Acta Mathematica Sinica

, Volume 11, Issue 2, pp 128–136 | Cite as

Bifurcation and chaotic dynamics of homoclinic systems in ℝ3

  • Sun Jianhua 
Article

Abstract

We consider perturbations which may or may not depend explicitly on time for the three-dimensional homoclinic systems. We obtain the existence and bifurcation theorems for transversal homoclinic points and homoclinic orbits, and illustrate our results with two examples.

Keywords

Chaotic Dynamic Homoclinic Orbit Homoclinic Point Bifurcation Theorem Transversal Homoclinic Point 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Sun Jianhua, A criterion for weak Homoclinic attractors in ℝ3, Chin. Sci. Bull.,37:21(1992), 1935–1937.Google Scholar
  2. [2]
    Wiggins, S. and Holmes, P.,Homoclinic orbits in slowly varying oscillators, SIAM. J. Math. Ana.l,18: 3(1987), 612–629.Google Scholar
  3. [3]
    Melnikov, V. K.,On the stability of the center for time periodic perturbations, Trans. Moscow Math. Soc.,12, 1–57.Google Scholar
  4. [4]
    Guckenheimer, J. and Holmes, P. J., Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer-Verlag, New York, 1983.Google Scholar
  5. [5]
    Sun Jianhua,Chaotic behavior in forced vibration systems containing a higher-degree nonlinear term, Ann. of Diff. Eqs.,2: 4(1986), 435–446.Google Scholar
  6. [6]
    Sun Jianhua,Chaotic motions of the pendulum systems, J. Nanjing University Math. Blgu.,4: 1(1987), 43–50.Google Scholar
  7. [7]
    Hirsch, M. W., Pugh, C. C. and Shub, M., Invariant Manifolds, Lecture Notes in Mathematics, No. 583, Springer-Verlag, New York.Google Scholar
  8. [8]
    Newhouse, S. E.,The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms, Publ. Math. IHES,50, 101–151.Google Scholar
  9. [9]
    Sil'nikov, L. P.,A case of the existence of a countable number of periodic motions, Soviet. Math. Dokl., 1–6 (1965).Google Scholar
  10. [10]
    J. H. Sun and D. J. Luo, Local and global bifurcation with nonhyperbolic equilibria, Science in China,37:5(1994), 523–534.Google Scholar
  11. [11]
    J. H. Sun, Heteroclinic bifurcation with nonhyporbolic equilibria in ℝ3, Science in China,24:11(1994), 1145–1151.Google Scholar
  12. [12]
    J. H. Sun, The equivalence for two classes of Melnikov functions, Chin Sci. Bull. (to appear).Google Scholar

Copyright information

© Science Press 1995

Authors and Affiliations

  • Sun Jianhua 
    • 1
  1. 1.Department of MathematicsNanjing UniversityNanjingPeople's Republic of China

Personalised recommendations