Diffusion-Induced Instabilities near a Canard

Abstract

We investigate diffusion-induced instabilities of phase waves in one spatial dimension for a two-variable model of the Belousov-Zhabotinsky reaction. We use as initial conditions small-amplitude phase waves which exist in the parametric range between a canard point and a supercritical Hopf bifurcation point. Closer to the canard point, the instability leads to initiation of trigger waves, usually at the zero flux boundary. Such induced trigger waves reflect from the boundary, and when they collide, a new trigger wave emerges at the location of the collision. When the parameters are chosen nearer to the Hopf point, the phase waves lose their regular pattern and become uncorrelated. Very close to the Hopf point, diffusion alters the phase wave profile into small-amplitude synchronized bulk oscillations. Different types of spatiotemporal behavior are observed when the wavelength of the phase waves, the overall size of the system, or the diffusion coefficients are changed. Comparison of the behavior near a canard and near a subcritical Hopf bifurcation shows that in the former case trigger waves can be initiated at all points of the excitable medium, whereas in the latter case trigger waves are generated only at the boundary.