  We examine the evolution of a bistable reaction in a one-dimensional
stretching flow, as a model for chaotic advection. We derive two reduced
systems of ordinary differential equations (ODE's) for the dynamics of the
governing advection--reaction--diffusion partial differential equation (PDE),
for pulse-like and for plateau-like solutions, based on a non-perturbative
approach. This reduction allows us to study the dynamics in two cases: first,
close to a saddle--node bifurcation at which a pair of nontrivial steady states
are born as the dimensionless reaction rate (Damkohler number) is increased,
and, second, for large Damkohler number, far away from the bifurcation. The
main aim is to investigate the initial-value problem and to determine when an
initial condition subject to chaotic stirring will decay to zero and when it
will give rise to a nonzero final state. Comparisons with full PDE simulations
show that the reduced pulse model accurately predicts the threshold amplitude
for a pulse initial condition to give rise to a nontrivial final steady state,
and that the reduced plateau model gives an accurate picture of the dynamics of
the system at large Damkohler number.
