Singularly perturbed system

The original model regarding surface intermediates is a system of ordinary differential equations. It corresponds to the detailed mechanism under an assumption that the surface diffusion factor can be neglected. Physico-chemical status of the QSSA is based on the presence of the small parameter, i.e. the total amount of the surface active sites is small in comparison with the total amount of gas molecules. Mathematically, the QSSA is a zero-order approximation of the original (singularly perturbed) system of differential equations by the system of the algebraic equations (see in detail Yablonskii et al., 1991). Then, in our analysis... 

The LEN approximation will be employed in various electro-diffusional contexts in Chapters 3, 4, and 6. In particular, in Chapter 4 we shall elaborate upon the limits of applicability of the LEN approximation. It will be further treated as a leading approximation for the singularly perturbed system (1.9), (l.lld) in Chapter 5. 

Nonstandard singularly perturbed systems with two time scales... 

Example 2.3. Depending on the mechanism, reacting systems with vastly different reaction rates can be modeled by either standard or nonstandard singularly perturbed systems of equations. Systems in which a reactant is involved in both slow and fast reactions belong to the latter category. Consider the reaction system in Example 2.2, with the difference that the reactant Ri also participates in the second reaction ... 

Time-scale decomposition and model reduction methods for multiply singularly perturbed systems typically involve the nested application of the procedures discussed so far. For the interested reader, an overview of existing research results concerning the dynamic behavior of multiply singularly perturbed systems is presented in Appendix B. 

Let us now consider an augmented representation of the standard and nonstandard singularly perturbed systems discussed above, namely... 

The majority of the existing literature on the control of singularly perturbed systems considers the two-time-scale, standard form (see, e.g., Kokotovic et al. 1986, Christofides and Daoutidis 1996a, 1996b). Nevertheless, the methods available for standard singularly perturbed systems can be extended to systems in nonstandard form, since these can be transformed into an equivalent standard form as mentioned above. 

This chapter has reviewed existing results in addressing the analysis and control of multiple-time-scale systems, modeled by singularly perturbed systems of ODEs. Several important concepts were introduced, amongst which the classification of perturbations to ODE systems into regular and singular, with the latter subdivided into standard and nonstandard forms. In each case, we discussed the derivation of reduced-order representations for the fast dynamics (in a newly defined stretched time scale, or boundary layer) and the corresponding equilibrium manifold, and for the slow dynamics. Illustrative examples were provided in each case. 

The remaining state variables in Equation (3.27) display a similar behavior. The fast and slow dynamics are thus not associated with any distinct subsets of the state variables, which is consistent with the statement that the model of the process under consideration is a nonstandard singularly perturbed system of equations. 

Saberi, A. and Khalil, H. (1985). Stabilization and regulation of non-linear singularly perturbed systems-composite control. IEEE Trans. Automat. Contr., 30, 739-747. 

This is a standard slow-fast system (singularly perturbed system) that combines singular and regular perturbations in the second equation the singular perturbation induces the small parameter e before the time derivative of v and this same small parameter, caused by the regular perturbation, appears before the second term on the right-hand side. 

Another kind of examples where our blue sky catastrophe may appear naturally is given by singularly perturbed systems i.e. the systems of the form... 

In fact, the triggering from one stable branch to another is the most typical phenomenon in singularly perturbed systems, so one may encounter for our blue sky catastrophe every time when jumps between the branches of fast periodic orbits and fast equilibrium states are observed. 

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