Abstract
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.
Original language | English (US) |
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Pages (from-to) | 111-132 |
Number of pages | 22 |
Journal | Dynamical Systems |
Volume | 25 |
Issue number | 1 |
DOIs | |
State | Published - Mar 2010 |
Bibliographical note
Funding Information:The authors are indebted to the anonymous referees for many helpful remarks and constructive criticism. Support by the National Science Foundation under grant NSF-DMS-0504271 is gratefully acknowledged.
Keywords
- Excitable medium
- Homoclinic saddle-node
- Pulse interaction
- Stability of travelling waves