Irving R. Epstein and Martin Golubitsky
Symmetric Patterns in Linear Arrays of Coupled Cells
Chaos 3(1) (1993) 1-5
In this note we show how to find patterned solutions in linear arrays of
coupled cells. The solutions are found by embedding the system in a circular
array with twice the number of cells. The individual cells have a unique
steady state, so that the patterned solutions represent a discrete analog of
Turing structures in continuous media. We then use the symmetry of the
circular array (and bifurcation from an invariant equilibrium) to identify
symmetric solutions of the circular array that restrict to solutions of the
original linear array. We apply these abstract results to a system of coupled
Brusselators to prove that patterned solutions exist. In addition, we show,
in certain instances, that these patterned solutions can be found by numerical
integration and hence are presumably asymptotically stable.