Waves in a simple, excitable or oscillatory, reaction-diffusion model

A simple one variable caricature for oscillating and excitable reaction-diffusion systems is introduced. It is shown that as a parameter, λ, varies the system dynamics change from oscillatory (λ > λ₀) to excitable (λ < λ₀) and the frequency of the oscillation vanishes as {Mathematical expression} for λ ↘ λ₀. When such dynamics are coupled by continuous diffusion in a ring geometry (1-space dimension), propagating wave trains may be found. On an infinite ring excitable devices lead to unique solitary waves which are analogous to "pulse" waves. A solvable example is presented, illustrating properties of dispersion, excitability, and waves. Finally it is shown that the caricature arises in a natural way from more general excitable/oscillatory systems.