Perturbations of geodesic flows by recurrent dynamics

Abstract

We present a mechanism for Arnold diffusion based on intertwining homoclinic orbits to a normally hyperbolic invariant manifold, followed for long time intervals, with orbits of the dynamics restricted to that manifold, followed for short time intervals. The resulting trajectories are rather fast, and their construction is explicit, so they can be used in concrete applications.

The method to construct such orbits relies mainly on correctly aligned windows and does not require the use of delicate analytical methods such as averaging or KAM, nor the convexity assumptions of variational methods. It only requires control of the inner dynamics over short times, and so it allows easy verification of the hypotheses with finite precision numerical calculations.

As an illustration of this mechanism, we consider a geodesic flow on a compact manifold with a generic (Riemannian, Finsler or Lorentz) metric, subject to time-dependent perturbations via a generic potential. The genericity conditions are given explicitly. The perturbation is driven by a flow on an external manifold which satisfies some mild recurrence condition. Periodic and quasiperiodic perturbations are included, and there are many others.We show the existence of trajectories whose energy grows to infinity, as well as of trajectories that follow prescribed energy paths. The trajectories can be constructed to diffuse at speeds that match upper bounds.

While the study of perturbations of the geodesic flow is a classical problem, we use it mainly to showcase the main ingredients of the method in a simple way. Other results related to ours exist in the literature, but the trajectories we obtain are different from the previously constructed ones. Our approach is flexible enough to be widely applicable; some applications to the restricted three-body problem involving also numerical calculations are in progress.