__Complex
dynamics in a simple model of solid flame microstructure__

__Jeffrey Beck__**,** Northwestern University

We present a simple model of condensed phase combustion which attempts
to elucidate the effects of spatially localized reaction sites on the frontal
propagation of a combustion wave. Heat
transfer is assumed to be uniform but the exothermic reactions which drive the
wave are allowed to occur only at discrete but evenly distributed
locations. Combined with ignition
temperature kinetics this results is a model similar to that of
integrate-and-fire neural networks.
Since micro-structure is of interest we do not apply the continuum
assumption, but rather investigate the stability of waves which propagate
through the discrete medium. 'Steady' wave speed is related to particle
ignition temperature, particle geometry and the ratio of heat diffusion to
reaction times through a single

transcendental equation.
Furthermore, the dynamics of this system can be related to a history
dependent implicit map $\vec{f}:\Rset^\infty \rightarrow \Rset^\infty$. Iteration of this map demonstrates that
average wave speed undergoes a period doubling cascade in one dimension and a
period doubling bifurcation with regions of bi-stability and chaos in two
dimensions. We also elucidate a
technique which allows for a linear stability analysis of the discrete system
capable of obtaining stability boundaries for both steady and periodic waves as
well as a discrete "wave number" associated with various growth
modes. The effects of boundary
locations which break the symmetry of the grid are also discussed.