Systems with a complete stirring

Koros and Noyes [13] suggested to use as the basic model for the Belousov-Zhabotinsky system a rather complicated set of chemical reactions with seven intermediate products. Its more global analysis based on macrokinetic stages and retaining still the principal features of this reaction [14] has led to the simplified scheme with three intermediate products only. This model called Oregonator [9, 15] is described by the following equations  

Here E and F are initial reactants whereas P and Q are final products. A, B and C are intermediate compounds HBr02, Br and Ce +. Concentrations riE and np of the initial reactants are assumed to be constant in an open system under study due to stationary matter source. Under well-stirring condition, the kinetic law of mass action leads to a set of the ordinary differential equations 

Along with this model [13], other basic models were presented which involve three intermediate products [10, 16, 17], as well as four products [10, 18]. 

A numerical solution of the basic equations demonstrated their ability to reproduce concentration oscillations. At the same time, for the systems possessing three and more intermediate products the standard method to prove existence of periodical solutions, using a phase portrait of a system (Section 2.1.1) fails. An additional reduction in a number of differential equations, e.g., using an idea that one of concentrations, say, [BrO j, serves as a rapid variable and thus the relevant kinetic equation (8.1.5) could be solved as the stationary [10], cannot be always justified due to uncertainty in the kinetic coefficients ki. 

As it was mentioned in Section 2.1.1, the concentration oscillations could be simulated quite well by a set of even two ordinary differential equations of the first order but paying the price of giving up the rigid condition imposed on interpretation of mechanisms of chemical reactions namely that they are based on mono- and bimolecular stages only (remember the Hanusse theorem [19]) An example of what Smoes [7] called the heuristic-topological model is the well-known Brusselator [2]. Its scheme was discussed in Section 2.1.1 see equations (2.1.33) to (2.1.35). 

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