Speaker:
Date of Talk:
9/30/10
Affiliation:
Department of Applied Mathematics, University of Colorado- Boulder
Title:
The Last Invariant Torus in Volume Preserving Maps
Abstract
Invariant tori are prominent features of both symplectic and volume
preserving maps. An integrable map with one "action" and /n/-1 "angle"
variables, has a family of /n/-1 dimensional tori that separate phase
space. When such a map is perturbed some tori are destroyed, but, when
KAM theory applies, some tori (a cantor set) persist. Typically,
however, when the perturbation is strong enough all of the tori are
destroyed, and for a critical strength there is one remaining,
most-robust torus. For the two-dimensional case, John Greene discovered
a remarkable self-similarity near this "last" invariant torus that is
reflected in the numerology of the golden mean.
Is there a similar phenomena in the higher-dimensional case? I have been
running numerical experiments to study the break-up of tori for
three-dimensional, volume-preserving maps and will report on several
discoveries and many frustrations so far.
