Mixing in excitable dynamics

A similar, but somewhat more complex system in the family of autocatalytic-type processes is that of an excitable reaction dynamics. This requires multiple reactions with significantly different characteristic timescales. We consider excitable dynamics occurring in the same two flows as in the previous cases (Neufeld et al., 2002c), and as a specific reaction example we focus on the FitzHugh-Nagumo 

there is a main transition at a critical Damkohler number, Da = Dac. For smaller values of Da the initial perturbation is quickly diluted and the activator decays to the C state, as in the bistable case. The same behavior is observed when the initial perturbation is not sufficiently large. For Da Dac the perturbation grows as in the bistable case, forming a growing filament that eventually fills the whole system (in the closed flow case), or covers the unstable manifold of the chaotic saddle (in open flows). The filament consists now of a pulse of the C concentration, with a maximum close to the excited state, and accompanied by a smaller pulse of C2. In the closed 

These observations can be interpreted again in terms of the filament model of Sect. 2.7.1. The interesting point is that there exists a stable steady state filament solution with the excited state in the center, even though in the homogeneous system the excited state is not steady. This can be explained qualitatively by the different timescales corresponding to the dynamics of the two reaction com- 

Examples of stable filament solutions of the excitable model (7.33), obtained numerically for the flow of the filament model of Sect. 2.7.1, are shown in Fig. 7.13 (Neufeld et al., 2002c Hernandez-Garcfa et ah, 2003). At Da = Dac 12.5 the stable filament solution collide with the unstable pulse (7.29) in a saddle-node bifurcation, so that no 

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