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.