#
Low dimensional structures in dynamical systems with variable time lags

June 7 to June 11, 2010
at the

American Institute of Mathematics,
Palo Alto, California

organized by

John Mallet-Paret,
Roger Nussbaum,
and Hans-Otto Walther

## Original Announcement

This workshop will focus on low (Hausdorff) dimensional structures in
differential delay equations with state dependent time
lags. The study of such structures seems to offer the
best hope of progress in the broader field. We mention
some examples below.
Periodic solutions of differential delay equations
often play a central role in understanding the dynamics
of the equations, but even proving existence of such
solutions may be nontrivial. Related problems include
uniqueness and stability questions, regularity (real
analyticity) of solutions, and limiting shapes of
periodic solutions under singular limits. Nontrivial
examples are already provided by equations such as
$ax'(t) = f(x(t), x(t-r))$, where $a > 0$ and $r := r(x(t))$,
and $f$ and $r$ are given functions.

The local theory of invariant manifolds is reasonably
well-developed for differential delay equations with
constant time lags, but extending the theory to the state
dependent case presents difficulties, e.g., in proving
higher order differentiability of invariant manifolds.
Related problems are also present for compact attractors,
which are typically known to have finite Hausdorff
dimension in the constant time lag case, and suspected
so for the variable time lag case.

The presence of a Morse decomposition for the maximal
attractor occurs for certain classes of systems with a
single delay, or a cyclic structure, with a signed
feedback. Many such systems, under generic conditions,
are of Morse-Smale type and much effort has gone into
studying their global structure. Broadening these results
to systems with multiple delays is a major open question
of much significance, both for the theory and scientific
applications.

One goal of the workshop will be to bring together
established figures in the field with younger researchers
who may become the next generation of leaders. Since many
of the equations which have been extensively studied have
their origin in simplified models from applications,
another objective is to facilitate conversations among
researchers with widely varying degrees of interest in
applications. We expect significant and fruitful advances
in the field to emerge from these interactions.

## Material from the workshop

A list of participants.
The workshop schedule.

A report on the workshop activities.

Papers arising from the workshop: