Characterizations of Complex Patterns
Gemunu H. Gunaratne
University of Houston
Local bifurcations from uniform states to structures consisting of one or more planforms have been the subject of many studies. However, patterns generated in large, finite systems exhibit a far more complicated structure that depends on the initial states and boundary conditions. In order to conduct theoretical analyses of these complex patterns, it is necessary to be able to characterize those features which change with system parameters, but are independent of the initial (typically noisy) configurations.
We show how equivariance under Euclidean transformations of the pattern can be used to identify several classes of such "configuration independent characteristics." As an application, we study properties of the relaxation of a noisy initial state under the Swift-Hohenberg equation. These conclusions are validated in an experiment on a vibrated layer of granular material. A second class of characteristics is used to analyze the growth of surfaces under a model system that represents epitaxial growth. For this case, the patterns do not have a labyrinthine structure.