List All Events

We consider an integrable Hamiltonian system subject to a small, time-periodic perturbation. We assume that the perturbed system has a normally hyperbolic invariant manifold (NHIM) whose stable and unstable manifolds intersect transversally. Associated to each transverse intersection one can define a scattering map, which gives the future asymptotic of a homoclinic orbit as a function of its past asymptotic.  We assume that we have a system of such scattering maps. We provide results on the geometric controllability of the system. We show that, under explicit conditions on the scattering maps and on the inner dynamics (restricted to the NHIM),  for any two points on the NHIM, there is an orbit of the Hamiltonian flow that goes from near the first point to near the second point.  Also, for any path on the NHIM, there is an orbit of the Hamiltonian flow that shadows that path.  The upshot is that we use the perturbation as a controller.