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Abstract and Applied Analysis
Volume 2014, Article ID 792439, 15 pages
http://dx.doi.org/10.1155/2014/792439
Research Article

Limit Cycles Bifurcated from Some -Equivariant Quintic Near-Hamiltonian Systems

Department of Applied Mathematics, Guangxi University of Finance and Economics, Nanning, Guangxi 530003, China

Received 3 December 2013; Accepted 14 January 2014; Published 3 March 2014

Academic Editor: Yonghuia Xia

Copyright © 2014 Simin Qu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. D. Hilbert, “Mathematical problems,” Bulletin of the American Mathematical Society, vol. 8, no. 10, pp. 437–479, 1902. View at Publisher · View at Google Scholar · View at MathSciNet
  2. J. Li, Chaos and Melnikov Method, Chongqin University Press, Chongqing, China, 1989, (Chinese).
  3. J. Li, “Hilbert's 16th problem and bifurcations of planar polynomial vector fields,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 13, no. 1, pp. 47–106, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. C. Li, “Abelian integrals and limit cycles,” Qualitative Theory of Dynamical Systems, vol. 11, no. 1, pp. 111–128, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. M. Han and J. Li, “Lower bounds for the Hilbert number of polynomial systems,” Journal of Differential Equations, vol. 252, no. 4, pp. 3278–3304, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. C. Li, C. Liu, and J. Yang, “A cubic system with thirteen limit cycles,” Journal of Differential Equations, vol. 246, no. 9, pp. 3609–3619, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. L. Zhao, “Some new distribution of 13 limit cycles of a cubic system,” Journal of Beijing Normal University, vol. 48, pp. 231–234, 2012. View at Google Scholar
  8. J. Li and Y. Liu, “New results on the study of Zq-equivariant planar polynomial vector fields,” Qualitative Theory of Dynamical Systems, vol. 9, no. 1-2, pp. 167–219, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  9. T. Zhang, M. Han, H. Zang, and X. Meng, “Bifurcations of limit cycles for a cubic Hamiltonian system under quartic perturbations,” Chaos, Solitons & Fractals, vol. 22, no. 5, pp. 1127–1138, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. C. Christopher, “Estimating limit cycle bifurcations from center,” in Differential Equations with Symbolic Computation, Trends in Mathematics, pp. 23–35, Birkhäuser, Basel, Switzerland, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. W. Xu and M. Han, “On the number of limit cycles of a Z4-equivariant quintic polynomial system,” Applied Mathematics and Computation, vol. 216, no. 10, pp. 3022–3034, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  12. J. Li, H. S. Y. Chan, and K. W. Chung, “Investigations of bifurcations of limit cycles in Z2-equivariant planar vector fields of degree 5,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 12, no. 10, pp. 2137–2157, 2002. View at Publisher · View at Google Scholar · View at MathSciNet
  13. Y. Wu, L. Tian, and Y. Hu, “On the limit cycles of a Hamiltonian under Z4-equivariant quintic perturbation,” Chaos, Solitons and Fractals, vol. 33, no. 1, pp. 298–307, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  14. J. Li, H. S. Y. Chan, and K. W. Chung, “Bifurcations of limit cycles in a Z6-equivariant planar vector field of degree 5,” Science in China. Series A, vol. 45, no. 7, pp. 817–826, 2002. View at Google Scholar · View at MathSciNet
  15. W. H. Yao and P. Yu, “Bifurcation of small limit cycles in Z5-equivariant planar vector fields of order 5,” Journal of Mathematical Analysis and Applications, vol. 328, no. 1, pp. 400–413, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  16. Y. H. Wu, X. D. Wang, and L. X. Tian, “Bifurcations of limit cycles in a Z4-equivariant quintic planar vector field,” Acta Mathematica Sinica. English Series, vol. 26, no. 4, pp. 779–798, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  17. Y. Wu and M. Han, “New configurations of 24 limit cycles in a quintic system,” Computers & Mathematics with Applications, vol. 55, no. 9, pp. 2064–2075, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. M. Han, “Asymptotic expansions of Melnikov functions and limit cycle bifurcations,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 22, Article ID 1250296, 30 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. M. Han, J. Yang, A.-A. Tarţa, and Y. Gao, “Limit cycles near homoclinic and heteroclinic loops,” Journal of Dynamics and Differential Equations, vol. 20, no. 4, pp. 923–944, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. Y. Hou and M. Han, “Melnikov functions for planar near-Hamiltonian systems and Hopf bifurcations,” Journal of Shanghai Normal University, vol. 35, no. 1, pp. 1–10, 2006. View at Google Scholar
  21. J. Yang, “On the limit cycles of a kind of Liénard system with a nilpotent center under perturbations,” The Journal of Applied Analysis and Computation, vol. 2, no. 3, pp. 325–339, 2012. View at Google Scholar · View at MathSciNet
  22. R. Kazemi and H. R. Z. Zangeneh, “Bifurcation of limit cycles in small perturbations of a hyper-elliptic Hamiltonian system with two nilpotent saddles,” The Journal of Applied Analysis and Computation, vol. 2, no. 4, pp. 395–413, 2012. View at Google Scholar · View at MathSciNet
  23. X. Sun, “Bifurcation of limit cycles from a Liénard system with a heteroclinic loop connecting two nilpotent saddles,” Nonlinear Dynamics, vol. 73, no. 1-2, pp. 869–880, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  24. M. Han and X. Sun, “General form of reversible equivariant system and its limit cycles,” Journal of Shanghai Normal University, vol. 40, no. 1, pp. 1–14, 2011. View at Google Scholar