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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 152518, 11 pages
http://dx.doi.org/10.1155/2013/152518
Research Article

The Twisting Bifurcations of Double Homoclinic Loops with Resonant Eigenvalues

1School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin 130024, China
2School of Information Engineering, China University of Geosciences (Beijing), Beijing 100083, China
3School of Mathematics and Statistics, Central China Normal University, Wuhan, Hubei 430079, China

Received 19 February 2013; Accepted 22 April 2013

Academic Editor: Maoan Han

Copyright © 2013 Xiaodong Li 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. Blázquez-Sanz and K. Yagasaki, “Analytic and algebraic conditions for bifurcations of homoclinic orbits I: saddle equilibria,” Journal of Differential Equations, vol. 253, no. 11, pp. 2916–2950, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  2. S.-N. Chow, B. Deng, and B. Fiedler, “Homoclinic bifurcation at resonant eigenvalues,” Journal of Dynamics and Differential Equations, vol. 2, no. 2, pp. 177–244, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. S.-N. Chow and X.-B. Lin, “Bifurcation of a homoclinic orbit with a saddle-node equilibrium,” Differential and Integral Equations, vol. 3, no. 3, pp. 435–466, 1990. View at Zentralblatt MATH · View at MathSciNet
  4. B. Deng, “Homoclinic twisting bifurcations and cusp horseshoe maps,” Journal of Dynamics and Differential Equations, vol. 5, no. 3, pp. 417–467, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. J. Gruendler, “Homoclinic solutions for autonomous dynamical systems in arbitrary dimension,” SIAM Journal on Mathematical Analysis, vol. 23, no. 3, pp. 702–721, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. J. Gruendler, “Homoclinic solutions for autonomous ordinary differential equations with nonautonomous perturbations,” Journal of Differential Equations, vol. 122, no. 1, pp. 1–26, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. A. J. Homburg, “Global aspects of homoclinic bifurcations of vector fields,” Memoirs of the American Mathematical Society, vol. 121, no. 578, 1996. View at Zentralblatt MATH · View at MathSciNet
  8. 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
  9. Y. L. Jin and D. M. Zhu, “Bifurcations of rough heteroclinic loop with two saddle points,” Science in China A, vol. 46, no. 4, pp. 459–468, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. 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
  11. H. Kokubu, M. Komuro, and H. Oka, “Multiple homoclinic bifurcations from orbit-flip. I. Successive homoclinic doublings,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 6, no. 5, pp. 833–850, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. J. Knobloch and T. Wagenknecht, “Homoclinic snaking near a heteroclinic cycle in reversible systems,” Physica D, vol. 206, no. 1-2, pp. 82–93, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. X. Liu and D. Zhu, “Bifurcation of degenerate homoclinic orbits to saddle-center in reversible systems,” Chinese Annals of Mathematics. Series B, vol. 29, no. 6, pp. 575–584, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. 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
  15. B. Sandstede, “Constructing dynamical systems having homoclinic bifurcation points of codimension two,” Journal of Dynamics and Differential Equations, vol. 9, no. 2, pp. 269–288, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. S. Schecter and C. Sourdis, “Heteroclinic orbits in slow-fast Hamiltonian systems with slow manifold bifurcations,” Journal of Dynamics and Differential Equations, vol. 22, no. 4, pp. 629–655, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. S. L. Shui and D. M. Zhu, “Codimension 3 nonresonant bifurcations of homoclinic orbits with two inclination flips,” Science in China A, vol. 48, no. 2, pp. 248–260, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. T. Wagenknecht, “Two-heteroclinic orbits emerging in the reversible homoclinic pitchfork bifurcation,” Nonlinearity, vol. 18, no. 2, pp. 527–542, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. S. Wiggins, Global Bifurcations and Chaos, vol. 73, Springer, New York, NY, USA, 1988. View at Publisher · View at Google Scholar · View at MathSciNet
  20. S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, vol. 2, Springer, New York, NY, USA, 1990. View at MathSciNet
  21. T. Zhang and D. Zhu, “Codimension 3 homoclinic bifurcation of orbit flip with resonant eigenvalues corresponding to the tangent directions,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 14, no. 12, pp. 4161–4175, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. D. M. Zhu, “Problems in homoclinic bifurcation with higher dimensions,” Acta Mathematica Sinica, vol. 14, no. 3, pp. 341–352, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. 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
  24. M. Han and P. Bi, “Bifurcation of limit cycles from a double homoclinic loop with a rough saddle,” Chinese Annals of Mathematics B, vol. 25, no. 2, pp. 233–242, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. M. A. Han and J. Chen, “On the number of limit cycles in double homoclinic bifurcations,” Science in China A, vol. 43, no. 9, pp. 914–928, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. A. J. Homburg and J. Knobloch, “Multiple homoclinic orbits in conservative and reversible systems,” Transactions of the American Mathematical Society, vol. 358, no. 4, pp. 1715–1740, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. C. A. Morales, M. J. Pacifico, and B. San Martin, “Expanding Lorenz attractors through resonant double homoclinic loops,” SIAM Journal on Mathematical Analysis, vol. 36, no. 6, pp. 1836–1861, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. C. A. Morales, M. J. Pacifico, and B. San Martin, “Contracting Lorenz attractors through resonant double homoclinic loops,” SIAM Journal on Mathematical Analysis, vol. 38, no. 1, pp. 309–332, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. Q. Lu, “Codimension 2 bifurcation of twisted double homoclinic loops,” Computers & Mathematics with Applications, vol. 57, no. 7, pp. 1127–1141, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. C. G. Ragazzo, “On the stability of double homoclinic loops,” Communications in Mathematical Physics, vol. 184, no. 2, pp. 251–272, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. W. P. Zhang and D. M. Zhu, “Codimension 2 bifurcations of double homoclinic loops,” Chaos, Solitons and Fractals, vol. 39, no. 1, pp. 295–303, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  32. W. Zhang, D. Zhu, and D. Liu, “Codimension 3 nontwisted double homoclinic loops bifurcations with resonant eigenvalues,” Journal of Dynamics and Differential Equations, vol. 20, no. 4, pp. 893–908, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet