Formal Stability Analysis

The established tools of nonlinear dynamics provide an elaborate and versatile mathematical framework to examine the dynamic properties of metabolic systems. In this context, the metabolic balance equation (Eq. 5) constitutes a deterministic nonlinear dynamic system, amenable to systematic formal analysis. We are interested in the asymptotic, the linear stability of metabolic states, and transitions between different dynamic regimes (bifurcations). For a more detailed account, see also the monographs of Strogatz [290], Kaplan and Glass [18], as well as several related works on the topic [291 293],... 

Aiming at a more formal analysis, the asymptotic stability of a steady-state value S° of a metabolic system upon an infinitesimal perturbation is determined by linear stability analysis. Given a metabolic system at a positive steady-state value... 

Providing a more formal analysis, several objective measures to rank the parameters according to their impact on the stability can be utilized. Possible measures of dependency are ... 

If the normal conversion is about 50%, a sudden small increase in the temperature could increase Qq by up to twice the amount predicted by steady-state analysis. In practice, there will be some decrease in concentration during a rise in temperature, and the change in Qq will depend on the order of the reaction and the nature of the upset. A formal analysis of reactor stability for a first-order reaction yields an additional criterion [4-6] ... 

The formal analysis of the closed-loop dynamics can be done with an extension of the nonlocal closed-loop stability analysis of a continuous reactor with temperature cascade controller presented before [22] in conjunction with the stability definitions (Eq. 7) given in section 2. Here it suffices to mention that, the closed-loop motion is stable if the filter and estimator gains are chosen not faster than the characteristic frequency Oj of the jacket hydraulic dynamics [26], and the secondary and primary control gains are chosen so that there is adequate dynamic separation between coj, coj and 0). This is,(Eq. 30),... 

The quote is from the third volume of Henri Poincare s New Methods of Celestial Mechanics, and is a description of his discovery of homoclinic orbits (see below) in the restricted three-body problem. It is also one of the earliest recorded formal observations that very complicated behavior may be found even in seemingly simple classical Hamiltonian systems. Although Hamiltonian (or conservative) chaos often involves fractal-like phase-space structures, the fractal character is of an altogether different kind from that arising in dissipative systems. An important common thread in the analysis of motion in either kind of dynamical system, however, is that of the stability of orbits. 

For the first time, the primary nitrone (formaldonitrone) generation and the comparative quantum chemical analysis of its relative stability by comparison with isomers (formaldoxime, nitrosomethane and oxaziridine) has been described (357). Both, experimental and theoretical data clearly show that the formal-donitrones, formed in the course of collision by electronic transfer, can hardly be molecularly isomerized into other [C,H3,N,0] molecules. Methods of quantum chemistry and molecular dynamics have made it possible to study the reactions of nitrone rearrangement into amides through the formation of oxaziridines (358). 

StabUity of the article/carrier mixture can be established in conjimction with the homogeneity assays of nonsolutions. Separate stabdity tests will, of course, be required for solutions. Formal stability trials sufficient to show long-term sta-bdity of the mixtures are not required rather, stabdity shoidd be estabhshed for a period that encompasses the period of use of the article/carrier mixture. Period of use shoidd be defined as whichever of the foUowing two time periods is longer, the time between preparation of the mixture and final administration of that mixture to the test system, or the time between preparation of the mixture and the analysis of the mixture as required by 58.113(a)(2). Often the period between preparation and analysis may be longer than the period between preparation and last administration to the test system. 

An analysis of Eqn. (11.19) reveals that w is positive for all values of k. Therefore, under the given boundary conditions, all perturbations grow with time. Thus, the result of the formal stability analysis agrees with the conclusions drawn in Section 11.2.1. The slab of oxide AO at the reducing surface is morphologically unstable as it moves under the action of an oxygen potential gradient (see Fig. 11-7). 

As was noted in Section 2.1.1, the concentration oscillations observed in the Lotka-Volterra model based on kinetic equations (2.1.28), (2.1.29) (or (2.2.59), (2.2.60)) are formally undamped. Perturbation of the model parameters, in particular constant k, leads to transitions between different orbits. However, the stability of solutions requires special analysis. Assume that in a given model relation between averages and fluctuations is very simple, e.g., (5NASNB) = f((NA), (A b)), where / is an arbitrary function. Therefore k in (2.2.67) is also a function of the mean values NA(t) and NB(t). Models of this kind are well developed in population dynamics in biophysics [70], Since non-linearity of kinetic equations is no longer quadratic, limitations of the Hanusse theorem [23] are lifted. Depending on the actual expression for / both stable and unstable stationary points could be obtained. Unstable stationary points are associated with such solutions as the limiting cycle in particular, solutions which are interpreted in biophysics as catastrophes (population death). Unlike phenomenological models treated in biophysics [70], in the Lotka-Volterra stochastic model the relation between fluctuations and mean values could be indeed calculated rather than postulated. 

This statement is not self-evident and needs some comments. A role of concentration degrees of freedom in terms of the formally-kinetic description was discussed in Section 2.1.1. Stochastic approach adds here a set of equations for the correlation dynamics where the correlation functions are field-type values. Due to very complicated form of the complete set of these equations, the analytical analysis of the stationary point stability is hardly possible. In its turn, a numerical study of stability was carried out independently for the correlation dynamics with the fixed particle concentrations. 

Earlier, a similar instanton analysis for a PES with two transition states was performed by Ivlev and Ovchinnikov [1987], in connection with tunneling in Josephson junctions. In the language of stability parameters introduced in Section 4.1, the appearance of two-dimensional tunneling paths is signaled by vanishing of the stability parameter. As follows from (4.24), the one-dimensional tunneling path formally becomes infinitely... 

Stability data (not only assay but also degradation products and other attributes as appropriate) should be evaluated using generally accepted statistical methods. The time at which the 95% one-sided confidence limit intersects the acceptable specification limit is usually determined. If statistical tests on the slopes of the regression lines and the zero-time intercepts for the individual batches show that batch-to-batch variability is small (e.g., p values for the level of significance of rejection are more than 0.25), data may be combined into one overall estimate. If the data show very little degradation and variability and it is apparent from visual inspection that the proposed expiration dating eriod will be met, formal statistical analysis may not be necessary. 

In such a small cluster, the surface coverage is 70%, much higher than the percentage measured on flat surfaces covered by arylthiolates (33%).206 The formation of the Au102(SR)44 nanoparticle may be explained with the electronic rules of cluster stability. Considering that each gold atom contributes one valence electron, and that each of the 44 SR thiyl radicals formally takes one electron, a 58 electron closed shell is obtained. A deep analysis and discussion on this landmark achievement have been recently published.207,208... 

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