Journal of Dynamics and Differential Equations

, Volume 21, Issue 1, pp 73–115

Pulse Dynamics in a Three-Component System: Existence Analysis

Authors

  • Arjen Doelman
    • Centrum voor Wiskunde en Informatica (CWI)
    • Korteweg-de Vries InstituutUniversiteit van Amsterdam
  • Peter van Heijster
    • Centrum voor Wiskunde en Informatica (CWI)
    • Department of Mathematics & Center for BioDynamicsBoston University
Open AccessArticle

DOI: 10.1007/s10884-008-9125-2

Cite this article as:
Doelman, A., van Heijster, P. & Kaper, T.J. J Dyn Diff Equat (2009) 21: 73. doi:10.1007/s10884-008-9125-2

Abstract

In this article, we analyze the three-component reaction-diffusion system originally developed by Schenk et al. (PRL 78:3781–3784, 1997). The system consists of bistable activator-inhibitor equations with an additional inhibitor that diffuses more rapidly than the standard inhibitor (or recovery variable). It has been used by several authors as a prototype three-component system that generates rich pulse dynamics and interactions, and this richness is the main motivation for the analysis we present. We demonstrate the existence of stationary one-pulse and two-pulse solutions, and travelling one-pulse solutions, on the real line, and we determine the parameter regimes in which they exist. Also, for one-pulse solutions, we analyze various bifurcations, including the saddle-node bifurcation in which they are created, as well as the bifurcation from a stationary to a travelling pulse, which we show can be either subcritical or supercritical. For two-pulse solutions, we show that the third component is essential, since the reduced bistable two-component system does not support them. We also analyze the saddle-node bifurcation in which two-pulse solutions are created. The analytical method used to construct all of these pulse solutions is geometric singular perturbation theory, which allows us to show that these solutions lie in the transverse intersections of invariant manifolds in the phase space of the associated six-dimensional travelling wave system. Finally, as we illustrate with numerical simulations, these solutions form the backbone of the rich pulse dynamics this system exhibits, including pulse replication, pulse annihilation, breathing pulses, and pulse scattering, among others.

Keywords

Three-component reaction-diffusion systems One-pulse solutions Travelling pulse solutions Two-pulse solutions Geometric singular perturbation theory Melnikov function

AMS (MOS) Subject Classifications

Primary: 35K55 35B32 34C37 Secondary: 35K40

Copyright information

© The Author(s) 2008