Basic Attractors in Dynamical Systems of Nonlinear PDE's

Bjorn Birnir, Mathematics, UC Santa Barbara

We describe mathematical methods that have been developed over the last ten years to describe the states that the solutions of nonlinear dissipative PDE's settle down to for almost every initial conditions. These methods combine global analysis of the so-called BASIC ATTRACTORS with local analysis; such as linearization about stationary or explicit solutions and perturbation methods. The rather surprising result is a complete qualitative analysis of almost all the solutions that are relevant in experiments and their BIFURCATIONS.

This theory seems to be generic, that is given some general features of the problem one should be able to tell what the results are before carrying out the analysis. We will discuss the possible impact on technology involving nonlinearities and dissipation, by way of examples.