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 $\nu < 0$ and complex with nonzero real part 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.