Bayliss and Matkowsky (1992) and Bayliss, Matkowsky and Riecke (1993) have published studies of numerical simulations using full equations that describe the thermodiffusive instability. In both studies they considered a one-dimensional circular (ribbon) flame concentric with a source of premixed gas directed outward along the radius. Steady ordered states with cell numbers from three to nine were found in the parameter range considered. Our preliminary experiments using a porous plug in the shape of a slot have also found that a one-dimensional ordered array of cells is steady and does not have chaotic motion, suggesting that the motion observed on the circular burner is a two-dimensional effect.
Nicolenko (1993) conducted numerical studies of the two-dimensional Kuramoto-Sivashinsky equation using a square geometry with periodic boundary conditions. His results capture four elements of the phenomenology of ordered states observed in our experiments: the metastability of a state with two inner cells, a maximum of six cells forming around a point, a transition from order to disorder, and chaotic dynamics of ordered states with motion concentrated around the cusps and folds.