Geometry, Topology and Dynamics Seminar
Marian Gidea
Northeastern Illinois University
Topological methods in the large gap problem of Arnold diffusion
Abstract: We extend and simplify some current proofs on the existence of
diffusing orbits in Hamiltonian systems. We assume the existence
of a normally hyperbolic manifold with transverse homoclinic
intersections and study the interplay between the dynamics
restricted to the normally hyperbolic manifold and another
dynamics given by the homoclinic excursions. We assume the
existence of almost invariant tori on the normally hyperbolic
manifold. These tori are whiskered tori in the full system and
have heteroclinic intersections. We establish that there are
orbits of the system which follow transition chains.
The main tool is an extension of the windowing mechanism introduced by Easton. The windowing argument shows that, if there exists sequences of windows whose images go through other windows, then there is an orbit that shadows the windows. One advantage of the method presented here is that, since the boxes are very robust and the argument is topological, it requires significantly fewer analytic arguments.
Monday March 6, 2006 at 3:00 PM in SEO 512