The second author is supported by Graduate Student's Research and Innovation Fund 2018YJSY047 and Doctoral Graduate Student's Academic Visit Fund of Sichuan University. The third author is supported by NSFC 11871355.

In this paper we investigate the bi-center problem and the total Hopf cyclicity of two center-foci for the general cubic Liénard system which has three distinct equilibria and is equivalent to the general Liénard equation with cubic damping and restoring force. The location of these three equilibria is arbitrary, specially without any kind of symmetry. We find the necessary and sufficient condition for the existence of bi-centers and prove that there is no case of a unique center. On the Hopf cyclicity we prove that there are totally $ 9 $ possible styles of small amplitude limit cycles surrounding these two center-foci and $ 6 $ styles of them can occur, from which the total Hopf cyclicity is no more than $ 4 $ and no less than $ 2 $.

Keywords: Bi-center, Hopf cyclicity, Liénard system, limit cycle.
Mathematics Subject Classification: Primary: 34C07, 34C23, 34D10, 37G10.
Citation: Min Hu, Tao Li, Xingwu Chen. Bi-center problem and Hopf cyclicity of a Cubic Liénard system. Discrete and Continuous Dynamical Systems - B, 2020, 25 (1) : 401-414. doi: 10.3934/dcdsb.2019187
References:
[1]

N. N. Bautin, On the number of limit cycles appearing with variation of the coefficients from an equilibrium state of the type of a focus or a center, Matematicheskii Sbornik N.S., 30 (1952), 181-196. 

[2]

X. Chen, J. Llibre, Z. Wang and W. Zhang, Restricted independence in displacement function for better estimation of cyclicity, J. Differential Equations, 262 (2017), 5773-5791.  doi: 10.1016/j.jde.2017.02.015.

[3]

L. Chen, Z. Lu and D. Wang, A class of cubic systems with two centers or two foci, J. Math. Anal. Appl., 242 (2000), 154-163.  doi: 10.1006/jmaa.1999.6630.

[4]

S. -N. Chow and J. K. Hale, Methods of Bifurcation Theory, Springer-Verlag, New York, 1982.

[5]

C. Christopher and C. Li, Limit Cycles of Differential Equations, Birkh$\ddot{a}$user Verlag, Basel, 2007.

[6]

C. Christopher and S. Lynch, Small-amplitude limit cycle bifurcations for Liénard systems with quadratic or cubic damping or restoring forces, Nonlinearity, 12 (1999), 1099-1112.  doi: 10.1088/0951-7715/12/4/321.

[7]

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

[8]

F. Dumortier, J. Llibre and J. Artés, Qualitative Theory of Planar Differential Systems, Springer-Verlag, New York, 2006.

[9]

I. A. García, J. Llibre and S. Maza, The Hopf cyclicity of the centers of a class of quintic polynomial vector fields, J. Differential Equations, 258 (2015), 1990-2009.  doi: 10.1016/j.jde.2014.11.018.

[10]

J. Giné, Center conditions for polynomial Liénard systems, Qual. Theory Dyn. Syst., 16 (2017), 119-126.  doi: 10.1007/s12346-016-0202-3.

[11]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, 3$^{rd}$ edition, Springer-Verlag, New York, 2004. doi: 10.1007/978-1-4757-3978-7.

[12]

C. Li, Hilbert's 16th problem and bifurcations of planar polynomial vector fields, Int. J. Bifurcation & Chaos, 13 (2003), 47-106.  doi: 10.1142/S0218127403006352.

[13]

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

[14]

Y. Liu and W. Huang, A cubic system with twelve small amplitude limit cycles, Bull. Sci. math., 129 (2005), 83-98.  doi: 10.1016/j.bulsci.2004.05.004.

[15]

Y. Liu and J. Li, Some Classical Problems for Planar Vector Fields(in Chinese), Science Press, Beijing, 2010.

[16]

Y. Liu and J. Li, Complete study on a bi-center problem for the Z$_2$-equivariant cubic vector fields, Acta Math. Sin. English Series, 27 (2011), 1379-1394.  doi: 10.1007/s10114-011-8412-8.

[17]

L. Peng, Z. Feng and C. Liu, Quadratic perturbations of a quadratic reversible Lotka-Volterra system with two centers, Disc. Contin. Dyn. Syst., 34 (2014), 4807-4826.  doi: 10.3934/dcds.2014.34.4807.

[18]

V. G. Romanovski, W. Fernandes and R. Oliveira, Bi-center problem for some classes of Z$_2$-equivariant systems, J. Comput. Appl. Math., 320 (2017), 61-75.  doi: 10.1016/j.cam.2017.02.003.

[19]

V. G. Romanovski and D. S. Shafer, The Center and Cyclicity Problems: A Computational Algebra Approach, Birkh$\ddot{a}$user Verlag, Boston, 2009. doi: 10.1007/978-0-8176-4727-8.

[20]

Y. Tian and M. Han, Hopf bifurcation for two types of Liénard systems, J. Differential Equations, 251 (2011), 834-859.  doi: 10.1016/j.jde.2011.05.029.

[21]

Y. Wu, G. Chen and X. Yang, Kukles system with two fine foci, Ann. of Diff. Eqs., 15 (1999), 422-437. 

[22]

P. Yu and M. Han, Twelve limit cycles in a cubic case of the 16th Hilbert problem, Int. J. Bifurcation & Chaos, 15 (2005), 2191-2205.  doi: 10.1142/S0218127405013289.

[23]

Z. Zhang, T. Ding, W. Huang and Z. Dong, Qualitative Theory of Differential Equations, Transl. Math. Monogr., Amer. Math. Soc., Providence, RI, 1992.

[24]

Z. Zhang, C. Li, Z. Zheng and W. Li, Elementary Theory of Bifurcations of Vector Fields(in Chinese), Higher Education Press, Beijing, 1997.

[25]

H. Żoładek, Eleven small limit cycles in a cubic vector field, Nonlinearity, 8 (1995), 843-860.  doi: 10.1088/0951-7715/8/5/011.

show all references

References:
[1]

N. N. Bautin, On the number of limit cycles appearing with variation of the coefficients from an equilibrium state of the type of a focus or a center, Matematicheskii Sbornik N.S., 30 (1952), 181-196. 

[2]

X. Chen, J. Llibre, Z. Wang and W. Zhang, Restricted independence in displacement function for better estimation of cyclicity, J. Differential Equations, 262 (2017), 5773-5791.  doi: 10.1016/j.jde.2017.02.015.

[3]

L. Chen, Z. Lu and D. Wang, A class of cubic systems with two centers or two foci, J. Math. Anal. Appl., 242 (2000), 154-163.  doi: 10.1006/jmaa.1999.6630.

[4]

S. -N. Chow and J. K. Hale, Methods of Bifurcation Theory, Springer-Verlag, New York, 1982.

[5]

C. Christopher and C. Li, Limit Cycles of Differential Equations, Birkh$\ddot{a}$user Verlag, Basel, 2007.

[6]

C. Christopher and S. Lynch, Small-amplitude limit cycle bifurcations for Liénard systems with quadratic or cubic damping or restoring forces, Nonlinearity, 12 (1999), 1099-1112.  doi: 10.1088/0951-7715/12/4/321.

[7]

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

[8]

F. Dumortier, J. Llibre and J. Artés, Qualitative Theory of Planar Differential Systems, Springer-Verlag, New York, 2006.

[9]

I. A. García, J. Llibre and S. Maza, The Hopf cyclicity of the centers of a class of quintic polynomial vector fields, J. Differential Equations, 258 (2015), 1990-2009.  doi: 10.1016/j.jde.2014.11.018.

[10]

J. Giné, Center conditions for polynomial Liénard systems, Qual. Theory Dyn. Syst., 16 (2017), 119-126.  doi: 10.1007/s12346-016-0202-3.

[11]

Y. A. Kuznetsov, Elements of Applied Bifurcation Theory, 3$^{rd}$ edition, Springer-Verlag, New York, 2004. doi: 10.1007/978-1-4757-3978-7.

[12]

C. Li, Hilbert's 16th problem and bifurcations of planar polynomial vector fields, Int. J. Bifurcation & Chaos, 13 (2003), 47-106.  doi: 10.1142/S0218127403006352.

[13]

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

[14]

Y. Liu and W. Huang, A cubic system with twelve small amplitude limit cycles, Bull. Sci. math., 129 (2005), 83-98.  doi: 10.1016/j.bulsci.2004.05.004.

[15]

Y. Liu and J. Li, Some Classical Problems for Planar Vector Fields(in Chinese), Science Press, Beijing, 2010.

[16]

Y. Liu and J. Li, Complete study on a bi-center problem for the Z$_2$-equivariant cubic vector fields, Acta Math. Sin. English Series, 27 (2011), 1379-1394.  doi: 10.1007/s10114-011-8412-8.

[17]

L. Peng, Z. Feng and C. Liu, Quadratic perturbations of a quadratic reversible Lotka-Volterra system with two centers, Disc. Contin. Dyn. Syst., 34 (2014), 4807-4826.  doi: 10.3934/dcds.2014.34.4807.

[18]

V. G. Romanovski, W. Fernandes and R. Oliveira, Bi-center problem for some classes of Z$_2$-equivariant systems, J. Comput. Appl. Math., 320 (2017), 61-75.  doi: 10.1016/j.cam.2017.02.003.

[19]

V. G. Romanovski and D. S. Shafer, The Center and Cyclicity Problems: A Computational Algebra Approach, Birkh$\ddot{a}$user Verlag, Boston, 2009. doi: 10.1007/978-0-8176-4727-8.

[20]

Y. Tian and M. Han, Hopf bifurcation for two types of Liénard systems, J. Differential Equations, 251 (2011), 834-859.  doi: 10.1016/j.jde.2011.05.029.

[21]

Y. Wu, G. Chen and X. Yang, Kukles system with two fine foci, Ann. of Diff. Eqs., 15 (1999), 422-437. 

[22]

P. Yu and M. Han, Twelve limit cycles in a cubic case of the 16th Hilbert problem, Int. J. Bifurcation & Chaos, 15 (2005), 2191-2205.  doi: 10.1142/S0218127405013289.

[23]

Z. Zhang, T. Ding, W. Huang and Z. Dong, Qualitative Theory of Differential Equations, Transl. Math. Monogr., Amer. Math. Soc., Providence, RI, 1992.

[24]

Z. Zhang, C. Li, Z. Zheng and W. Li, Elementary Theory of Bifurcations of Vector Fields(in Chinese), Higher Education Press, Beijing, 1997.

[25]

H. Żoładek, Eleven small limit cycles in a cubic vector field, Nonlinearity, 8 (1995), 843-860.  doi: 10.1088/0951-7715/8/5/011.

Figure 1.  Bi-centers
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