Reductive Perturbation Method

Small-amplitude oscillations near the Hopf bifurcation point are generally governed by a simple evolution equation. If such oscillators form a field through diffusion-coupling, the governing equation is a simple partial differential equation called the Ginzburg-Landau equation. 

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Taniuti, T. (1974) Reductive perturbation method and far fields of wave equations. Prog. Theor. Phys. Suppl. 55, 1... 

The reduction techniques which take advantage of this separation in scale are described below. They include the quasi-steady-state approximation (QSSA), the computational singular perturbation method (CSP), the slow manifold approach (intrinsic low-dimensional manifold, ILDM), repro-modelling and lumping in systems with time-scale separation. They are different in their approach but are all based on the assumption that there are certain modes in the equations which work on a much faster scale than others and, therefore, may be decoupled. We first describe the methods used to identify the range of time-scales present in a system of odes. 

G. Li and H. Rabitz, A Special Singular Perturbation Method for Kinetic Model Reduction With Application to an H2/O2 Oxidation Model, J. Chem. Phys. 105 (1996) 4065-4075. 

In summary, our conclusion is that by supplementing T ordering with nodal reduction and nodal approximation, and (for low p) using it in a simple mixed perturbation method, we can come a long way toward the goal enunciated in the first sentence of our introduction. 

Perturbation methods based on theory from physics and chemistry, electronic computer simulation studies, and careful comparisons with real fluid behavior are moving quickly toward producing an effective equation of state for liquids. These efforts are in the hands of the theoretician today, but further development and reduction to practice should be explored. 

Li, G., Rabitz, H. A special singular perturbation methods for kinetic model reduction with application to an H2/O2 oxidation model. J. Chem. Phys. 105, 4065 075 (1996a)... 

Timescales are important features of dynamical models. Whilst historically we may be used to identifying individual timescales with individual species within a mechanism, we demonstrated in Chap. 6 that within a nonlinear kinetic model, there is usually not a one-to-one relationship between them. Nevertheless, we showed that the relationship between species and timescales, and the dynamic changes in timescales during a model simulation, can be explored using perturbation methods. Timescales are related to the stiffness of dynamical models, which is an important feature for the selection of appropriate numerical simulation methods. However, the wide range of timescales and the timescale separation can be exploited within the context of model reduction, and therefore, there are important links between Chaps. 6 and 7 in this regard. 

I apply these computational methods to various aspects of the Earth system, including the responses of ocean and atmosphere to the combustion of fossil fuels, the influence of biological activity on the variation of seawater composition between ocean basins, the oxidation-reduction balance of the deep sea, perturbations of the climate system and their effect on surface temperatures, carbon isotopes and the influence of fossil fuel combustion, the effect of evaporation on the composition of seawater, and diagenesis in carbonate sediments. These applications have not been fully developed as research studies rather, they are presented as potentially interesting applications of the computational methods. 

A variety of physical methods has been used to ascertain whether or not surface ruthenation alters the structure of a protein. UV-vis, CD, EPR, and resonance Raman spectroscopies have demonstrated that myoglobin [14, 18], cytochrome c [5, 16, 19, 21], and azurin [13] are not perturbed structurally by the attachment of a ruthenium complex to a surface histidine. The reduction potential of the metal redox center of a protein and its temperature dependence are indicators of protein structure as well. Cyclic voltammetry [5, 13], differential pulse polarography [14,21], and spectroelectrochemistry [12,14,22] are commonly used for the determination of the ruthenium and protein redox center potentials in modified proteins. 

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