8901273 Mischaikow is developing an approach to studying the structure of isolated invariant sets which is applicable to a wide variety of problems, e.g. symmetric systems, travelling waves for reaction - diffusion systems, dynamic phase transitions, magnetohydrodynamic shock waves, delay equations, predator-prey models, to name a few. More specifically Mischaikow is concerned with deriving algebraic invariants which describe invariant set, determining how they can change through bifurcations, and determining which geometric structures are imposed on the invariant sets by the algebraic invariants. The guiding philosophy in this development comes from the fact that many differential equations that arise as models of biological, chemical, or physical systems share the following three characteristics: (1) they are derived from idealized descriptions of complicated phenomena, (2) they are parameterized systems and the exact values of the parameters are not known, and (3) they are non-linear. Thus, from an applied point of view describing the global and robust structures of the solutions to these differential equations is of great importance. Mischaikow's research makes extensive use of the Conley index and, in particular, of a relatively new tool called the connection matrix.
