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Degenerate Orbit Flip Homoclinic Bifurcations with Higher Dimensions

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Abstract

Bifurcations of a degenerate homoclinic orbit with orbit flip in high dimensional system are studied. By establishing a local coordinate system and a Poincaré map near the homoclinic orbit, the existence and uniqueness of 1–homoclinic orbit and 1–periodic orbit are given. Also considered is the existence of 2–homoclinic orbit and 2–periodic orbit. In additon, the corresponding bifurcation surfaces are given.

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References

  1. Deng, B.: Silnikov problem, exponential expansion, strong λ–lemma, C 1–linearization and homoclinic bifurcation. J. Diff. Equs., 79(2), 189–231 (1989)

    Article  MATH  Google Scholar 

  2. Chow, S. N., Deng, B., Fiedler, B.: Homoclinic bifurcation at resonant eignvalues. J. Dyna. Syst. and Diff. Equs., 2(2), 177–244 (1990)

    Article  MATH  Google Scholar 

  3. Palmer, K. J.: Exponential dichotomies and transversal homoclinic points. J. Diff. Equs., 55(2), 220–256 (1984)

    MathSciNet  Google Scholar 

  4. Gruendler, J.: Homoclinic solutions for autonomous dynamical systems in arbitrary dimension. SIAM J. Math. Anal., 23, 702–721 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  5. Gruendler, J.: Homoclinic solutions for autonomous ordinary differential equations with nonautonomous perturbations. J. Diff. Equs., 122(1), 1–26 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  6. Zhu, D. M.: Problems in homoclinic bifurcations with higher dimensions. Acta Mathematica Sinica, New Series, 14(3), 341–352 (1998)

    MATH  MathSciNet  Google Scholar 

  7. Zhu, D. M., Xia, Z. H.: Bifurcations of heteroclinic loops. Science in China, 41(8), 837–848 (1998)

    Article  MATH  Google Scholar 

  8. Zhu, D. M.: Exponential trichotomy and heteroclinic bifurcation. Nonlinear Analysis TMA, 28(3), 547–557 (1997)

    Article  MATH  Google Scholar 

  9. Zhu, D. M.: Melnikov type vectors and principal normals. Science in China, 37(9), 814–822 (1994)

    MATH  Google Scholar 

  10. Zhu, D. M.: Transversal heteroclinic orbits in general degenerate cases. Science in China, 39A(2), 113–121 (1996)

    Google Scholar 

  11. Jin, Y. L., Zhu, D. M.: Degenerated homoclinic bifurcations for degenerated case. Chin. Ann. of Math., 21B(2), 201–210 (2000)

    Article  Google Scholar 

  12. Jin, Y. L., Li, X. Y., Liu, X. B.: Nontwisted homoclinic bifurcations for degenerated case. Chin. Ann. of Math., 22A(4), 473–479 (2001)

    Google Scholar 

  13. Jin, Y. L., Zhu, D. M.: Bifurcations of rough heteroclinic loop with two saddle points. Science in China, 46A(4), 459–468 (2003)

    Article  Google Scholar 

  14. Sun, J. H., Luo, D.J.: Local and global bifurcations with nonhyperbolic equilibria. Science in China, 37A(5), 523–534 (1994)

    Google Scholar 

  15. Sun, J. H.: Heteroclinic bifurcations with nonhyperbolic equilibria in R n. Science in China, 24, 1145–1151 (1994)

    Google Scholar 

  16. Sun, J. H., Kooij, R. E.: Bifurcations to a heteroclinic manifold with nonhyperbolic equilibria in R n. Acta Mathematica Scientia, 18(3), 293–302 (1998)

    MATH  Google Scholar 

  17. Luo, D. J., Wang, X., Zhu, D. M., Han, M. A.: Bifurcation theory and methods of dynamical systems, Advanced Series in Dynamical Systems, Vol 15, World Sciectific, Singapore, 1997

  18. Wiggins, S.: Global bifurcations and chaos–analytical methods, Springer–Verlag, New York, 1988

  19. Sandstede, B.: Constructing dynamical systems having homoclinic bifurcation points of codimension two. J. Dyna. Syst. Diff. Equs., 9, 269–288 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  20. Kisaka, M., Kokubu, H., Oka, H.: Bifurcations to N– homoclinic orbits and N–periodic orbits in vector fields. J. Dyna. Syst. Diff. Equs., 5, 305–357 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  21. Homburg, A. J., Krauskopf, B.: Resonant homoclinic flip bifurcations. J. Dyna. Syst. Diff. Equs., 12, 807–850 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  22. Homburg, A. J., Kokubu, H., Naudot, V.: Homoclinic doubling cascades. Arch. Rational Mech. Anal., 160, 195–243 (2001)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Nanjing University, Nanjing 210093, P. R. China

    Ran Chao Wu & Jian Hua Sun

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  1. Ran Chao Wu
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  2. Jian Hua Sun
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Correspondence to Ran Chao Wu or Jian Hua Sun.

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Project supported by the National Natural Science Foundation of China (No: 10171044), the Natural Science Foundation of Jiangsu Province (No: BK2001024), the Foundation for University Key Teachers of the Ministry of Education of China

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Wu, R.C., Sun, J.H. Degenerate Orbit Flip Homoclinic Bifurcations with Higher Dimensions. Acta Math Sinica 22, 1651–1656 (2006). https://doi.org/10.1007/s10114-005-0705-3

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  • DOI: https://doi.org/10.1007/s10114-005-0705-3

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