Oscillations stable steady states

Most chemically reacting systems tliat we encounter are not tliennodynamically controlled since reactions are often carried out under non-equilibrium conditions where flows of matter or energy prevent tire system from relaxing to equilibrium. Almost all biochemical reactions in living systems are of tliis type as are industrial processes carried out in open chemical reactors. In addition, tire transient dynamics of closed systems may occur on long time scales and resemble tire sustained behaviour of systems in non-equilibrium conditions. A reacting system may behave in unusual ways tliere may be more tlian one stable steady state, tire system may oscillate, sometimes witli a complicated pattern of oscillations, or even show chaotic variations of chemical concentrations. 

As shown on Figs. 8.31 to 8.33 the rate and UWR (or 0) oscillations of CO oxidation can be started or stopped at will by imposition of appropriate currents.33 Thus on Fig. 8.31 the catalyst is initially at a stable steady state. Imposition of a negative current merely decreases the rate but imposition of a positive current of200 pA leads to an oscillatory state with a period of 80s. The effect is completely reversible and the catalyst returns to its initial steady state upon current interruption. 

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. 

The dynamic behavior of nonisothermal CSTRs is extremely complex and has received considerable academic study. Systems exist that have only a metastable state and no stable steady states. Included in this class are some chemical oscillators that operate in a reproducible limit cycle about their metastable... 

The steady-state design equations (i.e., Equations (14.1)-(14.3) with the accumulation terms zero) can be solved to find one or more steady states. However, the solution provides no direct information about stability. On the other hand, if a transient solution reaches a steady state, then that steady state is stable and physically achievable from the initial composition used in the calculations. If the same steady state is found for all possible initial compositions, then that steady state is unique and globally stable. This is the usual case for isothermal reactions in a CSTR. Example 14.2 and Problem 14.6 show that isothermal systems can have multiple steady states or may never achieve a steady state, but the chemistry of these examples is contrived. Multiple steady states are more common in nonisothermal reactors, although at least one steady state is usually stable. Systems with stable steady states may oscillate or be chaotic for some initial conditions. Example 14.9 gives an experimentally verified example. 

If the roots are, however, complex numbers, with one or two positive real parts, the system response will diverge with time in an oscillatory manner, since the analytical solution is then one involving sine and cosine terms. If both roots, however, have negative real parts, the sine and cosine terms still cause an oscillatory response, but the oscillation will decay with time, back to the original steady-state value, which, therefore remains a stable steady state. 

The experiments were performed at a constant inflow concentration of ascorbic acid ([H2A]) in the CSTR. Oscillations were found by changing the flow rate and the inflow concentration of the copper(II) ion systematically. At constant Cu(II) inflow concentration, the electrode potential measured on the Pt electrode showed hysteresis between two stable steady-states when first the flow-rate was increased, and then decreased to its original starting value. The results of the CSTR experiments were summarized in a phase diagram (Fig. 6). 

In Fig. 6, separate regions of bi-stability, oscillations and single stable steady-states can be noticed. This cross-shaped phase diagram is common for many non-linear chemical systems containing autocatalytic steps, and this was used as an argument to suggest that the Cu(II) ion catalyzed autoxidation of the ascorbic acid is also autocatalytic. The... 

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. 

Numerical simulations indicate that relay of cAMP pulses represents a different mode of dynamic behavior, closely related to oscillations. Just before autonomous oscillations break out, cells in a stable steady state can amplify suprathreshold variations in extracellular cAMP in a pulsatory manner. Thus, relay and oscillations of cAMP are produced by a unique mechanism in adjacent domains in parameter space. The two types of dynamic behavior are analogous to the excitable or pacemaker behavior of nerve cells. 

If the Lewis numbers are not chosen carefully for the apparatus, then the corresponding chemical/biological system may be subject to oscillations even when we use feedback control to limit the number of steady states to one optimal stable steady state as we have done since Figure 4.34. 

Some of these chlorite oscillators exhibit particularly interesting or exotic phenomena. Batch oscillations in the absence of flow may be obtained in the systems numbered 3, 10 a and 13, while the chlorite-iodide-malonic acid reaction gives rise to spatial wave patterns as well. These latter, which are strikingly similar to those observed in the BZ reaction61 are shown in Fig. 12. Addition of iodide to the original chlorite-iodate-arsenite oscillator produces a system with an extremely complex phase diagram58, shown in Fig. 13, which even contains a region of tristability, three possible stable steady-states for the same values of the constraints. 

We next verify that there exists a hot stable steady state when the system equations have three stationary solutions. With the temperature controller in manual, we lower the bypass rate from 12 percent to 5 percent. According to Fig. 5.21 this should create three steady states and we would expect the intermediate state to be unstable. Figure 5.30 shows what happens when we reduce the bypass rate. The reactor feed temperature goes up initially due to reduced bypassing. This makes the reactor exit temperature drop due to the inverse response. The drop in the reactor exit temperature causes a drop in the feed temperature below the intermediate steady state (point b in Fig. 5.21). However, this state is unstable so the temperatures continue to oscillate. Slowly, but surely, the temperatures trend toward the hot steady state (point c in Fig. 5.21) where the oscillations die out and the reactor remains stable. 

The start-up procedure for the experiments shown in Figure 11 involved filling the reactor with distilled, deionized water prior to starting the feed stream pumps. In these runs the conversion seems much more stable. The top curve, however, demonstrates that these apparently stable steady states may be subject to rapid change perhaps the seeking of a new, more stable, steady state or the beginning of oscillations. ... 

For those cases where only one stable steady state is possible, sustained oscillations cannot be excluded (the single steady state may be unstable). For a restricted class of reaction networks, stability (or lack thereof) of the single steady state can be demonstrated. 

Fig. 5.9. Variation of (a) concentration y and (b) temperature excess 6 in time for a value of in region of steady-state instability showing sustained oscillations (c) typical limit cycle lying around unstable steady-state o produced by plotting y against (d) for some parameter values there are two limit cycles surrounding a stable steady-state, one unstable (broken curve) the other stable (solid curve). In (c) and (d) example trajectories are shown (thin curves) that wind either onto the stable limit cycle or, in (d) onto the stable steady-state...

Stable, limit cycle. The latter occurs in the Salnikov case and the modified bifurcation diagram is shown in Fig. 5.11(b). The stable limit cycle born at the lower Hopf point overshoots the upper Hopf point but is extinguished by colliding with the unstable limit cycle born at the upper Hopf point which also grows in amplitude as )jl is increased. Over a, typically narrow range, then there are two limit cycles, one unstable and one stable around the (stable) steady-state point. If we start with the system at some large value of /r, so we settle onto the steady-state locus, and then decrease the parameter, we will first swap to oscillations at the Hopf point /r - At this point there is a stable limit cycle available as the system departs from the now unstable steady-state, but this stable limit cycle is not born at this point and so already has a relatively large amplitude. We would expect to... 

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