• Previous Article
    Convergence of Lindstedt series for the non linear wave equation
  • CPAA Home
  • This Issue
  • Next Article
    Asymptotic theory for disc-like crystal growth (II): interfacial instability and pattern formation at early stage of growth
2004, 3(3): 515-526. doi: 10.3934/cpaa.2004.3.515

Twelve limit cycles in a cubic order planar system with $Z_2$- symmetry

1. 

Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200030, China

2. 

Department of Mathematics, Shanghai Jiao Tong University, Shanghai, China 200030, China

Received  June 2003 Revised  June 2004 Published  June 2004

In this paper, we report the existence of twelve small limit cycles in a planar system with 3rd-degree polynomial functions. The system has $Z_2$-symmetry, with a saddle point, or a node, or a focus point at the origin, and two focus points which are symmetric about the origin. It is shown that such a $Z_2$-equivariant vector field can have twelve small limit cycles. Fourteen or sixteen small limit cycles, as expected before, cannot not exist.
Citation: P. Yu, M. Han. Twelve limit cycles in a cubic order planar system with $Z_2$- symmetry. Communications on Pure & Applied Analysis, 2004, 3 (3) : 515-526. doi: 10.3934/cpaa.2004.3.515
[1]

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

[2]

Matteo Franca, Russell Johnson, Victor Muñoz-Villarragut. On the nonautonomous Hopf bifurcation problem. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 1119-1148. doi: 10.3934/dcdss.2016045

[3]

Jisun Lim, Seongwon Lee, Yangjin Kim. Hopf bifurcation in a model of TGF-$\beta$ in regulation of the Th 17 phenotype. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3575-3602. doi: 10.3934/dcdsb.2016111

[4]

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

[5]

Eric Benoît. Bifurcation delay - the case of the sequence: Stable focus - unstable focus - unstable node. Discrete & Continuous Dynamical Systems - S, 2009, 2 (4) : 911-929. doi: 10.3934/dcdss.2009.2.911

[6]

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

[7]

Qiongwei Huang, Jiashi Tang. Bifurcation of a limit cycle in the ac-driven complex Ginzburg-Landau equation. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 129-141. doi: 10.3934/dcdsb.2010.14.129

[8]

Ryan T. Botts, Ale Jan Homburg, Todd R. Young. The Hopf bifurcation with bounded noise. Discrete & Continuous Dynamical Systems - A, 2012, 32 (8) : 2997-3007. doi: 10.3934/dcds.2012.32.2997

[9]

Chan-Gyun Kim, Yong-Hoon Lee. A bifurcation result for two point boundary value problem with a strong singularity. Conference Publications, 2011, 2011 (Special) : 834-843. doi: 10.3934/proc.2011.2011.834

[10]

Zvia Agur, L. Arakelyan, P. Daugulis, Y. Ginosar. Hopf point analysis for angiogenesis models. Discrete & Continuous Dynamical Systems - B, 2004, 4 (1) : 29-38. doi: 10.3934/dcdsb.2004.4.29

[11]

Hooton Edward, Balanov Zalman, Krawcewicz Wieslaw, Rachinskii Dmitrii. Sliding Hopf bifurcation in interval systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3545-3566. doi: 10.3934/dcds.2017152

[12]

Zhiqin Qiao, Deming Zhu, Qiuying Lu. Bifurcation of a heterodimensional cycle with weak inclination flip. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 1009-1025. doi: 10.3934/dcdsb.2012.17.1009

[13]

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

[14]

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

[15]

Dmitriy Yu. Volkov. The Hopf -- Hopf bifurcation with 2:1 resonance: Periodic solutions and invariant tori. Conference Publications, 2015, 2015 (special) : 1098-1104. doi: 10.3934/proc.2015.1098

[16]

Shao-Yuan Huang, Shin-Hwa Wang. On S-shaped bifurcation curves for a two-point boundary value problem arising in a theory of thermal explosion. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4839-4858. doi: 10.3934/dcds.2015.35.4839

[17]

Fernando Antoneli, Ana Paula S. Dias, Rui Paiva. Coupled cell networks: Hopf bifurcation and interior symmetry. Conference Publications, 2011, 2011 (Special) : 71-78. doi: 10.3934/proc.2011.2011.71

[18]

R. Ouifki, M. L. Hbid, O. Arino. Attractiveness and Hopf bifurcation for retarded differential equations. Communications on Pure & Applied Analysis, 2003, 2 (2) : 147-158. doi: 10.3934/cpaa.2003.2.147

[19]

Fatihcan M. Atay. Delayed feedback control near Hopf bifurcation. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 197-205. doi: 10.3934/dcdss.2008.1.197

[20]

Begoña Alarcón, Víctor Guíñez, Carlos Gutierrez. Hopf bifurcation at infinity for planar vector fields. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 247-258. doi: 10.3934/dcds.2007.17.247

2017 Impact Factor: 0.884

Metrics

  • PDF downloads (1)
  • HTML views (0)
  • Cited by (33)

Other articles
by authors

[Back to Top]