Thematic Year on Emerging Applications of Dynamical Systems

IMA Program for September 1997 - June 1998

Organizers:

Name                         Present Institution

Rafael de la Llave           Univ. of Texas, Austin
Eusebius Doedel              Concordia University
Martin Golubitsky            University of Houston
John Guckenheimer (Chair)    Cornell University
Yannis Kevrekedis            Princeton University
John Rinzel                  National Institutes of Health

The year is divided into three components:

Fall Quarter, September 1 - December 30, 1997:
Numerical Analysis of Dynamical Systems

Winter Quarter, Janurary 2 - March 31, 1998:
Dynamics in Physiology and Chemistry

Spring Quarter, April 1 - June 30, 1998:
Symmetry and Pattern Formation

Dynamical systems theory describes general patterns found in the solutions of systems of nonlinear differential equations. The theory focuses upon those equations representing the change of processes in time. Geometric and analytic study of simple examples has led to tremendous insight into universal aspects of nonlinear dynamics. Experimental studies in diverse areas ranging from fluid flows to chemical reactions to laser dynamics to cardiac rhythms to neural output have confirmed the ubiquity of these dynamical patterns. Harnessing theoretical advances in the mathematics for the solution of larger, more complex practical problems requires further effort in understanding algorithmic and computational issues related to dynamical systems, extensions of the theory to important classes of systems that arise in applications, and attention to the modeling of complex systems that are accessible to only limited measurements of their components.

Work at applying the methods developed by dynamical systems theory to ``real world'' problems has been a thoroughly interdisciplinary effort. For over fifteen years, there has been a lively dialogue between mathematicians, scientists and engineers concerning the observation and interpretation of dynamical patterns in laboratory and natural systems. To some extent, missing from this discussion has been a set of quantitative models that accurately represent the behavior of the observed systems. The patterns identified by the theory are qualitative, and frequently the theory has been used to classify patterns rather than to build models that can be used for purposes of design or prediction. Computational capabilities have been a limiting factor in constructing such models since they seldom lend themselves to solution solely with analytic methods.

This proposal offers a set of activities that address the issue of applying dynamical systems methods to a wider circle of problems. There are three components to our approach: a focus on the algorithms that underlie the computation of system behavior, a focus on particular application areas that appear timely for rapid scientific advances through the use of dynamical systems methods, and emphasis upon areas in which existing mathematical theory provides an inadequate substrate for work with applications. The application areas we have selected involve physiological and chemical processes.

The year has been divided into three segments, with a total of seven workshops and a further week long program of concentrated activity on a smaller scale than the workshops. We intend to work with the Geometry Center on sponsorship of the activities that fall into areas of mutual interest. The workshops are designed with a focal point that is complementary to those of other meetings that have been held in recent years. In each case, we endeavor to bring together groups whom we feel have overlapping interests but tend to move in disjoint scientific circles. Also, we will work to put traditional researchers in dynamical systems in contact with these new areas of activity.

Fall Program, September 1 - December 30, 1997:
Numerical Analysis of Dynamical Systems

Winter Program, Janurary 2 - March 31, 1998:
Dynamics in Physiology and Chemistry

Spring Program, April 1 - June 30, 1998:
Symmetry and Pattern Formation

Fall 1997

Numerical Analysis of Dynamical Systems