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
as considered in . 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. .
University of Minnesota Ph.D. dissertation. February 2009. Major: Mathematics. Advisor: Arnd Scheel. 1 computer file (PDF); vii, 105 pages.
Bellay, Jeremy Charles.
The stability and transitions of coherent structures on excitable and oscillatory media..
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