Speaker:
Prof. James Meiss, University of Colorado at Boulder

Title:
Coherent Structures and Transport in Transitory Dynamical Systems

Abstract:
The problem of transport is to determine how long it takes to get from one
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.