__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.