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

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

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 $.

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. ChenJ. LlibreZ. 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. ChenZ. 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. DumortierC. 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. GarcíaJ. 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. PengZ. 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. RomanovskiW. 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. WuG. 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. ChenJ. LlibreZ. 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. ChenZ. 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. DumortierC. 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. GarcíaJ. 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. PengZ. 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. RomanovskiW. 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. WuG. 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
[1]

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 and Applied Analysis, 2016, 15 (6) : 2509-2526. doi: 10.3934/cpaa.2016047

[2]

Na Li, Maoan Han, Valery G. Romanovski. Cyclicity of some Liénard Systems. Communications on Pure and Applied Analysis, 2015, 14 (6) : 2127-2150. doi: 10.3934/cpaa.2015.14.2127

[3]

Hong Li. Bifurcation of limit cycles from a Li$ \acute{E} $nard system with asymmetric figure eight-loop case. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022033

[4]

Jitsuro Sugie, Tadayuki Hara. Existence and non-existence of homoclinic trajectories of the Liénard system. Discrete and Continuous Dynamical Systems, 1996, 2 (2) : 237-254. doi: 10.3934/dcds.1996.2.237

[5]

Mats Gyllenberg, Yan Ping. The generalized Liénard systems. Discrete and Continuous Dynamical Systems, 2002, 8 (4) : 1043-1057. doi: 10.3934/dcds.2002.8.1043

[6]

Jianhe Shen, Maoan Han. Bifurcations of canard limit cycles in several singularly perturbed generalized polynomial Liénard systems. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 3085-3108. doi: 10.3934/dcds.2013.33.3085

[7]

Jaume Llibre, Claudia Valls. On the analytic integrability of the Liénard analytic differential systems. Discrete and Continuous Dynamical Systems - B, 2016, 21 (2) : 557-573. doi: 10.3934/dcdsb.2016.21.557

[8]

Bin Liu. Quasiperiodic solutions of semilinear Liénard equations. Discrete and Continuous Dynamical Systems, 2005, 12 (1) : 137-160. doi: 10.3934/dcds.2005.12.137

[9]

Robert Roussarie. Putting a boundary to the space of Liénard equations. Discrete and Continuous Dynamical Systems, 2007, 17 (2) : 441-448. doi: 10.3934/dcds.2007.17.441

[10]

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

[11]

Isaac A. García, Douglas S. Shafer. Cyclicity of a class of polynomial nilpotent center singularities. Discrete and Continuous Dynamical Systems, 2016, 36 (5) : 2497-2520. doi: 10.3934/dcds.2016.36.2497

[12]

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 and Applied Analysis, 2010, 9 (3) : 583-610. doi: 10.3934/cpaa.2010.9.583

[13]

Tomás Caraballo, David Cheban. Almost periodic and asymptotically almost periodic solutions of Liénard equations. Discrete and Continuous Dynamical Systems - B, 2011, 16 (3) : 703-717. doi: 10.3934/dcdsb.2011.16.703

[14]

Isaac A. García, Jaume Giné, Jaume Llibre. Liénard and Riccati differential equations related via Lie Algebras. Discrete and Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 485-494. doi: 10.3934/dcdsb.2008.10.485

[15]

Wenbin Liu, Zhaosheng Feng. Periodic solutions for $p$-Laplacian systems of Liénard-type. Communications on Pure and Applied Analysis, 2011, 10 (5) : 1393-1400. doi: 10.3934/cpaa.2011.10.1393

[16]

Tiantian Ma, Zaihong Wang. Periodic solutions of Liénard equations with resonant isochronous potentials. Discrete and Continuous Dynamical Systems, 2013, 33 (4) : 1563-1581. doi: 10.3934/dcds.2013.33.1563

[17]

John Guckenheimer, Hinke M. Osinga. The singular limit of a Hopf bifurcation. Discrete and Continuous Dynamical Systems, 2012, 32 (8) : 2805-2823. doi: 10.3934/dcds.2012.32.2805

[18]

Jihua Yang, Erli Zhang, Mei Liu. Limit cycle bifurcations of a piecewise smooth Hamiltonian system with a generalized heteroclinic loop through a cusp. Communications on Pure and Applied Analysis, 2017, 16 (6) : 2321-2336. doi: 10.3934/cpaa.2017114

[19]

A. Ghose Choudhury, Partha Guha. Chiellini integrability condition, planar isochronous systems and Hamiltonian structures of Liénard equation. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2465-2478. doi: 10.3934/dcdsb.2017126

[20]

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

2021 Impact Factor: 1.497

Metrics

  • PDF downloads (292)
  • HTML views (235)
  • Cited by (0)

Other articles
by authors

[Back to Top]