Coherent structures near the boundary between excitable and oscillatory media

We investigate reaction-diffusion systems near parameter values that mark the transition from an excitable to an oscillatory medium. We analyse the existence and stability of travelling waves near a steep pulse that arises as the limit of excitation pulses when parameters cross into the oscillatory regime. Travelling waves near this limiting profile are obtained by analysing a codimension-two homoclinic saddle-node/orbit-flip bifurcation. The main result shows that there are precisely two generic scenarios for such a transition, distinguished by the sign of an interaction coefficient between pulses. In both scenarios, we find stable fast fronts, unstable slow fronts, stable excitation pulses, and trigger and phase waves. Both trigger and phase waves are stable for repulsive interaction and both are unstable for attractive interaction.