Computation of Connection Matrices Using the Software Package Conley

In this paper we demonstrate the power of the computer algebra package conley, which enables one to compute connection and transition matrices, two of the main algebraic tools of the Conley index theory. In particular, we study the Cahn-Hilliard equation on the unit square and extend the results obtained in [Maier-Paape et al., 2007] to a bigger range of the bifurcation parameter. Besides providing several explicit computations using conley, the definition of connection matrices is reconsidered, simplified, and presented in a self-contained manner in the language of Conley index theory. Furthermore, we introduce so-called energy induced bifurcation intervals, which can be utilized by conley to differential equations with a parameter. These bifurcation intervals are used to automatically path-follow the set of connection matrices at bifurcation points of the underlying set of equilibria.