Stability oscillatory

Galerkin method becomes unstable and useless. It can also be seen that these oscillations become more intensified as a becomes larger (note that the factor affecting the stability is the magnitude of a and oscillatory solutions will also result using large negative coefficients). 

C.G. Vayenas, and J. Michaels, On the Stability Limit of Surface Platinum Oxide and its role in the oscillatory behavior of Platinum Catalyzed Oxidations, Surf. Sci. 120, L405-L408 (1982). 

R. W. Thatcher, A. A. Omon-Arancibia, and J. W. Dold, Oscillatory flame edge propagation, isolated flame tubes and stability in a non-premixed counterflow. Combust. Theory Model. 6 487-502,2002. 

It can easily be shown that for the upwind scheme all coefficients a appearing in Eq. (37) are positive [81]. Thus, no unphysical oscillatory solutions are foimd and stability problems with iterative equation solvers are usually avoided. The disadvantage of the upwind scheme is its low approximation order. The convective fluxes at the cell faces are only approximated up to corrections of order h, which leaves room for large errors on course grids. 

Study the simple, open-loop (KC = 0) and closed-loop responses (KC = -1 to 5, TSET = TDIM, and 300 to 350 K) and the resulting yields of B. Confirm the oscillatory behaviour and find appropriate values of KC and TSET to give maximum stable and maximum oscillatory yield. For the open-loop response, show that the stability of operation of the CSTR is dependent on the operating variables by carrying out a series of simulations with varying Tq in the range 300 to 350 K. 

At time t=212 h the continuous feeding was initiated at 5 L/d corresponding to a dilution rate of 0.45 d . Soon after continuous feeding started, a sharp increase in the viability was observed as a result of physically removing dead cells that had accumulated in the bioreactor. The viable cell density also increased as a result of the initiation of direct feeding. At time t 550 h a steady state appeared to have been reached as judged by the stability of the viable cell density and viability for a period of at least 4 days. Linardos et al. (1992) used the steady state measurements to analyze the dialyzed chemostat. Our objective here is to use the techniques developed in Chapter 7 to determine the specific monoclonal antibody production rate in the period 212 to 570 h where an oscillatory behavior of the MAb titer is observed and examine whether it differs from the value computed during the start-up phase. 

Furthermore, conjugate poles on the imaginary axis are BIBO stable—a step input leads to a sustained oscillation that is bounded in time. But we do not consider this oscillatory steady state as stable, and hence we exclude the entire imaginary axis. In an advanced class, you should find more mathematical definitions of stability. 

It is straightforward to show that the desired steady-state (i.e., the origin) is unstable. A robust tracking control law can be constructed to stabilize the CSTR under forced oscillatory operation. That is, we can derive a controller to track an oscillatory temperature profile (say, yr t) = a- - sin(47rt)), which can be generated by the exosystem (3) where... 

From a mathematical point of view, the onset of sustained oscillations generally corresponds to the passage through a Hopf bifurcation point [19] For a critical value of a control parameter, the steady state becomes unstable as a focus. Before the bifurcation point, the system displays damped oscillations and eventually reaches the steady state, which is a stable focus. Beyond the bifurcation point, a stable solution arises in the form of a small-amplitude limit cycle surrounding the unstable steady state [15, 17]. By reason of their stability or regularity, most biological rhythms correspond to oscillations of the limit cycle type rather than to Lotka-Volterra oscillations. Such is the case for the periodic phenomena in biochemical and cellular systems discussed in this chapter. The phase plane analysis of two-variable models indicates that the oscillatory dynamics of neurons also corresponds to the evolution toward a limit cycle [20]. A similar evolution is predicted [21] by models for predator-prey interactions in ecology. 

Bui, P.-A., D. G. Vlachos, and P. R. Westmoreland. 1999. On the local stability of multiple solutions and oscillatory dynamics of spatially distributed flames. Combustion Flame 117 307-22. 

Rheological observations of the UHMWPE pseudo-gels of different concentrations under oscillatory shear conditions at different temperatures showed that these systems exhibit considerable drawability at temperatures above ambient. The deformation of the crystalline phase of the gel-like system is not reversible and, as shown in the sequence of photographs Figure 2, for a pseudo-gel of 4% concentration, it was greater when the sample was sheared under the same oscillatory conditions at higher temperatures. The displaced crystals of the UHMWPE pseudo-gel showed remarkable dimensional stability after shear cessation and removal of any compression load in the optical rotary stage. 

The relative structural stability of a NFE metal is determined by an oscillatory pair potential of the type... 

If, however, we actually integrate the reaction rate equations numerically using the rate constants in Table 1.1 we find that the system does not always stick to, or even stay close to, these pseudo-steady loci. The actual behaviour is shown in Fig. 1.10. There is a short initial period during which d and b grow from zero to their appropriate pseudo-steady values. After this the evolution of the intermediate concentrations is well approximated by (1.41) and (1.42), but only for a while. After a certain time, the system moves spontaneously away from the pseudo-steady curves and oscillatory behaviour develops. We may think of the. steady state as being unstable or, in some sense repulsive , during this period in contrast to its stability or attractiveness beforehand. Thus we have met a bifurcation to oscillatory responses . The oscillations... 

Let us return to Fig. 1.12(c), where there are multiple intersections of the reaction rate and flow curves R and L. The details are shown on a larger scale in Fig. 1.15. Can we make any comments about the stability of each of the stationary states corresponding to the different intersections What, indeed, do we mean by stability in this case We have already seen one sort of instability in 1.6, where the pseudo-steady-state evolution gave way to oscillatory behaviour. Here we ask a slightly different question (although the possibility of transition to oscillatory states will also arise as we elaborate on the model). If the system is sitting at a particular stationary state, what will be the effect of a very small perturbation Will the perturbation die away, so the system returns to the same stationary state, or will it grow, so the system moves to a different stationary state If the former situation holds, the stationary state is stable in the latter case it would be unstable. 

The conditions under which the above stationary-state solution loses its stability can be determined following the recipe of 2.6. Again we find that instability may arise, and hence oscillatory behaviour is possible, in this reversible case. The condition for the onset of instability can be expressed in terms of the reactant concentration p < p p, where... 

Equations (3.20) and (3.21) with their stationary-state solutions (3.24) and (3.25) are simple enough to provide a good introduction to some of the mathematical techniques which can serve us so well in analysing these sorts of chemical models. In the next sections we will explain the ideas of local stability analysis ( 3.2) and then apply them to our specific model ( 3.3). After that we introduce the basic aspects of a technique known as the Hopf bifurcation analysis ( 3.4) which enables us to locate the conditions under which oscillatory states are likely to appear. We set out only those aspects that are required within this book, without any pretence at a complete... 

In fact we must work a little harder under these conditions as higher-order terms in the full expressions for d Aa/dr and d Ap/dx have now to be studied, and this simple oscillatory response may be modified or not realized at all in practice. Nevertheless, the change in local stability associated with the condition... 

These changes in stability and character are marked on the stationary-state loci in Fig. 3.5. This also shows clearly that the loss of stability occurs as the two loci cross, when /z = 1. At this point eqn (2.84) is also satisfied with tr(J) = 0, corresponding to the special case (f) of 3.2.1 and holding the promise of the onset of oscillatory behaviour. We return to this point later. 

In the previous sections we have implied that the loss of local stability which occurs for a stationary-state solution as the real part of the eigenvalues changes from negative to positive is closely linked to the conditions under which sustained oscillatory responses are born. 

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