Bifurcation parameters

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... 

Figure C3.6.6 The figure shows tire coordinate, for < 0, of tire family of trajectories intersecting tire Poincare surface at cq = 8.5 as a function of bifurcation parameter k 2- As tire ordinate k 2 decreases, tire first subhannonic cascade is visible between k 2 0.1, tire value of tire first subhannonic bifurcation to k 2 0.083, tire subhannonic limit of tire first cascade. Periodic orbits tliat arise by tire tangent bifurcation mechanism associated witli type-I intennittency (see tire text for references) can also be seen for values of k 2 smaller tlian tliis subhannonic limit. The left side of tire figure ends at k 2 = 0.072, tire value corresponding to tire chaotic attractor shown in figure C3.6.1(a). Otlier regions of chaos can also be seen.

Figure 6-12 is the bifurcation diagram, in which the quantity q is the bifurcation parameter. The ordinates of the curve represent radii of cycles, and stability and instability is indicated by and o, respectively. If one starts with negative values of q, the origin 0 is unstable, and the... 

In our computational model the strain rate and the equivalence ratio are the natural bifurcation parameters. If we denote either of these parameters by a, then the system of equations in (3.2) can be written in the form... 

Figure 28. The eigenvalues of the Jacobian of minimal glycolysis as a function of the influence of ATP on the first reaction V (ATP) (feedback strength). Shown is the largest real part of the eigenvalues (solid line), along with the corresponding imaginary part (dashed line). Different dynamic regimes are separated by vertical dashed lines, for > 0 the state is unstable. Transitions occur via a saddle node (SN) and a Hopf (HO) bifurcation. Parameters are v° 1, TP° 1, ATP0 0.5, At 1, and 6 0.8. See color insert.

Fig. 9. Linear stability diagram illustrating (21). The bifurcation parameter B is plotted against the wave number n. (a) and (b) are regions of complex eigenvalues < > (b) shows region corresponding to an unstable focus (c) region corresponding to a saddle point. The vertical lines indicate the allowed discrete values of n. A = 2, D, = 8 103 D2 = 1.6 10-3.

In Fig. 21 we have drawn the bifurcation diagram of the fundamental steady-state solutions for three values of p [ Kxn is plotted versus UK) as the bifurcation parameter]. There is a subcritical region in the upper or lower branch, depending on the relative height of the peaks in Fig. 20c. The asymptotes K and K" of these branches correspond to half-period solutions of infinite length. When p 2 the asymptote K merges with the w-axis therefore situation 2 above can be viewed as a particular case of situation 3 above, in which the bifurcation point moves to infinity. 

In the models discussed here we have considered primarily as bifurcation parameters the affinity of reaction as measured by the parameter B in model (1) or the length l as in Section VI. The results have illustrated that when B or l increase, the multiplicity of solutions increases. This is not astonishing, as a variation in length is a simple way through which the interactions of the reaction cell with its environment can be increased or decreased. 

The use of the length or size as the bifurcating parameter is, of course, quite natural in biological processes involving growth or morphogenetic effects. 

The various Hopf bifurcation parameters / 2, p2, and t2 can again be determined explicitly but have much more complex forms. We will discuss the details in the next chapter and only consider here the stability of the emerging limit cycle through P2. With the full Arrhenius form, this parameter is given by... 

Here F represents the functional form of the left-hand sides of the various stationary-state equations, x is the stationary-state solution such as the extent of reaction, the temperature excess, etc., and rres is the parameter we have singled out as the one which can be varied during a given experiment (the distinguished or bifurcation parameter). All the remaining parameters are represented by p, q, r, s,. For example, in eqn (7.21) the role of x could be played by the extent of reaction 1 — ass, with p = 0ad and q = tN for isothermal autocatalysis, x can again be the extent of reaction, with p = P0, q = k2, and r = jcu. 

The above scenario is accounted for by the normal form (4.9) truncated at fourth order in q with k = v = a = p = 0 and x < 0, taking p as the bifurcation parameter, which increases with energy (p thus plays a similar role as the total energy in the actual Hamiltonian dynamics). The antipitchfork bifurcation occurs at pa = 0. The fixed points of the mapping (4.8) are given by p = 0 and dv/dq = 0. Since the potential is quartic, there are either one or three fixed points that correspond to the shortest periodic orbits 0, 1, and 2 of the flow. 

The non-dimensionalization used in this work is perhaps the simplest, but it suffers from the defect that important physical bifurcation parameters are not isolated. The simple cuspoid diagrams are probably not those that would be obtained from experiments, where the residence time is a convenient parameter. Balakotaiah and Luss (1983) considered such a formulation for two parallel or simultaneous reactions the diagrams for the case of sequential reactions are similar, at least when the activation energies are equal. The maximum multiplicity question, however, is independent of the formulation and we conjecture that diagrams with seven steady states could be found in a small region of parameter space, though we have not looked for them. 

The effects of forced oscillations in the partial pressure of a reactant is studied in a simple isothermal, bimolecular surface reaction model in which two vacant sites are required for reaction. The forced oscillations are conducted in a region of parameter space where an autonomous limit cycle is observed, and the response of the system is characterized with the aid of the stroboscopic map where a two-parameter bifurcation diagram for the map is constructed by using the amplitude and frequency of the forcing as bifurcation parameters. The various responses include subharmonic, quasi-peri-odic, and chaotic solutions. In addition, bistability between one or more of these responses has been observed. Bifurcation features of the stroboscopic map for this system include folds in the sides of some resonance horns, period doubling, Hopf bifurcations including hard resonances, homoclinic tangles, and several different codimension-two bifurcations. 

But when the field is nonzero the trivial solution is not allowed. Instead, there is always one real nontrivial solution for all values of the bifurcation parameter X and a pair of other real solutions which exist only for values of X larger than a certain value Xc. However, there exists no bifurcation of new solutions from a given branch. This situation is described in Fig. 10 of the paper by I. Prigogine. It provides the basis for understanding the high sensitivity of the system in the vicinity of X, and the pattern selection introduced by the gravitational field. We come back to this problem in Section IV. 

The most intriguing predictions of the model come from its application to the developing egg. The egg does not increase in size therefore, size cannot be used as a bifurcation parameter to select different chemical patterns. However, as the nuclei migrate to the cortex of the egg, the cortical layer becomes viscous cytoplasm, rather than a relatively less... 

If we keep ko constant, then a is directly proportional to the volume V, and inversely proportional to the flow rate q. The values of a can reach very large magnitudes due to the nature of ko and its range of values. In general, a is the most widely varying parameter of the reaction. Therefore we usually investigate bifurcation with a chosen as the bifurcation parameter. 

To gain further and broader insights into the bifurcation behavior of nonadiabatic, nonisothermal CSTR systems, we again use the level-set method for nonalgebraic surfaces such as z = /(K,., y). This particular surface is defined via equation (3.14) as follows for a given constant value of yc with the bifurcation parameter Kc ... 

Nonadiabatic CSTR bifurcation curve for Kc as the bifurcation parameter... 

Compute and plot the bifurcation diagram with a as the bifurcation parameter when Kc = 0. Discuss the physical meaning of the diagram. 

The vectors X and Xo contain the state variables and the initial conditions vector, respectively, while Y)0 and Kc represent the bifurcation parameters for the open loop (uncontrolled, Kc = 0) and the closed loop (controlled, Y 0 = constant) systems, respectively, and f = t/eiiR-The column vectors X and Xo are... 

The bifurcation analysis, i.e., the analysis of the steady states and the dynamic solutions is carried out for the dilution rate D as the bifurcation parameter. We have chosen D as the bifurcation parameter since the flow rate q V/D is directly related to D and q is most easily manipulated during the operation of a fermentor. 

As before we use the dilution rate D as the bifurcation parameter for varying values of Cso The role of these two variables can easily be exchanged to achieve similar results with Cso as the bifurcation parameter for varying dilution rates D. 

Bifurcation diagram at Cso = 140 kg/rn3 with D as the bifurcation parameter Steady state branch — stable — unstable Periodic branch stable average of oscillations... 

Figure 7.30 shows bifurcation diagrams with D as the bifurcation parameter. 

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