Solution unstable steady-state

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. 

Fig. 5.11. Variation of the oscillatory (limit cycle) solution with ju for the simple Salnikov model showing that the stable limit cycle born at one supercritical Hopf bifurcation exists over the whole range of the unstable steady-state, shrinking to zero amplitude at the other Hopf point (b) in this case, each Hopf point gives rise to a different limit cycle, with a stable limit cycle born at fi growing as increases and an unstable limit cycle born at /x also increasing in size as fx increases. At some fx> fx the two limit cycles collide and are...

In addition to the performance variations with reactant concentration and gas hourly space velocity (GHSV), there can be multiple steady states observed. Generally, a reactor in which a single, exothermic reaction is occurring will operate in one of two stable steady states. Additionally, an unstable steady-state solution to the mass... 

We obtained three different steady states. The stability of these states can be verified by assigning these values as the initial conditions. If we start with a stable steady state solution as the initial condition, the process remains at the stable steady state solution. If we start with an unstable steady state solution, the process moves to one of the steady state solutions. 

Figure 13.19 presents the conversion vs. plant Damkdhler number, for two different reaction orders and different values of the recycle concentration. Solid and dashed lines refer to stable and unstable steady states, respectively. It is obvious that one solution is unfeasible, corresponding to zero conversion and infinite recycle flow rate (its stability is presented for 23= 1). The other one is feasible (0 [Pg.525]

The photoenolization of the quinone (286) can be carried by irradiation at 313 or 365 nm in acid solution. The steady state irradiation has identified the product as the unstable hydroxylated compound (287) which is formed via the enol (288). The presence of this intermediate was detected in a laser flash study of the reaction. The quinones (289) undergo cyclization when irradiated with visible light.The mechanism by which the compounds (289) are transformed into the derivatives (290) involves the production of an excited state that is either a zwitterion or a biradical. After the transfer of a hydrogen the intermediate (291) is formed. It is within this species that cyclization occurs to give the final products. (2+2)-Cyclo-adducts such as (292) and oxetanes can be obtained by the photochemical addition of quinones to homobenzvalene. Interest in the photo-SET in quinone systems has led to the synthesis of the pyropheophytin substituted naphthoquinone dyads (293). A pulse radiolysis study of vitamin K in solution has been reported. 

Stability analysis could prove to be useful for the identification of stable and unstable steady-state solutions. Obviously, the system will gravitate toward a stable steady-state operating point if there is a choice between stable and unstable steady states. If both steady-state solutions are stable, the actual path followed by the double-pipe reactor depends on the transient response prior to the achievement of steady state. Hill (1977, p. 509) and Churchill (1979a, p. 479 1979b, p. 915 1984 1985) describe multiple steady-state behavior in nonisothermal plug-flow tubular reactors. Hence, the classic phenomenon of multiple stationary (steady) states in perfect backmix CSTRs should be extended to differential reactors (i.e., PFRs). 

M, and c = 5.0 x 10 M. Find the chemically acceptable stationary states of the kinetic term in (4.136) and determine their stability. Show that (4.136) has a propagating front solution connecting the stable to the unstable steady state and determine the propagation velocity v. Compare your value with the experimental... 

The three solutions of Equation (8.18) can be either all real or one real and two conjugate complexes, depending on the parameters /c and A. In other words, this nonlinear system has one (Figure 8.1b) or three (Figure 8.1c) steady states for same external conditions. They can be stable or unstable. A steady state is stable if a small perturbation of the system tends to decay. It is unstable if the perturbation tends to grow, displacing the system in another state. The unstable steady state is always surrounded by the stable steady states (Figure 8.1c, Ai < A < A2). 

Figure 2. Dependence of the solution (y)Qo on the inlet concentration of oxygen in CO oxidation, obtained by the continuation. sSS-stable steady state, uSS-unstable steady state, sP-stable periodic oscillations (minimum and maximum values), Hopf BP-Hopf bifurca-tionpoint unstable periodic solutions are not presented. T=630 K( isothermal), 5=20 pm, Lhm=S0 mol.m, Lqsc-32 mol.m, no diffusiorml resistance in the washcoat.

More complex oscillations have been found when the full TWC microkinetic model (Eqs. 1-31 in Table 1) has been used in the computations, cf. Fig. 4. The complex spatiotemporal pattern of oxidation intermediate C2H2 (Fig. 4, right) illustrates that the oscillations result from the composition of two periodic processes with different time constants. For another set of parameters the coexistence of doubly periodic oscillations with stable and apparently unstable steady states has been found (cf. Fig. 5). Even if LSODE stiff integrator (Hindmarsh, 1983) has been succesfully employed in the solution of approx. 10 ODEs, in some cases the unstable steady state has been stabilised by the implicit integrator, particularly when the default value for maximum time-step (/imax) has been used (cf. Fig. 5 right and Fig. 3 bottom). Hence it is necessary to give care to the control of the step size used, otherwise false conclusions on the stability of steady states can be reached. 

Since this quantity is positive for sufficiently small nonzero q, the plane waves are always unstable. The origin of the instability of small-amplitude plane waves is also clear from the fact that the zero-amplitude plane wave state is nothing but the unstable steady state from which the time-periodic solution has bifurcated. 

Usually, this is a good approximation, but one should be careful, since it is not always guaranteed that there is only one set of coverages corresponding to a steady state. For cases where the coverages are time dependent due to an oscillation in rates, a steady-state approximation is still typically made. This relies on the (not proven) expectation for the time-averaged rate of the time-dependent microkinetic model to be similar to the rate of an unstable steady-state solution. 

The next step should clarify why the unstable growth of the variable x occurs through a stable state at the bifurcation point. To determine the stability of the bifurcation point, it is necessary to examine the linear stability of the steady-state solution. For Eq. (1), the steady-state solution at the bifurcation point is given as jc0 = 0. So, let us examine whether the solution is stable for a small fluctuation c(/). Substituting Jt = b + Ax(f) into Eq. (1), and neglecting the higher order of smallness, it follows that... 

Figure 10. Adsorbed cation coverage as a function of electrode potential, assuming a cation interaction parameter / = 6.18 The solid line is the steady-state solution, whereas the broken line is the quasi-steady solution. Open circles indicate the unstable area. (From G. L. Griffin, J. Electrochettu Soc. 131, 18, 1984, Fig. 1. Reproduced by permission of The Electrochemical Society, Inc.)...

Fig. 10.9 Diagram of steady states I and III are domains of existence of single solution, II is a domain of existence of three solutions (two stable and one unstable). Lines A and A2 correspond to two stable solutions. Reprinted from Yarin et al. (2002) with permission...

This set of first-order ODEs is easier to solve than the algebraic equations where all the time derivatives are zero. The initial conditions are that a ut = no, bout = bo,... at t = 0. The long-time solution to these ODEs will satisfy Equations (4.1) provided that a steady-state solution exists and is accessible from the assumed initial conditions. There may be no steady state. Recall the chemical oscillators of Chapter 2. Stirred tank reactors can also exhibit oscillations or more complex behavior known as chaos. It is also possible that the reactor has multiple steady states, some of which are unstable. Multiple steady states are fairly common in stirred tank reactors when the reaction exotherm is large. The method of false transients will go to a steady state that is stable but may not be desirable. Stirred tank reactors sometimes have one steady state where there is no reaction and another steady state where the reaction runs away. Think of the reaction A B —> C. The stable steady states may give all A or all C, and a control system is needed to stabilize operation at a middle steady state that gives reasonable amounts of B. This situation arises mainly in nonisothermal systems and is discussed in Chapter 5. 

If A,i and X.2 are real numbers and both or one of the roots are positive, the system response will diverge with time and the steady-state solution will therefore be unstable, corresponding to an unstable node. ... 

---