Transition fronts and localized structures in bistable reaction-diffusion equations

Models of chemical reactions that undergo subcritical Hopf bifurcations and have a regime of bistability are shown to exhibit some interesting behavior when diffusively coupled. Waves joining a stable steady state to a periodic wavetrain are constructed for low and high thresholds. For intermediate thresholds, localized oscillatory regions are found. These can interact at a distance and behave like weakly coupled oscillators. These patterns are found in both solvable and realistic models for chemical oscillations.