Local stability analysis procedure

Of the various methods of weighted residuals, the collocation method and, in particular, the orthogonal collocation technique have proved to be quite effective in the solution of complex, nonlinear problems of the type typically encountered in chemical reactors. The basic procedure was used by Stewart and Villadsen (1969) for the prediction of multiple steady states in catalyst particles, by Ferguson and Finlayson (1970) for the study of the transient heat and mass transfer in a catalyst pellet, and by McGowin and Perlmutter (1971) for local stability analysis of a nonadiabatic tubular reactor with axial mixing. Finlayson (1971, 1972, 1974) showed the importance of the orthogonal collocation technique for packed bed reactors. 

It is obvious, that in spite of its prognostic value the discussed procedure is based on integrated parameters of molecular structure and does not allow for judging the reason of stability of one isomer or instability of another. The attempts to work out some criteria of stability (J. Aihara et al.) have failed, because of the absence of analysis of local stability, i.e. the stability of substructures combination of which makes a fullerene molecule. In our opinion the synthesis of fullerenes proceeds by the way of selection of stable substructures. However, as a rule theoretical calculations consider molecule as a whole that seems to be a lack of existing approach of studying stability of fullerenes. 

An additional type of derivative that is often used for glycan profiling and analysis is the modification of the reducing end with a chromophore, usually achieved by the formation of a Schiff base with an aromatic amine, and subsequent reduction with sodium borohydride to stabilize the initially formed product by its conversion to a saturated amine. This procedure was initially developed for HPLC analysis with ultraviolet detection, but it is also appropriate for MS analysis because the substituted amino group represents a site for charge localization that simplifies the fragmentation pattern and also provides the opportunity to introduce a stable isotope label for quantitative purposes (54). 

Immobilized enzymes are defined as biocatalysts that are restrained or localized in a microenvironment yet retain their catalytic properties. Immobilization often increases stability and makes the reuse of the enzyme preparation very simple. The repeated analysis that can be performed by the use of enzymes in an immobilized form reduces the cost of the analysis. The ideal immobilization procedure for a given enzyme is one which permits a high turnover rate of the enzyme yet also retains a high catalytic activity over time. There are two major applications for immobilized enzymes in analytical systems. In one, the enzyme is immobilized onto a particulate solid support matrix, which is then packed into a small column and incorporated into a flow system. The other involves immobilization within or on the surface of an electrode, whereby the electrochemical transduction of enzymatic product is monitored. 

When a nonlinear system ewolwes under far-from-equilibrium conditions in the vicinity of a bifurcation point, a purely deterministic description often proved to be incomplete. The fluctuations of the dynamical variables can play an essential role and obstruct the observation of a transition expected by a deterministic analysis. In the framework of the deterministic approach, the stability of the different states according to the values of the control parameters is studied through a mathematical analysis of the velocity field. In particular, the theory of normal forms leads to the determination of the various kinds of attractors [l,2]. As far as we are concerned with the stochastic approach, the rrLa te.n. equation, has been widely used to analyze bifurcations of homogeneous or spatially ordered steady states or of limit cycles [3,4]. Our aim in the present contribution is to insist on the generality of the method to analyze various kinds of bifurcations in nonlinear nonequilibrium systems. The general procedure proposed to obtain a local description of the probability, which allows us to determine the system s attractors, turns out to display marked analogies with the theory of normal forms. 

---