Unstable steady states

The rabbit and l5mx problem does have stable steady states. A stable steady state is insensitive to small perturbations in the system parameters. Specifically, small changes in the initial conditions, inlet concentrations, flow rates, and rate constants lead to small changes in the observed response. It is usually possible to stabilize a reactor by using a control system. Controlhng the input rate of lynx can stabilize the rabbit population. Section 14.1.2 considers the more realistic control problem of stabilizing a nonisothermal CSTR at an unstable steady state. 

Kusumi, A., Koyama-Honda, I. and Suzuki, K. Molecular dynamics and interactions for creation of stimulation-induced stabilized rafts from small unstable steady-state rafts. Traffic 5 213-230, 2004. 

To evaluate model, we focus on the experimentally observed metabolic state of the pathway. However, in the case of sustained oscillations, the (unstable) steady state cannot be observed directly. We thus approximate the metabolic state by the average observed concentration and flux values, as reported in Refs. [101, 126], See Table VII for numeric values. 

D.D. Bruns and J.E. Bailey. Process operation near an unstable steady state using nonlinear feedback control. Chem. Eng. Sci., 30 755-762, 1975. 

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. 

When the steady state becomes unstable, the system moves away from it and often undergoes sustained oscillations around the unstable steady state. In the phase space defined by the system s variables, sustained oscillations generally correspond to the evolution toward a limit cycle (Fig. 1). Evolution toward a limit cycle is not the only possible behavior when a steady state becomes unstable in a spatially homogeneous system. The system may evolve toward another stable steady state— when such a state exists. The most common case of multiple steady states, referred to as bistability, is of two stable steady states separated by an unstable one. This phenomenon is thought to play a role in differentiation [30]. When spatial inhomogeneities develop, instabilities may lead to the emergence of spatial or spatiotemporal dissipative stmctures [15]. These can take the form of propagating concentration waves, which are closely related to oscillations. 

Figure 6-12 Transients in the adiabatic CSTR for the irreversibe reaction of the previous example. The upper panels show X(t) and T(t), while the lower panel displays X(T) for the same curves shown in the upper panel. The system converges on one of the stable steady states but never on the middle unstable steady state.

Fig. 17. Temporal dissipative structure after various time intervals during the period of oscillation. The reaction medium is a circle with zero flux boundary conditions. The lines correspond to isoconentrations. A =2, B = 5.4, Dt = 8 10 3, D2 = 4- 1G"3. Curves of equal concentration for Y are represented by full or broken lines when the concentration is, respectively, larger or smaller than the unstable steady state. The radius of the circle r0 = 0.5861. 

Steady-states can be stable, unstable, or indifferent. A stable steady-state holds the system fast that is, if the concentration C, slightly deviates from the steady-state, the system reacts such as to bring the concentration back to the steady-state. At an unstable steady-state, small excursions are self-perpetuating such that the system moves away from its original state. At indifferent steady-states the system does not respond to small excursions away from the steady-state. 

To demonstrate the difference between stable and unstable steady-states we compare the nontrivial steady states, C-2), of the cases shown in Fig. 21.66 and c, respectively. Imagine that the actual concentration is slightly smaller than C-2) (i.e., in Fig. 21.6 the system is to the left of Cj2)). Then for case b production p(Ct) is smaller than decay r(C,). According to Eq. 21-26, C, decreases and moves even further away from its original position until it finally comes to a halt at the other steady-state, Cf2 = 0. In contrast, for case c a slight deviation to the left causes the concentration to grow (p>r)... 

FIGURE 19 Phase-plane of a C controlled about the unstable steady state (initially a saddle-point) by cooling proportional to the temperature. 

This work is centred around the study of the response to periodic forcing of systems that, when autonomous, had a stable limit cycle surrounding an unstable steady state in their phase plane. For the sake of simplicity—and since many of the fundamental phenomena are the same—we studied two-dimensional systems. We chose two examples of isothermal reactor models the first is an autocatalytic homogeneous Brusselator (Glansdorff and Prigog-ine, 1971) ... 

As FA - 0, the projection of the torus on the phase plane [see Fig. 3(c)] is a very thin annulus surrounding the unperturbed limit cycle. Within this annulus there exists a small unstable periodic trajectory (a period 1) resulting from the forcing of the unperturbed unstable steady state within the limit cycle. This unstable period 1 will play a crucial role in the breaking of the torus as the forcing amplitude will become larger. 

If we consider the effect of the real values of the parameter Sh/Nu on the region of instabilities, then the value Lw decreases however, the inequality Lw > 1 is still valid (12). From the material presented it is apparent that the occurence of unstable steady states cannot be explained in terms of thermokinetic theory (12). 

Operating at the middle unstable steady state requires using some means of control for the plant, such as a stabilizing controller or nonadiabatic operation with carefully chosen parameters to stabilize the saddle-point type of the unstable steady state. 

If it is unstable, develop and use a MATLAB program for a nonadiabatic CSTR and find the cooling jacket parameters Kc and yc that will stabilize the unstable steady state. 

The maximum productivity of the desired product B usually occurs at the middle unstable saddle-type steady state. In order to stabilize the unstable steady state, a simple proportional-feedback-controlled system can be used, and we shall analyze such a controller now. A simple feedback-controlled bubbling fluidized bed is shown in Figure 4.25. 

Therefore we conclude not only that feedback control is useful to stabilize an optimal unstable steady state such as depicted in Figures 4.34 to 4.37 for the original set of parameter data, but feedback control can also help ensure the robustness of an otherwise stable optimal steady state over a larger region of parameters and system perturbations. Proper feedback control is also helpful in damping temperature explosions. 

The middle two plots show the dynamics of the reaction in the second tank. One steady state of tank 2 lies at (xAi(Tend), XBi(Tend), y Tend)) ss (0.33,0.67,1.28) and another at (xAi Tend), XBi Tend), y (Tend)) ss (0, 0.2,1.87). The latter gives the smaller yield of jg and results from the initial second tank conditions (x,i2(0), xb2(0), 2/2(0)) = (0.95,0,1.3) depicted in black. These two steady states are stable. There is another unstable steady state for this data, but our graphical method does obviously not allow us to find it because it is an unstable saddle-type steady state that will repel any profile that is near to it. It can be easily obtained from the steady-state equations, though. For a method to find all steady states of a three CSTR system, see Section 6.4.3. 

How can one distinguish between the stable and the unstable steady states found via the MATLAB code of problem 4 ... 

Since FCC units are usually operated at their middle unstable steady state, extensive efforts are needed to analyze the design and dynamic behavior of open loop and closed loop control systems to stabilize the desirable middle steady state. 

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