Journal of Dynamics and Differential Equations

, Volume 4, Issue 1, pp 95–126

Cascades of homoclinic orbits to, and chaos near, a Hamiltonian saddle-center

  • A. Mielke
  • P. Holmes
  • O. O'Reilly
Article

DOI: 10.1007/BF01048157

Cite this article as:
Mielke, A., Holmes, P. & O'Reilly, O. J Dyn Diff Equat (1992) 4: 95. doi:10.1007/BF01048157

Abstract

We consider a class of reversible, two-degree of freedom Hamiltonian systems possessing homoclinic orbits to a saddle-center: an equilibrium having two non-zero real and two nonzero imaginary eigenvalues. Under mild nondegeneracy conditions, we construct a two-parameter unfolding and show that there is a countable infinity of “secondary” homoclinic bifurcations in any neighborhood of the original system. We also demonstrate the existence of families of periodic orbits and of shifts on two symbols (horseshoes). The lack of hyperbolicity and the presence of conserved quantities make the analysis somewhat delicate. We discuss specific examples for which the nondegeneracy conditions can be explicitly checked but indicate that this is not always possible. We illustrate our results with numerical work.

Key words

Homoclinic orbitbifurcationreversibleHamiltoniansaddle-centerperiodic orbithorseshoeShil'nikov phenomena

AMS subject classifications

34 C 25-2858 F 0558 F 1458 F 30

Copyright information

© Plenum Publishing Corporation 1992

Authors and Affiliations

  • A. Mielke
    • 1
  • P. Holmes
    • 3
  • O. O'Reilly
    • 3
  1. 1.Mathematisches Institut AUniversität StuttgartStuttgart 80Germany
  2. 2.Mathematisches InstitutUniversität zu KölnKöln 41Germany
  3. 3.Department of Theoretical and Applied Mechanics and Mathematical Sciences InstituteCornell UniversityIthaca
  4. 4.Institut für MechanikETH-ZentrumZürichSwitzerland