Skip to main content
SpringerLink
Log in

Z 2-equivariant cubic system which yields 13 limit cycles

  • Published:
Acta Mathematicae Applicatae Sinica, English Series Aims and scope Submit manuscript

Abstract

For the planar Z 2-equivariant cubic systems having two elementary focuses, the characterization of a bi-center problem and shortened expressions of the first six Lyapunov constants are completely solved. The necessary and sufficient conditions for the existence of the bi-center are obtained. On the basis of this work, in this paper, we show that under small Z 2-equivariant cubic perturbations, this cubic system has at least 13 limit cycles with the scheme 1 ⊂ 6 ∪ 6.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Liu, Y.R., Li, J.B., Huang W.T. Singular Point Values, Center Problem and Bifurcations of Limit Cycles of Two Dimensional Differential Autonomous Systrms. Science Press, Beijing, 2008, 192–209

    Google Scholar 

  2. Li, J.B., Li, C.F. Planar cubic Hamiltonian systems and distribution of limit cycles of (E 3). Acta Math. Sinica, 28(4): 509–521 (1985)

    MATH  MathSciNet  Google Scholar 

  3. Li, J.B., Chen X.Q. Poincaré bifurcations in the quadratic differential system. Kexue Tongbao (English Ed.), 32(10): 655–660 (1987)

    MATH  MathSciNet  Google Scholar 

  4. Li, J.B., Huang, Q.M. Bifurcations of limit cycles forming compound eyes in the cubic system. Chin. Ann. Math., 8B: 391–403 (1987)

    Google Scholar 

  5. Li, J.B., Ou, Y.H. Global bifurcations and chaotic behavior in a disturbed quadratic system with two centers. Acta Math. Appl. Sinica, 11(3): 312–323 (1988)

    MATH  MathSciNet  Google Scholar 

  6. Li, J.B., Zhao, X.H. Rotation symmetry groups of planar Hamiltonian systems. Ann. Diff. Equs., 5: 25–33 (1989)

    MATH  Google Scholar 

  7. Li, J.B., Wan, S.D. Global bifurcations ina disturbed Hamiltonian vector field approaching a 3:1 resonant Poincaré map. Acta. Math. Appl. Sinica (Eng. Ser.), 7: 80–89 (1991)

    Article  MathSciNet  Google Scholar 

  8. Li, J.B., Liu, Z.R. Bifurcation set and limit cycles forming compound eyes in a perturbed Hamiltonian system. Publications Mathmatiques, 35: 487–506 (1991)

    Article  MATH  Google Scholar 

  9. Li, J.B., Lin Y.P. Normal form and critical points of the period of closed orbits for planar autonomous systems. Acta Mathematica Sinica, 34: 490–501 (1991) (in Chinese)

    MATH  MathSciNet  Google Scholar 

  10. Li, J.B., Liu, Z.R. Bifurcation set and compound eyes in a perturbed cubic Hamiltonian system. In Ordinary and Delay Differential Equations, π Pitman Research Notes in Math., Vol.272, Longman, England, 1992, 116–128

    Google Scholar 

  11. Li, J.B., Lin, Y.R. Global bifurcations in a perturbed cubic system with Z 2-symmetry. Acta Math. Appl. Sinica (English Ser.), 8(2): 131–143 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  12. Li, C.Z., Llibre, J., Zhang Z F. Abelian intrgrals of quadratic Hamiltonian vector fields with a invariant straight line. Publ. Math., 39(2): 355–366 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  13. Li, J.B., Feng, B.Y. Stability, Bifurcations and Chaos. Science and Technology Press of Yunnan, Kunming, 1995 (in Chinese)

    Google Scholar 

  14. Li, J.B., Chan, H.S.Y., Chung, K.W. Bifurcations of limit cycles in a Z 6-equivariant planar vector field of degree 5. Sci. in China (Ser. A), 45(7): 817–826 (2002)

    MATH  MathSciNet  Google Scholar 

  15. Li, J.B., Chan, H.S.Y., Chung K.W. Investigations of bifurcations of limit cycles in a Z 2-equivariant planar vector field of degree 5. Int. J. Bifur. Chaos, 12: 2137–2157 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  16. Li, J.B., Chan, H.S.Y., Chung, K.W. Some lower bounds for H(n) in Hilbert’s 16th problem. Qualitative Theory of Dynamical Systems, 44(3): 345–360 (2003)

    MathSciNet  Google Scholar 

  17. Li, J.B. Hilbert’s 16th problem and Bifurcations of Planar Polynomial vector Fields. Int. J. Bifur. Chaos, 13(1): 47–106 (2003)

    Article  MATH  Google Scholar 

  18. Li, J.B., Zhang, M.Q. Bifurcations of limit cycles of Z 8-equivariant planar vector field of degree 7. J. Dynamics and Differentil Equations, 16(4): 1123–1139 (2004)

    Article  MATH  Google Scholar 

  19. Li, J.B., Zhou, H.X. On the Control of Parameters of Distributions of Limit Cycles for a Z 2-Equivariant Perturbed Planar Hamiltonian Polynomial Vector Field. Int. J. Bifur. Chaos, 15(1): 137–155 (2005)

    Article  MATH  Google Scholar 

  20. Li, J.B., Zhang, M.J. Bifurcations of limit cycles in a Z 2-equivariant planar polynomial vector field of degree 7. Int. J. Bifur. Chaos, 16: 925–943 (2006)

    Article  MATH  Google Scholar 

  21. Liu, Y.R. Formulas of focal values, center conditions and center integrals for the system (E (3)3 ). Chinese Science Bulletin, 33(5): 357–359 (1988)

    MATH  Google Scholar 

  22. Liu, Y.R., Li, J.B. Theory of values of singular point in complex autonomous differential systems. Sci. in China 33(1): 10–23 (1989)

    Google Scholar 

  23. Liu, Y.R. Theory of center-focus in a class of high order singular points and infinity. Sci. in China, 31(1): 37–48 (2001)

    Google Scholar 

  24. Liu, Y.R., Li, J.B. Complete Study on a Bi-Center Problem for the Z 2 Equivariant Cubic Vector Fields. Acta Mathematica Sinica (English Series), 27(7): 1379–1394 (2011)

    Article  MathSciNet  Google Scholar 

  25. Liu, Y.R., Li, J.B. Center problem and multiple Hopf bifurcation for the Z 6 equivariant planar polynomial vector fields of degree 5. Int. J. Bifur. Chaos, 19(5): 1741–1749 (2009)

    Article  MATH  Google Scholar 

  26. Liu, Y.R., Li, J.B. Center problem and multiple Hopf bifurcation for the Z 5 equivariant planar polynomial vector fields of degree 5. Int. J. Bifur. Chaos, 19(6): 2115–2121 (2009)

    Article  MATH  Google Scholar 

  27. Liu, Y.R., Huang, W.T. A cubic system with twelve small amplitude limit cycles. Bull. Sci. Math., 129: 83–98 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  28. Yu, P., Han, M.A. Twelve limit cycles in 3rd-planar system with Z 2 symmetry. Communications on Appl. Pure Anal., 3: 515–526 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  29. Yu, P., Han, M.A. Small limit cycles bifurcating from fine focus points in cubic order Z 2-equivariant vector fields. Chaos, Solitons and Fractals, 24: 329–348 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  30. Yu, P., Han, M.A. Twelve limit cycles in a cubic case of the 16th Hilbert problem. Int. J. Bifur. Chaos, 15: 2192–2205 (2005)

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics, Central South University, Changsha, 410083, China

    Yi-rong Liu

  2. Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, China

    Ji-bin Li

  3. School of Science, Kunming University of Science and Technology, Kunming, 650093, China

    Ji-bin Li

Authors
  1. Yi-rong Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ji-bin Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yi-rong Liu.

Additional information

Supported by the National Natural Science Foundation of China (No. 11371373 and 10831003).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Liu, Yr., Li, Jb. Z 2-equivariant cubic system which yields 13 limit cycles. Acta Math. Appl. Sin. Engl. Ser. 30, 781–800 (2014). https://doi.org/10.1007/s10255-014-0420-x

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10255-014-0420-x

Keywords

2000 MR Subject Classification

Search

Navigation