Invariant Manifolds in the Hamiltonian–Hopf Bifurcation

Abstract

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 $$\nu $$. The eigenvalues of the linearized system are pure imaginary for $$\nu 0$$ (these are the same basic assumptions as found in the Hamiltonian–Hopf bifurcation theorem of the authors). For $$ \nu > 0 $$ the equilibrium has a two-dimensional stable manifold and a two-dimensional unstable manifold, but for $$ \nu < 0 $$ 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 $$\nu $$ 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 $${\mathcal {L}}_4$$ of the restricted three-body problem at the Routh critical value $$\mu _1$$. On the other hand if the sign of this term in the normal form is negative then for $$\nu =0$$ the stable and unstable manifolds persists and then as $$\nu $$ decreases from zero they detach from the equilibrium to follow a hyperbolic periodic solution.