An existing scaling model of a reaction-diffusion system is extended to a c
ircular trajectory. The equations describe the evolution of a slow (X) and
a fast (Y) concentration variable. The fast variable jumps between extreme
values across a reaction interface as the rate parameter becomes very large
. The model is reduced to one equation for the dynamics of the smooth (slow
) variable while the Y-jumps occur at two interfaces spatially located on a
ring. The equation is solved subject to 2 pi-periodicity conditions on the
ring and continuity conditions at the interfaces. Both static (with zero v
elocity) and moving (with velocity v) wave solutions are found. An analogy
is then drawn between our reaction-diffusion system and oscillating chemica
l reactions such as the Belousov-Zhabotinskii (BZ) reagent, confined to a t
orus-shaped container. A toroidal thin tube with a very small diameter coul
d simulate the ring geometry. The conjectured waves capture the oscillation
s of the catalyst (ferroin), with the maxima and minima corresponding to th
e ferroin and ferriin, spatial domains in the doughnut, respectively. The n
on-stationary wave solutions predict a migration of those domains yielding
swinging (back and forth) patterns along the ring. The azimuthal position o
f the interfaces exhibits temporal oscillations. Thus these simulations sug
gest interesting experiments on spatio-temporal patterns in excitable chemi
cal media in annular reactors.