OSCILLATORY REACTION-DIFFUSION EQUATIONS ON RINGS

We study the behavior of traveling waves in lambda-omega systems on bo th homogeneous and inhomogeneous rings. The stability regions in param eter space of lambda-omega waves were previously known [15, 19]; the r esults are extended here. We show the existence of Hopf bifurcations o f traveling waves and the stability of the limit cycles born at the Ho pf bifurcation for some parameter ranges. Using a Lindstedt-type pertu rbation scheme, we formally construct periodic solutions of the lambda -omega system near a Hopf bifurcation and show that the periodic solut ions superimposed on the original traveling wave have the effect of al tering its overall frequency and amplitude. We also study the lambda-o mega system on an annulus of variable width, which does not possess re flection symmetry about any axis. We formally construct traveling wave s on this variable-width annulus by a perturbation scheme, and iind th at perturbing the width of the annulus alters the amplitude and freque ncy of traveling waves on the domain by a small (order epsilon(2)) amo unt. 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 lambda-omega system on a variable-width domain cann ot account for directional preferences of traveling waves in biologica l systems.