Numerical Methods for Bifurcation Problems and Large-Scale Dynamical Systems

The Institute for Mathematics and its Applications (IMA) devoted its 1 997-1998 program to Emerging Applications of Dynamical Systems. Dynami cal systems theory and related numerical algorithms provide powerful t ools for studying the solution behavior of differential equations and mappings. In the past 25 years computational methods have been develop ed for calculating fixed points, limit cycles, and bifurcation points. A remaining challenge is to develop robust methods for calculating mo re complicated objects, such as higher-codimension bifurcations of fix ed points, periodic orbits, and connecting orbits, as well as the calc uation of invariant manifolds. Another challenge is to extend the appl icability of algorithms to the very large systems that result from dis cretizing partial differential equations. Even the calculation of stea dy states and their linear stability can be prohibitively expensive fo r large systems (e.g. 10_3-10_6 equations) if attempted by simple dire ct methods.