- 04-280 A. Delshams, R. de la Llave, T. M.-Seara
- Orbits of unbounded energy in quasi-periodic perturbations of
Sep 9, 04
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Abstract. We show that certain mechanical systems, including a geodesic
flow in any dimension plus a quasi-periodic perturbation by a
potential, have orbits of unbounded energy.
The assumptions we make in the case of geodesic flows are:
metric and the external perturbation are smooth enough.
geodesic flow has a hyperbolic periodic orbit such that its stable
and unstable manifolds have a tranverse homoclinic intersection.
The frequency of the external perturbation is Diophantine.
\item[d)] The external potential satisfies a generic condition
depending on the periodic orbit considered in b).
The assumptions on the metric are $\C^2$ open and are known to be
dense on many manifolds. The assumptions on the potential fail
only in infinite codimension spaces of potentials.
The proof is based on geometric considerations of invariant
manifolds and their intersections. The main tools include the
scattering map of normally hyperbolic invariant manifolds, as well
as standard perturbation theories (averaging, KAM and Melnikov
We do not need to assume that the metric is Riemannian and we
obtain results for Finsler or Lorentz metrics. Indeed, there is a
formulation for Hamiltonian systems satisfying scaling hypotheses.
We do not need to make assumptions on the global topology of the
manifold nor on its dimension.