Local separatrices for Hamiltonians with symmetries

Abstract

We consider dynamics generated by Hamiltonians with three degrees of freedom and symmetries. It is shown that locally, away from a possible saddle equilibrium, some codimension-1 invariant manifold may exist. They are stable/unstable manifolds of a codimension-2 hyperbolic invariant manifold. This structure appears when some periodic orbits constitutive of the Arnold web have bifurcated and become linearly unstable. This result generalizes the existence of normally hyperbolic invariant manifolds and their codimension-1 stable/unstable manifolds in the vicinity of an unstable ⊗ (stable)2 equilibrium point.