Spatio-Temporal Dynamics of Reaction-Diffusion Patterns

In this survey we look at parabolic partial differential equations from a dynamical systems point of view. With origins deeply rooted in celestial mechanics, and many modern aspects traceable to the monumental influence of Poincare, dynamical systems theory is mainly concerned with the global time evolution T(t)u 0 of points u 0 — and of sets of such points — in a more or less abstract phase space X. The success of dynamical concepts such as gradient flows, invariant manifolds, ergodicity, shift dynamics, etc. during the past century has been enormous — both as measured by achievement, and by vitality in terms of newly emerging questions and long-standing open problems.