The stability and transitions of coherent structures on excitable and oscillatory media.

Abstract

We investigate reaction-diffusion systems near parameter values that mark the transition from an excitable to an oscillatory medium. We analyze existence and stability of traveling waves near a steep pulse that arises as the limit of excitation pulses as parameters cross into the oscillatory regime. Traveling waves near this limiting profile are obtained by studying a codimension-two homoclinic saddle-node/orbit-flip bifurcation as considered in [1]. 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. Among others, we find stable fast fronts and unstable slow fronts in all scenarios, stable excitation pulses, trigger and phase waves. Trigger and phase waves are stable for repulsive interaction and unstable for attractive interaction. Finally, we study this transition numerically in the modified FitzHugh-Nagumo equations studied by Or-Guil et. al. [2].