Topological Analysis of Patterns

Abstract

We use computational homology to characterize the geometry of complicated time-dependent patterns. Homology provides very basic topological (geometrical) information about the patterns, such as the number of components (pieces) and the number of holes. For 3-dimensional patterns it also provides the number of voids. We apply these techniques to patterns generated by experiments on spiral defect chaos, as well as to numerically simulated patterns in the Cahn-Hilliard theory of phase separation and on spiral wave patterns in excitable media. These techniques allow us to distinguish patterns at different parameter values, to detect complicated dynamics through the computation of positive Lyapunov exponents and entropies, to compare experimental data with numerical simulations, to quantify boundary effects on finite size domains, among other things.