9501045 Kupiainen The project uses methods previously developed in Statistical Mechanics for the study of dynamical systems with spatially extended attractors. The first set of problems deals with the formation of patterns and fronts in nonlinear parabolic PDE's. The Renormalization Group will be used to study universal properties of fronts and patterns in the Cahn-Hilliard and related equations. The second set of problems deals with chaotic infinite dimensional dynamics. Expansion methods will be used to study the phenomena of spatio-temporal chaos and phase transitions in coupled map lattices and probabilistic cellular automata. Extended dynamical systems model regular and chaotic phenomena that are present in macroscopic systems that are composed of a very large number of identical parts such as liquids, gases and some biological and ecological systems. Both the formation of regular patterns (real life examples are waves in oceans, convective phenomena in atmosphere and patterns in clouds) and irregular, chaotic phenomena which exhibit chaos as irregular behaviour both in time and in space (for example turbulent motion of fluids) will be studied. Previously developed methods for problems in other areas of mathematical physics will be used to study universal aspectsof these phenomena, i.e. aspects that are rather insensitive to specific models that are being used. ***
