Invariant Manifolds in the Hamiltonian–Hopf Bifurcation

• Meyer, Kenneth R ^[1] ; Schmidt, Dieter S ^[1]
  1. [1] University of Cincinnati

     University of Cincinnati

     City of Cincinnati, Estados Unidos

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 19, Nº 1, 2020
• Idioma: inglés
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• Resumen

  • We study the evolution of the stable and unstable manifolds of an equilibrium point of a Hamiltonian system of two degrees of freedom which depends on a parameter ν. The eigenvalues of the linearized system are pure imaginary for ν<0 and complex with nonzero real part for ν>0 (these are the same basic assumptions as found in the Hamiltonian–Hopf bifurcation theorem of the authors). For ν>0 the equilibrium has a two-dimensional stable manifold and a two-dimensional unstable manifold, but for ν<0 there are no longer stable and unstable manifolds attached to the equilibrium. We study the evolution of these manifolds as the parameter is varied. If the sign of a certain term in the normal form is positive then for small positive ν the stable and unstable manifolds of the system are either identical or must have transverse intersection. Thus, either the system is totally degenerate or the system admits a suspended Smale horseshoe as an invariant set. This happens at the Lagrange equilibrium point L4 of the restricted three-body problem at the Routh critical value μ1. On the other hand if the sign of this term in the normal form is negative then for ν=0 the stable and unstable manifolds persists and then as ν decreases from zero they detach from the equilibrium to follow a hyperbolic periodic solution.