Application of Perturbation Theory to Chemical Kinetic Systems

3 Application of Perturbation Theory to Chemical Kinetic Systems 

For equation systems of low dimension, the investigation of the inherent timescales can be carried out through a non-dimensionalisation process. Small parameters can then often be identified indicating fast variables. A discussimi of non-dimensionalisation procedures for a simplified 4-variable model describing the horseradish peroxidase reaction can be found in Chap. 12 of Scott (1990). The 4-variable model can be described by the following reaction steps  

For larger systems such as those typically fotmd in complex chemical problems, non-dimensionalisation may be impractical, and hence, numerical perturbation methods are generally used to investigate system dynamics and to explore timescale separation. By studying the evolution of a small disturbance or perturbation to the nonlinear system, it is possible to reduce the problem to a locally linear one. The resulting set of linear equations is easier to solve, and information can be obtained about the local timescales and stability of the nonlinear system. Several books on mathematics and physics (see e.g. Pontryagin 1962) discuss the linear stability analysis of the stationary states of a dynamical system. In this case, the dynamical system, described by an ODE, is in stationary state, i.e. the values of its variables are constant in time. If the stationary concentrations are perturbed, one of the possible results is that the stationary state is asymptotically stable, which means that the perturbed system always returns to the stationary state. Another possible outcome is that the stationary point is unstable. In this case, it is possible that the system returns to the stationary state after perturbation towards some special directions but may permanently deviate after a perturbation to other directions. A full discussion of stationary state analysis in chemical systems is given in Scott (1990). 

Let us change the concentrations of several species during the course of the reaction at an arbitrarily selected time t(, = 0 according to the vector AY°  

The time derivative of Y (t) can be calculated in two ways. For the first method (a linearisation), a Taylor series expansion is used with higher-order terms neglected  

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