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. Joint work with Dieter Schmidt.
Invariant Manifolds in the Hamiltonian-Hopf Bifurcation
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. Joint work with Dieter Schmidt.
