Eusebius Doedel - Böcker

The Institute for Mathematics and its Applications (IMA) devoted its 1997-1998 program to Emerging Applications of Dynamical Systems. Dynamical systems theory and related numerical algorithms provide powerful tools for studying the solution behavior of differential equations and mappings. In the past 25 years computational methods have been developed for calculating fixed points, limit cycles, and bifurcation points. A remaining challenge is to develop robust methods for calculating more complicated objects, such as higher- codimension bifurcations of fixed points, periodic orbits, and connecting orbits, as well as the calcuation of invariant manifolds. Another challenge is to extend the applicability of algorithms to the very large systems that result from discretizing partial differential equations. Even the calculation of steady states and their linear stability can be prohibitively expensive for large systems (e.g. 10_3- -10_6 equations) if attempted by simple direct methods. Several of the papers in this volume treat computational methods for low and high dimensional systems and, in some cases, their incorporation into software packages.A few papers treat fundamental theoretical problems, including smooth factorization of matrices, self -organized criticality, and unfolding of singular heteroclinic cycles. Other papers treat applications of dynamical systems computations in various scientific fields, such as biology, chemical engineering, fluid mechanics, and mechanical engineering.

This IMA Volume in Mathematics and its Applications NUMERICAL METHODS FOR BIFURCATION PROBLEMS AND LARGE-SCALE DYNAMICAL SYSTEMS is based on the combined proceedings of two workshops devoted to compu- tational issues. The workshops were an integral part of the 1997-98 IMA program on "EMERGING APPLICATIONS OF DYNAMICAL SYSTEMS." I would like to thank Donald G. Aronson, University of Minnesota (Mathematics); Wolf-Juergen Beyn, Universitaet Bielefeld (Fakultaet fuer Mathematik); Eusebius Doedel, California Institute of Technology (Applied Mathematics); Bernold Fiedler, Free University of Berlin (Mathematics); H.B. Keller, Caltech (Applied Mathematics); Yannis Kevrekidis, Princeton University (Chemical Enginering); Jens Lorenz, University of New Mex- ico (Mathematics and Statistics); Edriss S. Titi, University of California (Mathematics); Laurette S. Tuckerman, Laboratoire d'Informatique pour la Mecanique et les Sciences de l'Ingenieur (LIMSI) for their excellent work as organizers of the meeting. Special appreciation to Eusebius Doedel and Laurette S. Tuckerman for serving as editors of the proceedings.I also take this opportunity to thank the National Science Foundation (NSF), and the National Security Agency (NSA), whose financial support made the workshop possible.