Conley index techniques for computing global structure: a study of the Swift-Hohenberg equation

In this talk, we will discuss a rigorous numerical method for the study and verification of global dynamics for gradient systems. The procedure involved relies on first verifying the structure of the set of stationary solutions, as often depicted in a bifurcation diagram produced via continuation methods. This includes proving the existence and uniqueness of computed branches of the diagram as well as showing the nonexistence of additional stationary solutions. Topological information in the form of the Conley index is then used to build a model for the set of bounded solutions that form an attractor for the system. As illustration, this method is used to produce a (semi-) conjugacy between an attractor for the Swift-Hohenberg equation and a constructed model system.