Computational and Applied Math Proseminar

Department of Mathematics and Statistics
Arizona State University

Thursday, March 6, 2003, 12:15 p.m. in GWC 604

Tim Callahan

Department of Mathematics and Statistics

Hopf Bifurcations on Cubic Lattices

Abstract Systems that spontaneously form patterns periodic in two dimensions have been known for over a century, and the large number of such systems has spurred extensive research. Reaction-diffusion systems, such as the Turing instability, differ from many other such systems in that their characteristic length scale is intrinsic, i.e., dependent upon the chemical reaction rates and diffusivities and independent of the geometry of the experimental apparatus. These systems can therefore be studied far away from any boundary, and three-dimensional patterns can result.

We analyze three-dimensional pattern forming bifurcations with the spatial periodicity of several cubic lattices. This is an equivariant bifurcation problem with a large symmetry group. The Equivariant Branching Lemma and Equivariant Hopf Theorem guarantee that certain isotropy subgroups have unique smooth primary solution branches in the bifurcation diagram, but finding all such solutions is a daunting task. Fortunately, it can be made easier by first extending the group to an even larger one. We use this method to find all the solutions guaranteed by the EBL and EHT and their branching solutions and stability criteria. We also show how to choose appropriate group extensions.

For further information please contact: mittelmann@asu.edu