A Retarded Differential Equation Model of Wave Propagation in a Thin Ring

We propose a mathematical model for wave propagation in a narrow ring filled with an excitable medium. The speed of the wave fronts is assumed to depend on the time that has passed since the last impulse. From this assumption we derive a system of nonlinear functional differential equations. We prove 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 distributed uniformly. Any initial distribution of the fronts (apart from certain exceptional cases) tends to this distribution; in this sense it is the "asymptotic state." That result is in agreement with chemical experimental observations, namely, that the long-term distribution of the fronts is uniform in an annular reactor. Our functional differential equation is transformed into a system of delay differential equations. After this transformation a global stability theorem is proved.