Bifurcation behaviour

Abstract. A model of the conformational transitions of the nucleic acid molecule during the water adsorption-desorption cycle is proposed. The nucleic acid-water system is considered as an open system. The model describes the transitions between three main conformations of wet nucleic acid samples A-, B- and unordered forms. The analysis of kinetic equations shows the non-trivial bifurcation behaviour of the system which leads to the multistability. This fact allows one to explain the hysteresis phenomena observed experimentally in the nucleic acid-water system. The problem of self-organization in the nucleic acid-water system is of great importance for revealing physical mechanisms of the functioning of nucleic acids and for many specific practical fields. 

For given values of the rate constants kx — and K, the behaviour of this reduced system is determined by the constant value of c. If the concentration of C is maintained at a sufficiently low value, the unique stationary-state solution (ass, bss) is locally stable. As c is increased, however, there may be a Hopf bifurcation, at c = c say. For c > c, then, the stationary state will be unstable and surrounded by a stable limit cycle whose amplitude grows in some non-linear fashion with c. This bifurcation behaviour is illustrated in Fig. 13.18. 

A major breakthrough with regard to the understanding of this phenomenon in the field of chemical reaction engineering was achieved by Ray and co-workers (Uppal et ai, 1974, 1976) when in one stroke they uncovered a large variety of possible bifurcation behaviours in non-adiabatic continuous stirred tank reactors. In addition to the usual hysteresis type bifurcation, Uppal et al. (1976) uncovered different types of bifurcation diagrams, the most important of which is the isola which is a closed curve disconnected from the rest of the continuum of steady states. 

J) The work of Uppal et al. 1974, 1976) and that of Ray (1977) on the bifurcation behaviour of continuous stirred tank reactors. 

Chapter 4 deals with the practical relevance of bifurcation behaviour and instability problems in chemically reacting systems. The chapter is brief and simple, however it covers a wide spectrum of information regarding this topic. The importance of this chapter relies on the fact that although theoretical studies on bifurcation behaviour has advanced tremendously during the last three decades, the industrial appreciation remains very limited. 

Later development in the singularity theory, especially the pioneering work of Golubitsky and Schaeffer (1985), provided a powerful tool for the analysis of bifurcation behaviour of chemically reactive systems. These techniques have been used extensively, elegantly and successfully by Luss and his co-workers (Balakotaiah and Luss, I982a,b 1983a,b 1984) to uncover a large number of possible types of bifurcation. They were also able to apply the technique successfully to complex reaction networks as well as distributed parameter systems. 

Many laboratory experiments were also designed to confirm the existence of bifurcation behaviour in chemically reactive systems (e.g. Root and Schmitz, 1969 Chang and Schimtz, 1975 Hlavacek el al., 1980 Harold and Luss, 1985 Marb and Vortmeyer, 1988) as well as in enzyme systems (e.g. Fribaulel ef ai, 1981). 

Multiplicity (or bifurcation) behaviour was found to occur in other systems such as distillation (Widagdo et al., 1989), absorption with chemical reaction (White and Johns, 1985), polymerization of olefins in fluidized beds (Choi and Ray, 1985), char combustion (Hsuen and Sotrichos, 1989a, 1989b), heating of wires (Luss, 1978) and recently in a number of processes used for the manufacturing and processing of electronic components (Kushner, 1982 Okamoto and Serikawa, 1986 Jensen, 1990). 

Nonetheless, it is still possible to perform the bifurcation analysis on the multiresonance Hamiltonian. In fact, the existence of the polyad number makes this almost as easy, despite the presence of chaos, as in the case of an isolated single Fermi or Darling-Dennison resonance. It is found [ ] that most often (though not always), the same qualitative bifurcation behaviour is seen as in the single resonance case, explaining why the simplified individual resonance analysis very often is justified. The bifurcation analysis has now been performed for triatomics with two resonances [60] and for C2H2 with a number of resonances [ ]. 

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. 

We first examine how chaos arises in tire WR model using tire rate constant k 2 as tire bifurcation parameter. However, another parameter or set of parameters could be used to explore tire behaviour. (Independent variation of p parameters... 

Generally it was found that resolution R is practically the same for isoeluotropic mixtures methanol and acetonitrile with water. The dependencies were obtained between capacity factors for derivatives of 3-chloro-l,4-naphtoquinone at their retention with methanol and acetonitrile. Previous prediction of RP-HPLC behaviour of the compounds was made by ChromDream softwai e. Some complications ai e observed at weak acetonitrile eluent with 40 % w content when for some substances the existence of peak bifurcation. 

Conditions at which there are qualitative changes in behaviour, such as the jumping between different branches or the onset of oscillations, as the flow rate is varied for instance, are known as bifurcations , with various subclassifications into different types. We will be very much concerned with understanding and learning how to predict these different bifurcation phenomena and whether the different types can be related. 

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... 

As well as the five possibilities for and /2 discussed above, there are two special cases which will be particularly important in locating and classifying the conditions under which the behaviour exhibited by our models changes ( bifurcates ). 

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. 

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. 

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. 

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 ... 

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... 

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). 

The Hopf bifurcation approach is a mathematically rigorous technique for locating and analysing the onset of oscillatory behaviour in general dynamical systems. Another approach which has been particularly well exploited for chemical systems is that of looking for relaxation oscillations. Typically, the wave form for such a response can be broken down into distinct periods,... 

We should also consider the behaviour along the top of the isola, on the part of the branch lying at longer residence times than the Hopf point. For Tres > t s, and with k2 still in the above range, the uppermost stationary state is unstable and is not surrounded by a stable limit cycle. The system cannot sit on this part of the branch, so it must eventually move to the only stable state, that of no conversion. Thus we fall off the top of the isola not at the long residence time turning point, but earlier as we pass the Hopf bifurcation point. 

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