Prof. James Meiss, University of Colorado at Boulder

region in phase space to another. As an example, think of the effective

mixing of a passive tracer in a fluid due to stirring. For chaotic dynamical

systems, there are no closed form solutions and numerical methods are

untrustworthy, so the best one can hope for is a statistical

characterization of transport. A region of phase space is deemed "coherent"

if the typical exit time for trajectories is "long". For autonomous

dynamical systems, invariant manifolds can form effective, low-flux

boundaries of such coherent structures.

The study of transport in nonautonomous systems is much more problematic. A

common approach is to use "finite time Lyapunov exponents" to characterize

regions that are approximately coherent. I will discuss the special case of

"transitory" systems---where the time-dependence is confined to a compact

interval--and show that invariant manifolds of past- and future-invariant

regions determine the transport from one to the other. For laminar,

incompressible flows, I will show that a generalized Lagrangian and action

permit the computation of the transported flux. Examples include

two-dimensional flows modeling an oceanic double-gyre and a model particle

accelerator, as well as a three-dimensional model of a microdroplet moving

through a microfluidic channel mixer.