Oscillatory reactions bifurcation analysis

The empty-site requirement in Eq. (28) can be physically interpreted in one of two different ways either the adsorbed A and B have to rearrange prior to reaction, or they are bound to more than one adsorption site. For the latter case, the intermediate concentration is low, thus allowing a pseudo-steady-state assumption. Through the application of bifurcation analysis and catastrophe theory this model was found to predict a very rich bifurcation and dynamic behavior. For certain parameter values, sub- and supercritical Hopf bifurcations as well as homoclinic bifurcations were discovered with this simple model. The oscillation cycle predicted by such a model is sketched in Fig. 6c. This model was also used to analyze how white noise would affect the behavior of an oscillatory reaction system... 

It has been established [59] that any activated exothermic reaction in an open system can give rise to bistable, excitable and oscillatory behavior. The principle features can be obtained using just a single unimolecular step. A detailed bifurcation analysis of such a system has been carried out by Vance and Ross [60] the bifurcation set consists of an unfolding of a (codimension-3) Takens-Bogdanov-cusp point [61]. (The bifurcation fine... 

This argument shows that for the first-order reaction model the stationary state always has some sort of stability to perturbations. In fact, this is only a first step and will not reveal Hopf bifurcations or oscillatory solutions, should they occur-. A full stability analysis of typical flow-reaction schemes will appear in the next chapter. 

The appearance and disappearance of the limit cycle behavior of this reaction are discussed in Field and Noyes (1974) in particular as related to the variables and parameters of the reaction system. Marek and Svobodova (1973) also discussed the appearance and disappearance of oscillatory behavior as well as the transition from one oscillatory solution to another. The onset of oscillations is further discussed in Berger and Koros (1980). These studies constitute in a way a nonsystematic analysis of the possible bifurcations of the system. 

A last type of dynamic phenomenon introduced by the recycling reaction is that of a multiplicity of oscillatory domains as a function of a control parameter. This phenomenon is apparent in the bifurcation diagram of fig. 3.6h. Here again, the interest of the phenomenon stems from its relationship to the behaviour of certain neurons the model provides a straightforward explanation for the neuronal behaviour in terms of phase plane analysis. 

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