Multiple time-scale perturbations

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. 

The presence of a singular perturbation induces multiple-time-scale behavior in dynamical systems, which is characterized by the presence of both fast and slow transients in their time response. The slow response is approximated by the reduced model (2.14), while the difference between the response of the reduced model (2.14) and that of the full model (2.7)-(2.8) is the fast transient. 

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. 

Section 2.4 alluded to the possibility of expanding the methods presented in Chapter 2 to account for the presence of multiple singular perturbation parameters in a system of differential equations. This appendix is concerned with this topic, and, to this end, let us consider a multiple-time-scale (multiply perturbed) system in the standard form... 

Condition (B.2) implies that m is the smallest singular perturbation parameter (that would yield the fastest fast time scale), while e is the largest singular perturbation parameter, and is responsible for the slowest fast time scale. Consequently the variable yJ+i will be faster than the variable yj, for j = 1,..., M — 1. It is this hierarchy of fast subsystems (boundary layers) that distinguishes multiple-time-scale systems from two-time-scale systems. 

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... 

Ladde, G. S. and Siljak, D.D. (1983). Multiparameter singular perturbation of linear systems with multiple time scales. Automatica, 19, 385-394. 

Vora, N.P., Contou-Carrere, M.N., and Daoutidis, P. (2006). Model reduction of multiple time scale processes in non-standard singularly perturbed form. In A. N. Gorban, N. Kazantzis, I.G. Kevrekidis, H.C. Ottinger, and K. Theodor-opoulos, eds., Coarse Graining and Model Reduction Approaches for Multiscale Phenomena, pp. 99-116. Berlin Springer-Verlag. 

The exploitation of multiple time-scales for systematic order reduction requires the transformation of a model to the standard two-time-scale form of singular perturbation theory. In the standard two-time-scale form, equation (1) is separated into fast and slow species. The model of the reaction system is expressed as... 

Free Energy Perturbation Calculations Molecular Dynamics and Hybrid Monte Carlo in Systems with Multiple Time Scales and Long-range Forces Reference System Propagator Algorithms Molecular Dynamics DNA Molecular Dynamics Simulations of Nucleic Acids Molecular Dynamics Studies of Lipid Bilayers. 

To derive the macroscopic transport equations, the conservation Relation [10] and [11] must be converted to differential equations. The main assumption needed is that the mean density and the mean velocity vary slowly in space and in time. Starting from Eq. [10] and [11], the macro dynamic equations describing the large-scale behavior of the lattice gas are obtained by multiple-scale perturbation expansion technique (Frisch et al., 1987). We shall not derive this formalism here. In the continuous limit, Eq. [10] leads to the macro dynamical conservation of mass or Euler equation... 

A. Kumar, P. D. Christofides, and P. Daoutidis. Singular perturbation modeling of nonlinear processes with nonexplicit time-scale multiplicity. Chem. Eng. Set, 53 8) 1491-1504,1998. 

It is in the documentation of the nature of the short-term perturbations in radiocarbon values that the two sets of data do not agree completely. Although there is a consensus that short-term episodes do exist, their magnitude is in dispute. There is also a consensus that the shortterm variations generate an additional set of problems for the archaeologist in the use and interpretation of radiocarbon values (61). In the Suess (52) and MASCA (53,54) plots, the interval of time from A.D. 1800 to the middle of the 4th millenium exhibits 12 major temporal episodes where radiocarbon values have multiple age equivalents. In these intervals, a given radiocarbon value may reflect two or more points in real time. Figure 5 illustrates this problem on an expanded scale. 

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