Prof. James Meiss, University of Colorado at Boulder

Coherent Structures and Transport in Transitory Dynamical Systems

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.