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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 720830, 28 pages
http://dx.doi.org/10.1155/2012/720830
Research Article

Limit Cycle Bifurcations from a Nilpotent Focus or Center of Planar Systems

1Department of Mathematics, Shanghai Normal University, Shanghai 200234, China
2Faculty of Natural Science and Mathematics, University of Maribor, 2000 Maribor, Slovenia
3Center for Applied Mathematics and Theoretical Physics, University of Maribor, 2000 Maribor, Slovenia

Received 11 August 2012; Accepted 28 October 2012

Academic Editor: Jaume Giné

Copyright © 2012 Maoan Han and Valery G. Romanovski. 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. 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,” American Mathematical Society Translations, vol. 30, pp. 181–196, 1952. View at Google Scholar
  2. A. F. Andreev, “Investigation of the behaviour of the integral curves of a system of two differential equations in the neighbourhood of a singular point,” American Mathematical Society Translations, vol. 8, pp. 183–207, 1958. View at Google Scholar
  3. A. M. Liapunov, “Studies of one special case of the problem of stability of motion,” Matematicheskiĭ Sbornik, vol. 17, pp. 253–333, 1983 (Russian). View at Google Scholar
  4. A. M. Liapunov, Stability of Motion, Mathematics in Science and Engineering, Academic Press, London, UK, 1966.
  5. A. P. Sadovskiĭ, “On the problem of discrimination of center and focus,” Differencial’nye Uravnenija, vol. 5, pp. 326–330, 1969 (Russian). View at Google Scholar
  6. V. V. Amel’kin, N. A. Lukashevich, and A. P. Sadovskiĭ, Nonlinear Oscillations in Second Order Systems, Belorusskogo Gosudarstvennogo Universiteta, Minsk, Russia, 1982.
  7. R. Moussu, “Symétrie et forme normale des centres et foyers dégénérés,” Ergodic Theory and Dynamical Systems, vol. 2, no. 2, pp. 241–251, 1982. View at Google Scholar · View at Zentralblatt MATH
  8. J. Chavarriga, H. Giacomin, J. Giné, and J. Llibre, “Local analytic integrability for nilpotent centers,” Ergodic Theory and Dynamical Systems, vol. 23, no. 2, pp. 417–428, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. H. Giacomini, J. Giné, and J. Llibre, “The problem of distinguishing between a center and a focus for nilpotent and degenerate analytic systems,” Journal of Differential Equations, vol. 227, no. 2, pp. 406–426, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. M. J. Álvarez and A. Gasull, “Monodromy and stability for nilpotent critical points,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 15, no. 4, pp. 1253–1265, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. M. J. Álvarez and A. Gasull, “Generating limit cycles from a nilpotent critical point via normal forms,” Journal of Mathematical Analysis and Applications, vol. 318, no. 1, pp. 271–287, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. Y. Liu and J. Li, “New study on the center problem and bifurcations of limit cycles for the Lyapunov system—I,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 19, no. 11, pp. 3791–3801, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. Y. Liu and J. Li, “New study on the center problem and bifurcations of limit cycles for the Lyapunov system—II,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 19, no. 9, pp. 3087–3099, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. Y. Liu and J. Li, “Bifurcations of limit cycles and center problem for a class of cubic nilpotent system,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 20, no. 8, pp. 2579–2584, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. Y. Liu and J. Li, “Bifurcations of limit cycles created by a multiple nilpotent critical point of planar dynamical systems,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 21, no. 2, pp. 497–504, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. F. Takens, “Singularities of vector fields,” Institut des Hautes Études Scientifiques. Publications Mathématiques, no. 43, pp. 47–100, 1974. View at Google Scholar
  17. E. Stróżyna and H. Żołądek, “The analytic and formal normal form for the nilpotent singularity,” Journal of Differential Equations, vol. 179, no. 2, pp. 479–537, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. 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.
  19. C. Chicone and M. Jacobs, “Bifurcation of critical periods for plane vector fields,” Transactions of the American Mathematical Society, vol. 312, no. 2, pp. 433–486, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  20. R. Roussarie, Bifurcation of Planar Vector Fields and Hilbert's Sixteenth Problem, vol. 164 of Progress in Mathematics, Birkhäuser, Basel, Switzerland, 1998. View at Publisher · View at Google Scholar
  21. V. G. Romanovski and D. S. Shafer, The Center and Cyclicity Problems: A Computational Algebra Approach, Birkhäuser, Boston, Mass, USA, 2009. View at Publisher · View at Google Scholar
  22. M. Han, “The Hopf cyclicity of Lienard systems,” Applied Mathematics Letters, vol. 14, no. 2, pp. 183–188, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  23. M. Han, “On some properties and limit cycles of Lienard systems,” Discrete and Continuous Dynamical Systems, pp. 426–434, 2001. View at Google Scholar
  24. Y. Liu, “Theory of center-focus in a class of high order singular points and infinity,” Science in China, vol. 31, pp. 37–48, 2001. View at Google Scholar