Perturbation theory, singular

The review in the previous chapter pointed out that, while long acknowledged, the multiple-time-scale dynamic behavior of integrated chemical plants has been dealt with mostly empirically, both from an analysis and from a control point of view. In the remainder of the book, we will develop a mathematically rigorous approach for identifying the causes, and for understanding and mitigating the effects of time-scale multiplicity at the process system level. 

The present chapter introduces the reader to singular perturbation theory as the framework for modeling and analyzing systems with multiple-time-scale dynamics, which we will make extensive use of throughout the text. 

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The mathematical theory of dissipative structures is mainly based on approximate methods such as bifurcation theory of singular perturbation theory. Situations like those described in Section VI and that permit an exact solution are rather exceptional. 

Tyson, J. J. and Keener, J. P. (1988). Singular perturbation theory of traveling waves in excitable media (a review). Physica, D 32, 327-61. (December)... 

We shall now solve the Kramers equation (7.4) approximately for large y by means of a systematic expansion in powers of y-1. Straightforward perturbation theory is not possible because the time derivative occurs among the small terms. This makes it a problem of singular perturbation theory, but the way to handle it can be learned from the solution method invented by Hilbert and by Chapman and Enskog for the Boltzmann equation.To simplify the writing I eliminate the coefficient kT/M by rescaling the variables,... 

An iterative method for developing higher order approximations to the solution to Equation 1 can be devised by using some of the ideas of singular perturbation theory (15). To display the systematics of the procedure let us rewrite Equation 7 as... 

The rate at which x2 approaches x2 can be very large, since dx2/dt = (l/e)g, and e —> 0. Singular perturbation theory relies on defining a stretched time variable r = t/e, with r = 0 at t = 0, to analyze such fast transient phenomena. The term stretched refers to the behavior of the new time variable r, which tends to 00 even for t only slightly larger than 0. Note that, while x2 and r vary very rapidly, xi stays near its initial value x°. 

Such nested applications of single-parameter singular perturbation theory (i.e., the extension of the analysis of two-time-scale systems presented in Chapter 2 to multiple-time-scale systems) have been used for stability analysis of linear (Ladde and Siljak 1983) and nonlinear (Desoer and Shahruz 1986) systems in the standard form. However, as emphasized above (Section 2.3), the ODE models of chemical processes are most often in the nonstandard singularly perturbed form, with the general multiple-perturbation representation... 

Fenichel, N. (1979). Geometric singular perturbation theory for ordinary differential equations. J. Diff. Equat., 31, 53. 

However, since the QSSA has been used to elucidate most reaction mechanisms and to determine most rate coefficients of elementary processes, a fundamental answer to the question of the validity of the approximation seems desirable. The true mathematical significance of QSSA was elucidated for the first time by Bowen et al. [163] (see also refs. 164 and 165 for history and other references) by means of the theory of singular perturbations, but only in the case of very simple reaction mechanisms. The singular perturbation theory has been applied by Come to reaction mechanisms of any complexity with isothermal CFSTR [118] and batch or plug flow reactors [148, 149]. The main conclusions arrived at for a free radical straight chain reaction (with only quadratic terminations) carried out in an isothermal reactor can be summarized as follows. 

It has been seen that singular perturbation theory constitutes the natural framework justifying the use of QSSA. Classically, the QSSA allows kineticists to obtain explicit mathematical relationships. 

The basic ideas that are necessary for the first program stage are explained in Sections II, III, and IV. In Section II, we formulate the problem of how to analyze a system that has a gap in characteristic time scales. Our method is to use perturbation theory with respect to a parameter that is the ratio between a long time scale and a short time scale, which is a version of singular perturbation theory. The reason will be explained in Section II. In Section III, the concept of NHIMs is introduced in the context of singular perturbation theory. We will give an intuitive description of NHIMs and explain how the description is implemented, leaving the precise formulation of the NHIM concept to the literature in mathematics. In Section IV, we will show how Lie perturbation theory can be used to transform the system into the Fenichel normal form locally near a NHIM with a saddle with index 1. Our explanation is brief, since a detailed exposition has already been published [2]. 

In this chapter, focus attention on a different method to analyze those systems that consist of fast and slow variables. It is called singular perturbation theory, and is suitable for understanding chaotic behavior in systems with many degrees of freedom. The reason for the term singular will be explained later. 

In this and the next sections, we will present the basic ideas of singular perturbation theory without going into mathematical rigor. 

Until now, we have discussed NHIMs in general dynamical systems. In this section, we limit our argument to Hamiltonian systems and show how singular perturbation theory works. In particular, we discuss NHIMs in the context of reaction dynamics. First, we explain how NHIMs appear in conventional reaction theory. Then, we will show that Lie permrbation theory applied to the Hamiltonian near a saddle with index 1 acmally transforms the equation of motion near the saddle to the Fenichel normal form. This normal form can be considered as an extension of the Birkhoff normal form from stable fixed points to saddles with index 1 [2]. Finally, we discuss the transformation near saddles with index larger than 1. 

The boundary conditions follow naturally from the conventions Uq = = 0, and similarly for v, which say that there are no microorganisms in the two reservoirs. They are justified by the agreement between the numerically computed rest points of (5.1) and the solutions of (5.2) obtained by using singular perturbation theory (as in [S9]). 

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