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Abstract and Applied Analysis
Volume 2013, Article ID 267826, 7 pages
http://dx.doi.org/10.1155/2013/267826
Research Article

Resonant Homoclinic Flips Bifurcation in Principal Eigendirections

1College of Science, University of Shanghai for Science and Technology, Shanghai 200093, China
2Department of Mathematics, East China Normal University, Shanghai 200062, China

Received 12 September 2013; Accepted 19 November 2013

Academic Editor: Svatoslav Staněk

Copyright © 2013 Tiansi Zhang 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. M. Kisaka, H. Kokubu, and H. Oka, “Bifurcations to N-homoclinic orbits and N-periodic orbits in vector fields,” Journal of Dynamics and Differential Equations, vol. 5, no. 2, pp. 305–357, 1993. View at Publisher · View at Google Scholar · View at MathSciNet
  2. M. Kisaka, H. Kokubu, and H. Oka, “Supplement to homoclinic-doubling bifurcation in vector fields,” in Handbook of Dynamical Systems, pp. 92–116, 1993. View at Google Scholar
  3. V. Naudot, “Strange attractor in the unfolding of an inclination-flip homoclinic orbit,” Ergodic Theory and Dynamical Systems, vol. 16, no. 5, pp. 1071–1086, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. C. A. Morales and M. J. Pacifico, “Inclination-flip homoclinic orbits arising from orbit-flip,” Nonlinearity, vol. 14, no. 2, pp. 379–393, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. V. Naudot, “A strange attractor in the unfolding of an orbit-flip homoclinic orbit,” Dynamical Systems, vol. 17, no. 1, pp. 45–63, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. A. J. Homburg and B. Krauskopf, “Resonant homoclinic flip bifurcations,” Journal of Dynamics and Differential Equations, vol. 12, no. 4, pp. 807–850, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. B. E. Oldeman, B. Krauskopf, and A. R. Champneys, “Numerical unfoldings of codimension-three resonant homoclinic flip bifurcations,” Nonlinearity, vol. 14, no. 3, pp. 597–621, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. T. S. Zhang and D. M. Zhu, “Homoclinic bifurcation of orbit flip with resonant principal eigenvalues,” Acta Mathematica Sinica, vol. 22, no. 3, pp. 855–864, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. T. Zhang and D. Zhu, “Bifurcations of homoclinic orbit connecting two nonleading eigendirections,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 17, no. 3, pp. 823–836, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. T. S. Zhang and D. M. Zhu, “Bifurcation analysis of homoclinic flips at principal eigenvalues resonance,” Applied Mathematics, vol. 4, pp. 271–278, 2013. View at Google Scholar
  11. F. Geng, D. Liu, and D. Zhu, “Bifurcations of generic heteroclinic loop accompanied by transcritical bifurcation,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 18, no. 4, pp. 1069–1083, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. Q. Lu, Z. Qiao, T. Zhang, and D. Zhu, “Heterodimensional cycle bifurcation with orbit-flip,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 20, no. 2, pp. 491–508, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. X. Liu, “Homoclinic flip bifurcations accompanied by transcritical bifurcation,” Chinese Annals of Mathematics B, vol. 32, no. 6, pp. 905–916, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. D. Zhu and Z. Xia, “Bifurcations of heteroclinic loops,” Science in China A, vol. 41, no. 8, pp. 837–848, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. S. Wiggins, Global Bifurcations and Chaos, vol. 73, Springer, New York, NY, USA, 1988. View at Publisher · View at Google Scholar · View at MathSciNet