Braided connecting orbits in parabolic equations via computational homology

Abstract

We develop and present a computational method for producing forcing theorems for stationary and periodic solutions and connecting orbits in scalar parabolic equations with periodic boundary conditions. This method is based on prior work by van den Berg, Ghrist, and Vandervorst on a Conley index theory for solutions braided through a collection of known stationary solutions. Essentially, the topological structure of the stationary solutions forces the existence of additional solutions with a specified topological type. In particular, this paper studies connecting orbits and develops and implements the algorithms required to compute the index, providing sample results as illustrations.