Stationary bifurcation

In their subsequent works, the authors treated directly the nonlinear equations of evolution (e.g., the equations of chemical kinetics). Even though these equations cannot be solved explicitly, some powerful mathematical methods can be used to determine the nature of their solutions (rather than their analytical form). In these equations, one can generally identify a certain parameter k, which measures the strength of the external constraints that prevent the system from reaching thermodynamic equilibrium. The system then tends to a nonequilibrium stationary state. Near equilibrium, the latter state is unique and close to the former its characteristics, plotted against k, lie on a continuous curve (the thermodynamic branch). It may happen, however, that on increasing k, one reaches a critical bifurcation value k, beyond which the appearance of the... 

Even though the bifurcation behavior exhibits a Z-shaped curve, it is more complicated due to the existence of the HB. For example, upon ignition, the system is expected to oscillate because no locally stable stationary solutions are found (an oscillatory ignition). Time-dependent simulations confirm the existence of self-sustained oscillations [7, 12]. The envelope of the oscillations (amplitude of H2 mole fraction) is shown in circles (a so-called continuation in periodic orbits). 

The VRIs are chemically relevant features on a PE hypersurface even though they do not happen to be stationary points. They represent perplexing places for traditional kinetic models, such as TST, because these models have no way of predicting what fraction of molecules will choose one path or the other at the bifurcation. In other words, TST cannot tell you what the product ratio will be in a reaction that occurs via a VRI. Several examples of such reactions are now known. 

We shall not specify here the initial conditions on C(x,t) since in what follows we shall only be preoccupied with the limit state resulting from a Hopf bifurcation from the following stationary solution of the above system... 

Thus, to summarize, a Hopf bifurcation of the stationary solution Wq, given by (6.3.16), occurs at... 

We thus observe that M is positive for all w, A > 0 and, therefore, the bifurcation is supercritical. This stands in contrast with the result for the classical Teorell model discussed in the previous section. Recall that there the type of bifurcation depended on the properties of the postulated stationary resistance function /( ), specifically on the sign of / "(0) whenever... 

The stationary-state response curves, or bifurcation diagrams shown in Figs 1.13(b) and 1.12(f), represent two of the simplest possible patterns monotonic variation and a single hysteresis loop respectively. These are the only qualitatively different responses possible for the cubic autocatalytic step on its own. They are also found for a first-order exothermic reaction in an adiabatic flow reactor (see chapter 6). With only slightly more complex chemical mechanisms a whole array of extra exotic patterns can be found, such as those displayed in Fig. 1.14. The origins of these shapes will be determined in chapter 4. 

Fig. 1.14. Four more of the possible stationary-state bifurcation diagrams for chemical systems (see also Fig. 1.2) in flow reactors (a) isola (b) mushroom (c) isola + hysteresis loop ...

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... 

Fig. 3.6. The dependence of local stability and character of the stationary-state solution on the parameters /i amd ku. (a) The locus of Hopf bifurcation points with tr(J) = 0 beneath...

Fig. 3.7. Stationary-state loci a( ) and / (/<) showing changes in local stability when the uncatalysed step is included in the model and ku <, showing two Hopf bifurcation points H and n. Particular numerical values correspond to parameter values in Table 3.1.

The question of what happens to the system in the range of instability, and how the concentrations of A and B vary as they move away from the unstable stationary state, leads us to the study of sustained oscillatory behaviour. Before a full appreciation of the latter can be obtained, however, we must rehearse the relevant theoretical background. Fortunately the autocatalytic model is again an exemplary system with which to introduce at least the basic aspects of the Hopf bifurcation, and we will do this in the next section. 

We must also examine the stability of the periodic solution and its limit cycle as it emerges from the bifurcation point. Just as stationary states may be stable or unstable, so may oscillatory solutions. If they are stable they may be observable in practice if they are unstable they will not be directly observable although their existence still has some physical relevance. We will give the recipe for evaluating the stability and character of a Hopf bifurcation in the... 

As the dimensionless concentration of the reactant decreases so that pi just passes through the upper Hopf bifurcation point pi in Fig. 3.8, so a stable limit cycle appears in the phase plane to surround what is now an unstable stationary state. Exactly at the bifurcation point, the limit cycle has zero size. The corresponding oscillations have zero amplitude but are born with a finite period. The limit cycle and the amplitude grow smoothly as pi is decreased. Just below the bifurcation, the oscillations are essentially sinusoidal. The amplitude continues to increase, as does the period, as pi decreases further, but eventually attains a maximum somewhere within the range pi% < pi < pi. As pi approaches the lower bifurcation point /zf from above, the oscillations decrease in size and period. The amplitude falls to zero at this lower bifurcation point, but the period remains non-zero. 

There are no unstable limit cycles in this model, and the oscillatory solution born at one bifurcation point exists over the whole range of stationary-state instability, disappearing again at the other Hopf bifurcation. Both bifurcations have the same character (stable limit cycle emerging from zero amplitude), although they are mirror images, and are called supercritical Hopf bifurcations. 

The second significant difference between the predictions and the actual results is that oscillations survive beyond the time. This arises because the pseudo-stationary state has focal character just after the second Hopf bifurcation (i.e. the slowly varying eigenvalues i1>2 are complex conjugates with now negative real parts) so there is a damped oscillatory return to the locus. In Fig. 3.10(a) this can be seen after t 3966, whilst t = 3891. 

In the next few sections we will concentrate on the form of the governing equations (4.24) and (4.25) with the exponential approximation to f(0) as given by (4.27). We will determine the stationary-state solution and its dependence on the parameters fi and k, the changes which occur in the local stability, and the conditions for Hopf bifurcation. Then we shall go on and use the full power of the Hopf analysis, to which we alluded in the previous chapter, to obtain expressions for the growth in amplitude and period of the emerging oscillatory solutions. 

The condition for a change in the local stability of the stationary state in this model is that the trace of the Jacobian matrix should be zero. We can also recognize this as the first requirement for Hopf bifurcation, about which we shall have more to say in the next section. The condition tr(J) = 0 is also most easily handled parametrically by replacing n by k0 wherever possible in eqn (4.42). This leads to... 

There are then, also, two values of the dimensionless concentration of the reactant, /if and /if with /if > /if say, on this locus. For our example these are /if = 0.05797 and /if = 0.2070. In between these solutions, the stationary state is unstable. For any other particular system with a different value of k, the appropriate Hopf bifurcation points can be calculated in a similar way, as given in Table 4.3, or read off Fig. 4.3. However, if k is small, we can also estimate /if and /if directly by using an approximate, but quite accurate,... 

We have already determined the following information about the behaviour of the pool chemical model with the exponential approximation. There is a unique stationary-state solution for ass, the concentration of the intermediate A, and 0SS, the temperature rise, for any given combination of the experimental conditions /r and k. If the dimensionless reaction rate constant k is larger than the value e-2, then the stationary state is always stable. If heat transfer is more efficient, so that k Hopf bifurcation points along the stationary-state locus as /r varies (Fig. 4.4). If these bifurcation points are /r and /z (with the stationary state... 

The behaviour exhibited by this model is relatively simple. There is only ever one limit cycle. This is born at one bifurcation point, grows as the system traverses the range of unstable stationary states, and then disappears at the second bifurcation point. Thus there is a qualitative similarity between the present model and the isothermal autocatalysis of the previous chapter. The limit cycle is always stable and no oscillatory solutions are found outside the region of instability. 

In this section, therefore, we briefly investigate the stationary-state and Hopf bifurcation patterns that are found with the exact Arrhenius temperature dependence. 

We may view eqns (4.71)—(4.73) in another way. Choose a system with y < Next choose the dimensionless rate constant k. If k is less than (1 — 4y)e-2, eqn (4.71) can be solved to yield two positive roots 9 and 9. From these values for the stationary-state temperature excess we calculate the reactant concentration required for Hopf bifurcation from eqn (4.72) whilst (4.73) gives the stationary-state concentration of the intermediate A. 

First, can we expect any oscillatory behaviour Instability is possible only if k < e 2. This requirement is satisfied here. From the data in Table 4.4, the Hopf bifurcation points for this system occur for n = 0.207 and n = 0 058. For our example, the initial value /r0 = 0.5 exceeds the upper bifurcation point, so the system at first has a stable pseudo-stationary state to approach, with dss x 10 and ass x 4.54 x 10 4. From Fig. 4.3 we may also estimate that the approach to this state will be monotonic since the initial conditions lie outside the region of damped oscillations. 

In this chapter we give an introduction and recipe for the full Hopf bifurcation analysis for chemical systems. Rather than work in completely general and abstract terms, we will illustrate the various stages by using the thermokinetic model of the previous chapter, with the exponential approximation for simplicity. We can draw many quantitative conclusions about the oscillatory solutions in that model. In particular we will be able to show (i)that the parameter values given by eqns (4.49) and (4.50) for tr(J) = 0 satisfy all the requirements of the. Hopf theorem (ii)that oscillatory behaviour is completely confined to the conditions for which the stationary state is... 

The locus described by these equations in the K-fx parameter plane is reproduced in Fig. 5.1, which also shows a typical stationary-state bifurcation diagram for fixed k. Hopf bifurcations occur at two values of the precursor reactant concentration /if, 2, with /if < /if, for any given k less than a... 

Fig. 5.1. (a) A typical parameter plane showing a locus of Hopf bifurcation points. For any given value of the parameter k on the ordinate we may construct a horizontal (broken line) the Hopf bifurcation points, pi and pi, are then located as shown. The corresponding stationary-state loci, shown in (b) and (c), have unstable solutions between pi and pi. ... 

When we come to look at the stability of the limit cycle which is born at the Hopf bifurcation point, we shall meet a quantity known as the Floquet multiplier , conventionally denoted p2, which plays a role similar to that played for the stationary state by the eigenvalues and k2. If / 2 is negative, the limit cycle will be stable and should correspond to observable oscillations if P2 is positive the limit cycle will be unstable. 

Fig. 5.3. Locus of Hopf bifurcation points in K-fi parameter plane for thermokinetic model with the full Arrhenius temperature dependence and y = 0.21. The nature of the Hopf bifurcation point and, hence, the stability of the emerging limit cycle changes along this locus at k = 2.77 x 10 3. Supercritical bifurcations are denoted by the solid curve, subcritical bifurcations occur along the broken segment, i.e. at the upper bifurcation point for the lowest k. The stationary-state solution is unstable and surrounded by a stable limit cycle for all parameter values within the enclosed region. Oscillatory behaviour also occurs in the small shaded region below the Hopf curve, where the stable stationary state is surrounded by both an unstable and...

When the Hopf bifurcation at p is supercritical (/ 2 < 0) the system has just a single stable limit cycle. This emerges at p and exists across the range p < p < p, within which it surrounds the unstable stationary-state solution. The limit cycle shrinks back to zero amplitude at the lower bifurcation point p%. This behaviour is qualitatively the same as that shown with the simplifying exponential approximation and is illustrated in Fig. 5.4(a). 

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