Computational Challenges of Nonlinear Systems

Abstract

We survey some of the major types of dynamical-systems computations that can be carried out for two or three-dimensional systems of partial differential equations. In order of increasing complexity, we describe methods for calculating steady states and bifurcation diagrams, linear stability and Floquet analysis, and heteroclinic orbits. These are illustrated by computations for Rayleigh–Benard convection in a cylindrical geometry, the Faraday instability of a fluid layer, the flow past a cylinder and over a square cavity, flow in a cylindrical container with counter-rotating lids, and Bose–Einstein condensation. We discuss some mathematical questions raised by these computations and the need for improved numerical tools.