  In order to describe excitable reaction-diffusion systems, we derive a
two-dimensional model with a Hopf and a semilocal saddle-node homoclinic
bifurcation. This model gives the theoretical framework for the analysis of the
saddle-node homoclinic bifurcation as observed in chemical experiments, and for
the concepts of excitability and excitability threshold. We show that if
diffusion drives an extended system across the excitability threshold then,
depending on the initial conditions, wave trains, propagating solitary pulses
and propagating pulse packets can exist in the same extended system. The
extended model shows chemical turbulence for equal diffusion coefficients and
presents all the known types of topologically distinct activity waves observed
in chemical experiments. In particular, the approach presented here enables to
design experiments in order to decide between excitable systems with sharp and
finite width thresholds.
