Tikhonov theorem

The conditions for which a reduction of the system (4.36) described above is allowed, are given by the Tikhonov theorem. 

A solution to the overall system (4.39) may be proved to approach, for e - 0, the solution to the following system of equations  

Conditions (a), (c), (d) are generally fulfilled in chemical systems whereas condition (b) may be unsatisfied in a case of processes in which oscillations in concentration occur. In accordance with our general methodology, the situation wherein with a continuous variation in control parameters condition (b) ceases to be met may be regarded as a catastrophe. The examination of catastrophes of this type (in general, these are not elementary catastrophes) will be described in Chapter 6. 

The Tikhonov theorem has an important generalization, called the centre manifold theorem, which will be discussed in Sections 5.4.5-5.4.7. In classification of catastrophes occurring in dynamical systems and represented by systems of autonomous equations, the centre manifold theorem plays the role of the splitting lemma (see Section 2.3.4). 

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At e = SI V -> 0, the Tikhonov theorem is applicable, hence starting from sufficiently small , we can use the quasi-steady-state approximation. 

Application of the Tikhonov theorem to modelling of dynamical systems and chemical reactions... 

Using the Tikhonov theorem, the desired behaviour of dynamical system may be modelled or, alternatively, one may obtain from the standard system of chemical kinetics equations (4.27) effective systems of equations which cannot be represented in tms form (for example, of an autocatalytic type or termolecular). 

We will now employ the Tikhonov theorem to designing standard systems whose slow dynamics is of an autocatalytic or termolecular character. We shall begin with modelling an autocatalytic reaction (such as, for example, Edelstein reaction (4.21)). 

The system of equations (4.45) is linear and may be readily solved however, we will apply the Tikhonov theorem to show how to use it. Assume also that the following relationship... 

We may now apply the Tikhonov theorem to the system (4.50) equation (4.50b) is a rapid system whereas (4.50a) is a slow equation and describes slow processes. 

Application of the Tikhonov theorem leads to the system of equations... 

As a second application of the Tikhonov theorem we will present the derivation of an effective kinetic equation containing the termolecular interaction from a standard system of kinetic equations. Such an interaction occurs in the sequence of reactions (4.23) and in the system of kinetic equations the cubic term x2y is present. We shall demonstrate that such a term may appear in slow dynamics of a system of elementary reactions at most bimolecular (a standard system). Consider the following sequence of elementary reactions... 

Applying now the Tikhonov theorem, fast equation (4.55c) may be replaced with... 

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... 

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. 

An additional material concerning the relation of the Tikhonov theorem with catastrophe theory is presented in a paper by Feinn and Ortoleva. 

Relation between the Tikhonov theorem and the centre manifold theorem... 

The centre manifold theorem may be regarded as a generalization of the Tikhonov theorem to a case when the time hierarchy does not explicitly occur in the examined system, i.e. when the equations cannot be divided into... 

The system (5.40) meets assumptions of the Tikhonov theorem. Equation (5.40c) is a fast equation while (5.40a, b) are slow equations. In accordance with the Tikhonov theorem (5.40) is replaced by the approximate system... 

A paper by Feinn and Ortoleva deals with the application of the Tikhonov theorem to chemical systems, similarly bo books by Romanovskii, Stepanova and Chernavskii, which additionally provides information on the more general Shoshitishvili theorem (covering essentially a part of Guckenheimer s results). Information on an application of the centre manifold theorem can be found in a book by Carr or by Guckenheimer and Holmes. In the Carr book, the Tikhonov theorem is compared with the centre manifold theorem. 

After making a number of assumptions on the magnitude of rate constants of the reaction and on the initial concentrations Sel kov reduced, using the Tikhonov theorem, the system of kinetic equations corresponding to the mechanism (6.66), bringing it to the following final form ... 

The system (6.98) has a form well suited for an application of the Tikhonov theorem and we already know that by reducing the system to a system in two variables we do not lose any catastrophe connected with the loss of stability of the stationary point (x2> y2, z2). It follows from equality... 

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