Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2012, Article ID 690453, 27 pages
http://dx.doi.org/10.1155/2012/690453
Research Article

Hopf Bifurcation of Limit Cycles in Discontinuous Liénard Systems

Department of Mathematics, Shanghai Normal University, Shanghai 200234, China

Received 26 April 2012; Accepted 8 August 2012

Academic Editor: Alberto D'Onofrio

Copyright © 2012 Yanqin Xiong and Maoan Han. 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. N. Glade, L. Forest, and J. Demongeot, “Liénard systems and potential-Hamiltonian decomposition. III. Applications,” Comptes Rendus Mathématique, vol. 344, no. 4, pp. 253–258, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. J. Llibre, “A survey on the limit cycles of the generalization polynomial Linard differential equations. Mathematical models in engineering, biology and medicine,” in Proceedings of the AIP International Conference on Boundary Value Problems: Mathematical Models in Engineering, Biology and Medicine, vol. 1124, pp. 224–233, The American Institute of Physics, Melville, NY, USA, 2009.
  3. M. Han, H. Zang, and T. Zhang, “A new proof to Bautin's theorem,” Chaos, Solitons and Fractals, vol. 31, no. 1, pp. 218–223, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. L. A. Cherkas, “Conditions for the center for certain equations of the form yy=P(x)+Q(x)y+R(x)y2,” Difference Equations, vol. 8, pp. 1104–1107, 1972. View at Google Scholar
  5. A. Gasull, “Differential equations that can be transformed into equations of Liénard type,” 1989.
  6. J. Giné and J. Llibre, “Weierstrass integrability in Liénard differential systems,” Journal of Mathematical Analysis and Applications, vol. 377, no. 1, pp. 362–369, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. L. A. Cherkas, “Conditions for a Lienard equation to have a center,” Differential Equations, vol. 12, pp. 201–206, 1976. View at Google Scholar
  8. L. A. Cherkas and V. G. Romanovski, “The center conditions for a Liénard system,” Computers and Mathematics with Applications, vol. 52, no. 3-4, pp. 363–374, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. A. Gasull and J. Torregrosa, “Center problem for several differential equations via Cherkas' method,” Journal of Mathematical Analysis and Applications, vol. 228, no. 2, pp. 322–343, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. M. A. Han, “Criteria for the existence of closed orbits of autonomous systems in the plane,” Journal of Nanjing University, vol. 21, no. 2, pp. 233–244, 1985 (Chinese). View at Google Scholar
  11. M. Han, “Liapunov constants and Hopf cyclicity of Liénard systems,” Annals of Differential Equations, vol. 15, no. 2, pp. 113–126, 1999. View at Google Scholar
  12. M. Han and W. Zhang, “On Hopf bifurcation in non-smooth planar systems,” Journal of Differential Equations, vol. 248, no. 9, pp. 2399–2416, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. X. Liu and M. Han, “Hopf bifurcation for nonsmooth Liénard systems,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 19, no. 7, pp. 2401–2415, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. J. Llibre and C. Valls, “On the number of limit cycles of a class of polynomial differential systems,” Proceedings of the Royal Society, vol. 468, no. 2144, pp. 2347–2360, 2012. View at Publisher · View at Google Scholar
  15. Y. Q. Ye, S. L. Cai, L. S. Chen et al., Theory of Limit Cycles, vol. 66 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, USA, 2nd edition, 1986.
  16. M. Han, J. Yang, and P. Yu, “Hopf bifurcations for near-Hamiltonian systems,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 19, no. 12, pp. 4117–4130, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. M. Han, C. Li, and J. Li, “Limit cycles of planar polynomial vector fields,” Scholarpedia, vol. 5, no. 8, article 9648, 2010. View at Google Scholar