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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.
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