Bifurcation oscillatory

The next problem to consider is how chaotic attractors evolve from tire steady state or oscillatory behaviour of chemical systems. There are, effectively, an infinite number of routes to chaos [25]. However, only some of tliese have been examined carefully. In tire simplest models tliey depend on a single control or bifurcation parameter. In more complicated models or in experimental systems, variations along a suitable curve in the control parameter space allow at least a partial observation of tliese well known routes. For chemical systems we describe period doubling, mixed-mode oscillations, intennittency, and tire quasi-periodic route to chaos. 

Once the parametric representation of the Jacobian is obtained, the possible dynamics of the system can be evaluated. As detailed in Sections VILA and VII.B, the Jacobian matrix and its associated eigenvalues define the response of the system to (small) perturbations, possible transitions to instability, as well as the existence of (at least transient) oscillatory dynamics. Moreover, by taking bifurcations of higher codimension into account, the existence of complex dynamics can be predicted. See Refs. [293, 299] for a more detailed discussion. 

Q dehydrocyclization, 29 311 ring enlargement, 29 311-316 Bifunctional Fisher-Tropsch/hydroformylation catalysts, 39 282 Bifunctional mechanism, 30 4 Bifurcation diagram, oscillatory CO/O, 37 233-234... 

TNC.48. G. Nicolis and I. Prigogine, Thermodynamic aspects and bifurcation analysis of spatio-temporal dissipative structures, in Proceedings, Faraday Symposium Chemical Society, no. 9, Physical Chemistry of Oscillatory Phenomena, 1975, pp. 7—20. 

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. 

Whereas two bifurcation values for the glucose input rate define the domain of oscillations in yeast extracts [40], only a single bifurcation value below which oscillations occur is found in intact yeast cells [47]. This does not necessarily imply a difference in oscillatory mechanism but merely indicates that in intact cells the glucose transporter becomes saturated before the intracellular glucose input has reached the upper bifurcation value above which oscillations disappear in yeast extracts [38]. 

Lakin, M. B. Oscillatory Flow Simulation in an Idealized Bifurcation. Ph.D. Thesis. Denver University of Denver. 1973. 223 pp. 

Interactions between the flame and the surrounding wall (in a combustion chamber) could influence the contaminant production. This is examined by Dionisios Vlachos and his group at the University of Delaware (formerly at the University of Massachusetts at Amherst) using numerical bifurcation techniques (Chapter 26). For the first time, oscillatory instabilities have been found and control methodologies have been proposed to reduce flame temperatures and NO2 emissions. 

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). 

Fig. 26.1a). At first, multistage ignitions and extinctions occur followed by a relaxation (long period) mode [7]. Oscillations die a few degrees below the ignition temperature at a saddle-loop infinite-period homoclinic orbit bifurcation point. This is an example where both ignition and extinction are oscillatory. 

If, however, we actually integrate the reaction rate equations numerically using the rate constants in Table 1.1 we find that the system does not always stick to, or even stay close to, these pseudo-steady loci. The actual behaviour is shown in Fig. 1.10. There is a short initial period during which d and b grow from zero to their appropriate pseudo-steady values. After this the evolution of the intermediate concentrations is well approximated by (1.41) and (1.42), but only for a while. After a certain time, the system moves spontaneously away from the pseudo-steady curves and oscillatory behaviour develops. We may think of the. steady state as being unstable or, in some sense repulsive , during this period in contrast to its stability or attractiveness beforehand. Thus we have met a bifurcation to oscillatory responses . The oscillations... 

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... 

Because of the importance of points where the trace of the Jacobian matrix vanishes, we will denote such values of the parameters by a superscript asterisk. Equation (3.65) has two real roots, provided the dimensionless rate constant for the uncatalysed step has a value less than In terms of the original rate constants this requires k2 > 8ku as presented previously ( 2.5). For the data in Table 2.3, n = 0.9847 and /rf = 0.1021. As k u increases, these two bifurcation points move closer together for ku = 0.1, // = 0.790 and n f = 0.420, so the oscillatory range is smaller. 

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. 

Before we can conclude, in general, that a given system will begin to show oscillatory behaviour between two Hopf bifurcation points we must attend to a few additional requirements of the theorem. 

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... 

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. 

Note that the dimensionless time rf, which gives the length of the pre-oscillatory period, will only be positive if the initial concentration /i0 exceeds the upper Hopf bifurcation value /if. If we start with a lower initial reactant concentration, so that /i0 < /if (but still with /i0 > /if), there will be no pre-oscillatory period the system will jump straight into oscillations which will persist until time rf. 

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. 

Because oscillatory behaviour persists only for a finite length of time, only a finite number of excursions can occur. We can estimate this number by obtaining an approximate value for the mean oscillatory period, im. For this we take a geometric mean of the periods at the two Hopf bifurcation points. These latter quantities can be evaluated from the frequency co0 defined by... 

We have now seen how local stability analysis can give us useful information about any given state in terms of the experimental conditions (i.e. in terms of the parameters p and ku for the present isothermal autocatalytic model). The methods are powerful and for low-dimensional systems their application is not difficult. In particular we can recognize the range of conditions over which damped oscillatory behaviour or even sustained oscillations might be observed. The Hopf bifurcation condition, in terms of the eigenvalues k2 and k2, enabled us to locate the onset or death of oscillatory behaviour. Some comments have been made about the stability and growth of the oscillations, but the details of this part of the analysis will have to wait until the next chapter. 

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. 

For all physically acceptable conditions, the determinant of J is positive, so we will not find saddle points or saddle-node bifurcations. We can, however, expect to find conditions under which nodal states become focal (damped oscillatory responses), i.e. where A = 0, and where focal states lose stability at Hopf bifurcations, i.e. where tr(J) = 0 and where we shall look for the onset of sustained oscillations. 

Fig. 4.4. The change in stability and growth of oscillatory solutions in the intermediate concentration and temperature excess as functions of the reactant concentration showing Hopf bifurcations at fi and n (parameter details as given in Table 4.1 except for y = 0) (a)...

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. 

Some typical oscillatory records are shown in Fig. 4.6. For conditions close to the Hopf bifurcation points the excursions are almost sinusoidal, but this simple shape becomes distorted as the oscillations grow. For all cases shown in Fig. 4.6, the oscillations will last indefinitely as we have ignored the effects of reactant consumption by holding /i constant. We can use these computations to construct the full envelope of the limit cycle in /r-a-0 phase space, which will have a similar form to that shown in Fig. 2.7 for the previous autocatalytic model. As in that chapter, we can think of the time-dependent... 

Fig. 4.9. The development of oscillatory amplitude Ae and period T across the range of instability, 4.2 x 10 3 = n < n < jx = 0.0195, for the pool chemical model with k = 2x 10-3 and y = 0.21, typical of a system with a subcritical Hopf bifurcation at which an unstable limit cycle emerges at The broken curves give the limiting forms predicted by eqns (4.59)—(4.61).

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. 

Hopf bifurcation analysis commonly signals the onset of oscillatory behaviour. This chapter uses a particular two-variable example to illustrate the essential features of the approach and to explore the relationship to relaxation oscillations. After a careful study of this chapter the reader should be able to ... 

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