Phases, local stability

Stationary-state solutions correspond to conditions for which both numerator and denominator of (3.54) vanish, giving doc/dp = 0/0, and so are singular points in the phase plane. There will be one singular point for each stationary state each of the different local stabilities and characters found in the previous section corresponds to a different type of singularity. In fact the terms node, focus, and saddle point, as well as limit cycle, come from the patterns on the phase plane made by the trajectories as they approach or diverge. Stable stationary states or limit cycles are often refered to as attractors , unstable ones as repellors or sources . The different phase plane patterns are shown in Fig. 3.4. 

To all intents and purposes, the Hopf prediction is the exact result. It is not difficult to construct an intersection between the maximum and minimum which is a stable stationary state. For instance, with y = 0.02, k = 0.12, and /i = 0.2 we have 0X = 1.042, 9C = 2400, and 0SS = i/k = 5/3. The corresponding phase plane nullclines are shown in Fig. 5.10, together with a trajectory spiralling in to the stationary-state intersection. The trace of the Jacobian matrix is negative for this solution (tr(J) = —4.1 x 10 2) indicating its local stability. This is not, however, a particularly fair test of the relaxation analysis because the parameters /i and k are not especially small. In the vicinity of the origin (where is small) both approaches converge. 

Such was the state of the art when Amundson and Bilous s paper was published in the first volume of the newly founded A.I.CH.E. Journal (Bilous and Amundson, 1955). This for the first time treated the reactor as a dynamical system and, using Lyapounov s method of linearization, gave a pair of algebraic conditions for local stability. One of these corresponded to the slope condition of previous analyses, and there was a brief flurry of attempts to invest the other with a similarly physical explanation. For the global picture they introduced the phase plane (another feature of the theory of dynamical systems) and, with consummate skill, Bilous conjured the now classic figures from a Reeves electronic analogue computer. Even in this early paper, they had touched upon the consecutive reaction scheme A - B - C and had shown that up to five steady states might be expected under some conditions. 

So far we have defined the local stability ("there exists such <3 as. . . ). Now let us define the global stability for rest points. The rest point c0 is called globally asymptotically stable (as a whole) within the phase space D if it is stable according to Lyapunov, and for any initial conditions d0e D the solution c(t, k, cLa) tends to approach c 0 at t - oo. 

In principle, to study the local stability of a stationary point from a linear approximation is not difficult. Some difficulties are met only in those cases where the real parts of characteristic roots are equal to zero. More complicated is the study of its global stability (in the large) either in a particular preset region or throughout the whole phase space. In most cases the global stability can be proved by using the properly selected Lyapunov function (a so-called second Lyapunov method). Let us consider the function V(c) having first-order partial derivatives dY/dCf. The expression... 

For convected vortical disturbance field in the freestream, the growth process is also qualitatively different as compared to the growth of disturbances that are created at the wall or inside the shear layer- as reported in Schubauer Skramstad (1947). For monochromatic wall excitation, the real frequency of the disturbance field is held fixed and the phase speed adjusts itself continually to the local stability property of the shear layer. Contrarily, for the disturbance held generated by convecting vortices, it is the phase speed or group velocity that is an invariant of the input distur-... 

Multiple Steady States and Local Stability in CSTR.—In the two decades since the seminal work of van Heerden and Amimdson, there has been vast output of papers conoemed with the dynamic behaviour of stirred-tank reactors. Bilous and Amundson put the van He den analysis of local stability of the equilibrium state on a rigorous basis by use of linear stability theory. Their method is similar to the phase-plane treatments of thermokinetic ignitions and oscillations discussed here in Sections 4 and 3 (and preceded them dironologically). The mass and energy balance for the CSTR having a single reactant as feedstock may be expressed as ... 

An analysis of the local stability of (c.,7. ) in the phase plane provides the necessary and sufficient conditions for stability as... 

Marangoni effects] nonequilibrated phases/local mass transfer leads to local changes in surface tension and stability analysis yields stable interfadal movement. 

The local extremum at S=0 represents the isotropic phase. For stability this must be a minimum. The other two values of S represent a local maximum and a local minimum for non-zero values of S. If S=0 is a local minimmn, then the solution using the negative sign must be either a local maximum or local minimum with the solution using the positive sign being the other. Let us look at the temperature for which the free energies per unit volume of the nematic and isotropic phases are equal. For this value of the temperature T, both the S=0 solution and the proper S>0 solution must be local minima with the... 

In case (a) near a concentration we have local stability, but we may still have an instability with respect to another branch of the phase diagram. This will become apparent in the plots of the Flory-Huggins free energy [eq. (IV.6)] (Fig. IV. 3). 

Now redo the linearization adding feedback control. For a value of K = 1, is the system locally stable What is the minimum value of K required for local stability Determine if the nonlinear system is actually stable for this value of K by developing a phase-plane plot using various starting conditions. 

When the second derivative is positive, the mixed system is stable and does not form separate phases. When the second derivative is negative, the system does form separate phases. To decide whether the system is stable, look at whether the free energy function is concave upward (]d F)/ dx ) > 0) or concave downward ( d F) j(dx ) < 0). This stability criterion can be applied to models for G(x) (when pressure is constant) or F(x) (when volume is constant) to predict phase boundaries. W e return to questions of local stabilities and small fluctuations on page 477. First, we consider global stabilities. 

The condition of local stability of a single component bulk phase consists in the positiveness of the compressibility or, in other words, d p./dP = dV/dP < 0 [64]. Thus, the entire branches LL and GG in Fig. 5b are concave. On the contrary, the unstable branch L G is convex. Change of sign of the value d p/dP = dV/dP occurs at the spinodal points (the boundaries of the region of the stable phases) L and G, where this value becomes infinite (cf... 

The analysis of the previous section provides detailed information only of the local stability of the individual stationary-states. The overall, global stability of the system is also of interest, especially when the highest extent of conversion is unstable. In particular, there is the possibility of sustained oscillatory responses. These correspond to movement around closed curves or limit cycles in the a-3 phase-plane (as opposed to isolas which are closed curves in the a - i... 

Sob, M., Wang, L. G., and Vitek, V., 1997, Local stability of higher-energy phases in metallic materials and its relation to the structure of extended defects. Comp. Mater. Sci. 8,100-106. 

The points (a,u) — (0,0) and (1,0) are singular points of the phase plane, at which dujda — (d /dz)/(do /d2 ) = 0/0. The evolution of the system in the vicinity of such singularities can be determined without a full solution of the differential equations. If Aa and Au represent the displacements of the concentration and gradient from one of these singularities, then the development of Aa and Au with 2 will depend on the local stability of the singularity and will follow an exponential growth or decay, with... 

An important aspect of the stabilization of emulsions by adsorbed films is that of the role played by the film in resisting the coalescence of two droplets of inner phase. Such coalescence involves a local mechanical compression at the point of encounter that would be resisted (much as in the approach of two boundary lubricated surfaces discussed in Section XII-7B) and then, if coalescence is to occur, the discharge from the surface region of some of the surfactant material. 

FIGURE 10 6 Confor mations and electron delo calization in 1 3 butadiene The s CIS and the s trans con formations permit the 2p or bitalsto be aligned parallel to one another for maxi mum TT electron delocaliza tion The s trans conformation is more stable than the s CIS Stabilization resulting from tt electron de localization is least in the perpendicular conformation which IS a transition state for rotation about the C 2—C 3 single bond The green and yellow colors are meant to differentiate the orbitals and do not indicate their phases... 

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