Results in Mathematics

, Volume 21, Issue 1–2, pp 211–223 | Cite as

Bifurcation of degenerate homoclinics

  • André Vanderbauwhede


We analyze the continuation and bifurcation of homoclinic orbits near a given degenerate homoclinic orbit. We show that the existence of such degenerate homoclinic orbit is a codimension three phenomenon, and that generically the set of parametervalues at which a nearby homoclinic exists forms a codimension one surface which shows a singularity of Whitney umbrella type at the critical parametervalue. The line of self-intersecting points of such surface corresponds to systems which have two nearby homoclinics.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    W.A. Coppel, Dichotomies in Stability Theory. Lect. Not. in Math. 629, Springer-Verlag, 1978.Google Scholar
  2. [2]
    B. Deng, The Sil’nikov Problem, Exponential Expansion, Strong λ-lemma, C1-Linearization, and Homoclinic Bifurcation. J. Diff. Eqns. 79 (1989), 189-231.Google Scholar
  3. [3]
    C.G. Gibson, Singular Points of Smooth Mappings. Research Not. in Math. 25, Pitman, London, 1979.Google Scholar
  4. [4]
    X.-B. Lin, Using Melnikov’s Method to Solve Silnikov’s Problems. Proc. Roy. Soc. Edinburgh 116A (1990), 295–325.CrossRefGoogle Scholar
  5. [5]
    A. Vanderbauwhede and B. Fiedler, Homoclinic Period Blow-up in Reversible and Conservative Systems. Z. Angew. Math. Phys. (ZAMP), to appear.Google Scholar
  6. [6]
    S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos. Texts in Appl. Math. 2, Springer-Verlag, 1990.Google Scholar

Copyright information

© Birkhäuser Verlag, Basel 1992

Authors and Affiliations

  • André Vanderbauwhede
    • 1
  1. 1.Instituut voor Theoretische MechanicaUniversiteit GentGentBelgium

Personalised recommendations