Slow dynamical systems and chemical kinetics equations

5 Slow dynamical systems and chemical kinetics equations 

The desired behaviour of a chemical dynamical system can be modelled by an effective system of kinetic equations in the way similar to that described in Section 3.5 for modelling the heartbeat. The method involves designing a system of differential equations having the desired slow dynamics (the proper slow surface). We should now answer the question whether application of the Tikhonov theorem to the standard kinetic system (4.27) may yield a completely arbitrary slow dynamical system (4.40b ). A partial answer to this question is provided by the Korzukhin theorem Each dynamical system of the form 

From the Korzukhin theorem follows an important conclusion. Any dynamical systems of the form (4.58) may be regarded as those corresponding to slow dynamics of a standard kinetic system. In other words, the behaviour of dynamical systems can be modelled using chemical reactions. In particular, any of the gradient systems may be modelled in this way. As will be shown in Chapter 5, catastrophes occurring in complex dynamical systems are equivalent to catastrophes appearing in much simpler systems. The latter can be classified — these are so-called standard forms. The standard forms are of the form (4.58) and it follows from the Korzukhin theorem that they can be modelled by the standard equations of chemical kinetics (4.27), corresponding to a realistic mechanism of chemical reactions. 

Fundamental information on chemical kinetics and kinetics equations can be found in the books of Avery, or Berry, Rice and Ross, as well as books by Tyson and Murray. 

Much information on kinetic equations and their reduction using the Tikhonov theorem and catastrophe theory is provided in a book by Romanovskii, Stepanova and Chernavskii. 

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