Oscillatory reaction-diffusion equations on rings

Abstract

We study the behavior of traveling waves in λ-ω systems on both homogeneous and inhomogeneous rings. The stability regions in parameter space of λ-ω waves were previously known [15, 19]; the results are extended here. We show the existence of Hopf bifurcations of traveling waves and the stability of the limit cycles born at the Hopf bifurcation for some parameter ranges. Using a Lindstedt-type perturbation scheme, we formally construct periodic solutions of the λ-ω system near a Hopf bifurcation and show that the periodic solutions superimposed on the original traveling wave have the effect of altering its overall frequency and amplitude. We also study the λ-ω system on an annulus ofvariable width, which does not possess reflection symmetry about any axis. We formally construct traveling waves on this variable-width annulus by a perturbation scheme, and find that perturbing the width of the annulus alters the amplitude and frequency of traveling waves on the domain by a small (order e2) amount. For typical parameter values, we find that the speed, frequency, and stability are unaffected by the direction of travel of the wave on the annulus, despite the rotationally asymmetric inhomogeneity. This indicates that the λ-ω system on a variable-width domain cannot account for directional preferences of traveling waves in biological systems.