Fast-slow system

Transformation of the Langfnuir System to a Slow-Fast System... 

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. 

Figure 13.7 A two-variable slow-fast system, (a) Constraint-response plot showing dependence of steady states of x on y, treated as a parameter (b) trajectory of oscillation in x-y plane (c) x oscillations (d) y oscillations. (Adapted from Rinzel, 1981.)...

For pressure, slow and fast systems are compared. An example of a slow system is control of the pressure of compressible gas in a large process volume. A fast system could be the pressure of fuel oil (incompressible liquid) supply to a burner. For flow, a single set of settings is recommended. 

Drugs are administered by various means from oral to intravenous to topical. The oral route is a relatively slow process where a drug must be absorbed across the GI tract and then passed through the liver and metabolized before it becomes available to bind to receptors and perform its intended function. On the other hand, intravenous application is quick but has the potential of fast systemic reaction if adverse reactions occur. In the case of topical administration, the effects of the drug are localized. 

Fast-fill CNG refueling systems need a dispenser to direct the CNG into the vehicles. Dispensers include a meter to measure the amount of the CNG that went into the vehicle and, for public refueling stations, the cost of that gas. In slow-fill systems, a hose and a connector is provided for each vehicle, and metering to determine the amount of CNG going into the vehicle is usually cost prohibitive. For this reason, slow-fill is usually considered practical only for vehicles of a single fleet. 

All the above mechanisms can be called the simplest catalytic oscillators. In all these mechanisms self-oscillations of the reaction rate are realized due to the combination of the fast system of steps (adsorption mechanism) leading to the sharp change in the number of unoccupied surface sites and of the "slow reversible step ensuring self-oscillations of their concentration. If the parameters of the "buffer step are sufficiently small compared with those of the main mechanism, all these oscillations will be typically relaxa-tional. 

Remark 2.4. In a general nonlinear system, there may be several distinct solutions X2i G 1,..., k. In such a case, one focuses on a particular solution and the corresponding representation for the slow subsystem (2.13) in an appropriate neighborhood. The choice of a particular quasi-steady-state solution depends on the initial condition x x . The solution of the fast system will... 

Figure 8 (a) A slow exchange system, (b) A fast exchange system (the broken lines represent where the uncomplexed and complexed signals would be in a slow exchange system). 

In photochromic systems, the number, nature, and kinetic and spectral properties of the transient species formed under irradiation cannot be established without a good knowledge of the reaction mechanism. Flash photolysis is often used for analysis of this type of problem.1-9 However, fast methods are not well suited to the study of slow photochromic systems such as those used in variable-transmission glasses. In practice, the fast time scales of observation (10-14-10-3 s) in flash photolysis are completely different from those of the slower reactions (10 -10+6s) characterizing the accumulation of a slow photoisomer under a low-powered photon flux.10 Among the different photoisomers formed, only those with the longest lifetimes (> 10 s) will be observed by techniques using continuous irradiation (see Appendix 1 of this chapter). 

Eaker and Muzyka [39] have performed a trajectory-surface hopping calculation on the D+ — H2 system, but restricted to the two lowest potential energy surfaces. They observed the electron transfer between the two molecules prior to dissociation, but their CID cross section for D+(v = 3) + H2 at 4 eV was only 0.9 A2, about three times smaller than our result. In addition, their calculated ratio for H+ to D+ products was 8 1, whereas our result was 1 1. A possible explanation of the discrepancy [15] is a mechanism in which the second excited state of the [H2H2]+ system is excited during the collision. This state, which was not considered by Eaker, is repulsive for the H2+ moiety, and would give slow (fast) H+ ions if electron transfer did (did not) occur during the first part of the collision. This suggestion remains to be tested by theory. 

The word multiplex has been used indiscriminately by many authors to describe both simultaneous and rapid sequential (e.g. rotating filters, fast-scan or slow-scan systems) multi-wavelength detection methods [37]. Other specialists use this designation to refer to spectroscopic methods based on the truly simultaneous measurement of several optical signals at different wavelengths. Finally, others reserve this term for those methods which use a single detector for multi-wavelength detection [17]. 

The nitration data of this investigation were first used to discriminate between the kinetic models developed for the slow, fast, and instantaneous reaction systems (or regimes) and second to pick the best kinetic model. In order to do this, values had to be estimated for the concentration of the nitrating species (or entity) in the acid phase, the interfacial concentration of benzene, the kinetic rate constants for acid-phase nitrations, interfacial area (between two liquid phases), and diffusivity values or mass transfer coefficients for benzene. In analyzing the data, the concentration of the nitrating species was assumed to be that of the nitric acid in the acid phase it is appreciated that this assumption is open to question as will be discussed later but based on available information is probably the best assumption possible. The interfacial concentration of benzene (hC., .) was estimated based in part on the results obtained in making distribution measurements h is the distribution coefficient for benzene and C. - is the benzene concentration in the hydrocarbon phase. KinetTc rate constants (for the second-order nitration reactions occurring in the acid phase) were estimated based on the results of Deno and Stein (10),... 

Thus for a method taking one step of the fast system followed by a step of the slow system, using fixed step size /i > 0, we can write the update scheme for z = [q,pV as... 

To provide a neutronic system employing a fissionable material as the source of fast neutrons and an efficient iluid moderator capable of slowing fast neutrons to ther-5 mal energy ... 

Numerous terms are put to use in the field of electrochemical kinetics to characterise typical situations which are limiting cases with particular shapes for the corresponding current-potential curves. In scientific literature, these terms are not always applied with the greatest rigour. In the forthcoming sections we will give a precise definition for the common terms nernstian redox systems in section 4.3.2.4 reversible/irreversible redox reactions in section 4.3.2.S slow/fast redox systems in section 4.3.2.6. 

When dealing with the same redox couple, it can behave either like a slow system or a fast system, depending on the experimental conditions chosen ... 

In all these descriptions, a separation of slow and fast systems is assumed on a short time scale, the cation oscillates on a given site, while on a long time scale, it diffuses. 

Recent theoretical analyses (Carreras et al. 2005), (Schlapfer et al. 2008), have shown the sensitivity to parameter variations in the dynamic degradation of networks. Breakdowns of such complex networks are often the result of relatively slow initial system degradation escalating into a fast avalanche of component... 

---