Rigorous Numerics Using Conley Index Theory

Conley index theory as a tool to qualitatively study dynamical systems has been around already for several decades. In recent years more and more applications emerge concerning stable numerical computations. The Conley index is predestined for those applications since it is robust against perturbations. The rigorous numerics method presented in this book is one of these new applications and goes back to an idea of K. Mischaikow and P. Zgliczy 'nski. It allows to determine steady states of a dynamical system given as a partial differential equation (PDE). The method is called rigorous since the calculation does not only give approximations but also implies existence (and sometimes even uniqueness) of these steady states within a small box neighborhood delivered by the algorithm. In this book this method is generalized to PDE with two-dimensional domains and combined with a path-following algorithm in order to rigorously calculate branches of equilibria of parameter dependent equations. All numerical computations are given within the Cahn-Hilliard equation, but the general method and in particular the necessary estimates for cubical nonlinearities generalizes easily to other PDE.