Multiple time scale behavior

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. 

Dynamic simulations were aimed at capturing the multiple-time-scale behavior revealed by the theoretical developments presented above. Figures 7.5 and 7.6 show the evolution of the mole fraction of n-butane and of the temperature on selected column stages for a small step change in the reboiler duty. Visual inspection of the plots indicates that the temperatures exhibit a fast transient,... 

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. 

Owing to the presence of the small parameter , the model in Equation (7.1) is stiff and can potentially exhibit a dynamic behavior with multiple time scales. Proceeding in a manner similar to the one adopted in Chapter 6, we define the fast time scale r = t/e and rewrite (7.1) as... 

Most observed kinetics of multibasin protein dynamics is a consequence of averaging over an ensemble of many activated barrier crossings with multiple time scales. The direct observation [53-55] of dynamical behavior of a single molecule, so far buried in an ensemble average, should provide us with a new magnifying glass, which enables us to explicitly see the dynamical behavior inherent to the composite molecules in complex systems. 

Similar to the spatial features, the temporal patterns such as the oscillation behaviors should also be understood with the collective or emergent properties across multiple time scales in connection with multiple spatial levels. Across-scale detections in all scopes are necessary to achieve a thorough understanding (see Fig. 2). 

Rabitz s use of a multiple-time-scale representation of the collision dynamics is somewhat different. The separation of timescales in this theory is based on the rate of phase accumulation, since in the semiclassical limit this is related to the time needed for transfer of a quantum of ener r. When the interactions are such as to generate rapid-phase accumulation, as in the description of deflected translational motion, classical mechanics is appropriate, and when the interactions generate slow-phase accumulation, as in vibrational depopulation, quantum mechanics must be used. The effect of interactions on rotational motion spans the range of behavior between these two limits. The stochastic assumption introduced by Rabitz asserts that large and rapidly varying phases permit use of a random phase approximation. Reduction of the equations of motion to a useful form requires further approximations the reader is referred to the original paper for full discussion of these. The theory described has some very interest-... 

Ortoleva, P., Ross, J. (1975) Theory of propagation of discontinuities in kinetic systems with multiple time scales Fronts, front multiplicity, and pulses. J. Chem. Phys. 63, 3398 Pavlidis, T. (1973) Biological Oscillators - Their Mathematical Analysis (Academic, New York) Pomeau, Y., Roux, J. C., Rossi, A., Bachelart, S., Vidal, C. (1981) Intermittent behavior in the Belousov-Zhabotinsky reaction. J. Phys. (Paris) Lett 42, L-271 Rashevsky, N. (1940) An approach to the mathematical biophysics of biological self-regulation and of cell polarity. Bull. Math. Biophys. 2, 15... 

A related topic is the issue of time scales. Dynamic simulations of atomic behavior generally require time steps that are short enough to capture the vibrational modes of the system, whereas changes at the macroscopic scale usually occur over vastly longer time scales. Coupling between such widely varying time scales is a very important challenge, but it is not within the scope of this review. However, the problem of multiple-time-scale simulations will be discussed briefly in the discussion of dynamical methods. 

Finally, it is expected that in addition to thermal and chemical explosion similar behavior will arise in a host of other problems involving transient evolution and multiple time scales. The formation and propagation of fractures in a variety of materials [12] and the switching phenomena in lasers [13 1 are two tempting examples. 

The dynamic behavior of reaction systems often contains multiple time-scales after an initial transient period, some fast reactions can be considered nearly instantaneous relative to the... 

The simplest one-constant limitation concept cannot be applied to all systems. There is another very simple case based on exclusion of "fast equilibria" A Ay. In this limit, the ratio of reaction constants Kij — kij/kji is bounded, 0equilibrium constant", even if there is no relevant thermodynamics.) Ray (1983) discussed that case systematically for some real examples. Of course, it is possible to create the theory for that case very similarly to the theory presented above. This should be done, but it is worth to mention now that the limitation concept can be applied to any modular structure of reaction network. Let for the reaction network if the set of elementary reactions is partitioned on some modules — U j. We can consider the related multiscale ensemble of reaction constants let the ratio of any two-rate constants inside each module be bounded (and separated from zero, of course), but the ratios between modules form a well-separated ensemble. This can be formalized by multiplication of rate constants of each module on a timescale coefficient fc,. If we assume that In fc, are uniformly and independently distributed on a real line (or fc, are independently and log-uniformly distributed on a sufficiently large interval) then we come to the problem of modular limitation. The problem is quite general describe the typical behavior of multiscale ensembles for systems with given modular structure each module has its own timescale and these time scales are well separated. 

It is instructive to study a much simpler mathematical equation that exhibits the essential features of boundary-layer behavior. There is a certain analogy between stiffness in initial-value problems and boundary-layer behavior in steady boundary-value problems. Stiffness occurs when a system of differential equations represents coupled phenomena with vastly different characteristic time scales. In the case of boundary layers, the governing equations involve multiple physical phenomena that occur on vastly different length scales. Consider, for example, the following contrived second-order, linear, boundary-value problem ... 

Subjective (e.g., Epworth Sleepiness Scale) and objective [e.g., Multiple Sleep Latency Test (MSLT)] daytime somnolence quantification does not seem to provide valuable information on patients risks. This could be explained by the fact that sleep-related accidents occur at certain times when behavioral and chronobiological factors play an important role. Medical and legal issues could nevertheless require an objective test, such as the Maintenance of Wakefulness Test (MWT), to confirm that treated apneic patients present a normal level of vigilance. 

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... 

It can be further extended to describe simultaneous diffusion of multiple species in a multicomponent solid. However, Pick s law may not effectively handle individual atomic jumps at short spatial scales or on very short time scales (as determined by some spectroscopic methods or computer simulations), nor can it address cases where diffusion occurs in a medium whose structure is changing, like the glass transition region. These may lead to what is termed non-Fickian behavior (e g. Crank 1975). 

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