A retarded differential equation model of wave propagation in a thin ring

We propose a mathematical model for wave propagation in a narrow ring fille d with an excitable medium. The speed of the wave fronts is assumed to depe nd on the time that has passed since the last impulse. From this assumption we derive a system of nonlinear functional differential equations. We prov e that it has a special solution, for which the speed of the fronts is the same constant (determined by the dispersion relation), and the fronts are d istributed uniformly. Any initial distribution of the fronts (apart from ce rtain exceptional cases) tends to this distribution; in this sense it is th e "asymptotic state". That result is in agreement with chemical experimenta l observations, namely, that the long-term distribution of the fronts is un iform in an annular reactor. Our functional differential equation is transf ormed into a system of delay differential equations. After this transformat ion a global stability theorem is proved.