Bifurcations multiple

Regarding stability, the five steady states y, . .., ys depicted in Figure 6 (A-2) alternate in their stability behavior the low temperature y is stable, yi is unstable, ys is stable, 2/4 is unstable and the high temperature steady state ys is stable. This can be deduced as before from the graph. The bifurcation multiple [Pg.557]

Most of the theory of diffusion and chemical reaction in gas-solid catalytic systems has been developed for these simple, unimolecular and irreversible reactions (SUIR). Of course this is understandable due to the obvious simplicity associated with this simple network both conceptually and practically. However, most industrial reactions are more complex than this SUIR, and this complexity varies considerably from single irreversible but bimolecular reactions to multiple reversible multimolecular reactions. For single reactions which are bimolecular but still irreversible, one of the added complexities associated with this case is the non-monotonic kinetics which lead to bifurcation (multiplicity) behaviour even under isothermal conditions. When the diffusivities of the different components are close to each other that added complexity may be the only one. However, when the diffusiv-ities of the different components are appreciably different, then extra complexities may arise. For reversible reactions added phenomena are introduced one of them is discussed in connection with the ammonia synthesis reaction in chapter 6. 

In this manner, instability, bifurcation, multiplicity of solutions and symmetry are all interrelated. We shall now give a few detailed examples of instability of the thermodynamic branch leading to dissipative structures. 

The condition that gives rise to multiple shock fronts (i.e., allows a shock wave to bifurcate as indicated in Fig. 4.10(b)) will occur when the second wave propagation velocity (with respect to the laboratory) is given by (4.39). How-... 

The mechanism of these transitions is nontrivial and has been discussed in detail elsewhere Q, 12) it involves the development of a homoclinic tangencv and subsequently of a homoclinic tangle between the stable and unstable manifolds of the saddle-type periodic solution S. This tangle is accompanied by nontrivial dynamics (chaotic transients, large multiplicity of solutions etc.). It is impossible to locate and analyze these phenomena without computing the unstable, saddle-tvpe periodic frequency locked solution as well as its stable and unstable manifolds. It is precisely the interactions of such manifolds that are termed global bifurcations and cause in this case the loss of the quasiperiodic solution. 

Expanding the sample size to 2Xc admits the other shape families shown on Fig. 6 into the analysis and leads to additional codimension-two interactions between the shapes is the (1A<.)- family and shapes with other numbers of cells in the sample. The bifurcation diagram computed for this sample size with System I and k = 0.865 is shown as Fig. 11. The (lAc)- and (Ac/2)-families are exactly as computed in the smaller sample size, but the stability of the cell shapes is altered by perturbations that are admissible is the larger sample. The secondary bifurcation between the (lAc)- and (2Ae/3)-families is also a result of a codimension two interaction of these families at a slightly different wavelength. Two other secondary bifurcation points are located along the (lAc)-family and may be intersections with the (4Ac and (4A<./7) families, as is expected because of the nearly multiple eigenvalues for these families. 

Because of nonlinear Interactions between buoyancy, viscous and Inertia terms multiple stable flow fields may exist for the same parameter values as also predicted by Kusumoto et al (M.). The bifurcations underlying this phenomenon may be computed by the techniques described In the numerical analysis section. The solution structure Is Illustrated In Figure 7 In terms of the Nusselt number (Nu, a measure of the growth rate) for varying Inlet flow rate and susceptor temperature. Here the Nusselt number Is defined as ... 

Procedures enabling the calculation of bifurcation and limit points for systems of nonlinear equations have been discussed, for example, by Keller (13) Heinemann et al. (14-15) and Chan (16). In particular, in the work of Heineman et al., a version of Keller s pseudo-arclength continuation method was used to calculate the multiple steady-states of a model one-step, nonadiabatic, premixed laminar flame (Heinemann et al., (14)) a premixed, nonadiabatic, hydrogen-air system (Heinemann et al., (15)). 

This device is based on multiple parallel bi-lamination using bifurcation cascade for generating multiple thin fluid olamellae [25]. The first feed stream is split into multiple sub-streams via a bifurcation cascade in a similar way this is done for the second feed stream in another level. The corresponding sub-streams enter via nozzles into the first level. Here, the end of the channels of the bifurcation cascade and the nozzles lie next to each other. Thereby, bi-laminated sub-streams are formed and enter many parallel channels of an inverse-bifurcation cascade. These are recombined to multilayered stream in one main channel which has a serpentine shape, i.e. comprises extended length. 

From there, the reaction flow either leaves the total system to be quenched or, more commonly, enters the next plate which contains a delay loop, a spiral channel [56]. Leaving that plate, the streams flow to the last structured plate containing a bifurcation-mini mixer unit. The streams are distributed in multiple streams and contacted with a likewise split water stream. This leads to fast dilution, e.g., of a concentrated sulfuric acid stream, and rapidly cools the reaction stream. The reaction is quenched more or less initially. The final plate is unstructured and acts as a cover plate with holes for liquid withdrawal (Figure 4.28). 

It must be realized that the basic reason for bifurcation is that the function F is multiple-valued and therefore non-linear. Other sources of non-linearity, like auto-catalysis have been explored systematically and have proven to be the starting point of geochemical catastrophes (e.g., Ortoleva, 1994). 

Dendrimers have a star-like centre (functionality e.g. 4) in contrast to a star however, the ends of the polymer chains emerging from the centre again carry multifunctional centres that allow for a bifurcation into a new generation of chains. Multiple repetition of this sequence describes dendrimers of increasing generation number g. The dynamics of such objects has been addressed by Chen and Cai [305] using a semi-analytical treatment. They treat diffusion coefficients, intrinsic viscosities and the spectrum of internal modes. However, no expression for S(Q,t) was given, therefore, up to now the analysis of NSE data has stayed on a more elementary level. 

Figure 26.1 shows the mole fraction of H2 just above the surface vs. the surface temperature for a mixture of 10% H2 in air at various pressures. At atmospheric pressure (Fig. 26.1a), the mole fraction of H2 is almost insensitive to surface temperature until a turning point, called an ignition (/i), is reached, where the system jumps from an unreactive state to a reactive one. As the surface temperature decreases from high values, the H2 mole fraction increases, and a Hopf bifurcation (HB) point is first found at 980 K, outside the multiplicity regime. The solution branch between the HBi and the extinction is locally unstable (dashed curve). 

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. 

Guckenheimer, J. (1986). Multiple bifurcation problem for chemical reactors. Physica, D20, 1-20. 

We also have the hint of a new type of degeneracy associated with systems possessing multiple stationary states. It is possible for both the trace and the determinant of the Jacobian matrix to become zero simultaneously this gives the system two eigenvalues which are both equal to zero. These double-zero eigenvalue situations are important because they represent conditions at which a Hopf bifurcation point with an associated homoclinic orbit first appears. In the present case, tr(J) = det(J) = 0 only when k2 = Vg, but then the isola has shrunk to a point. 

Fig. 8.9. The locus A of double-zero eigenvalue degeneracies of the Hopf bifurcation for cubic autocatalysis with decay. Also shown, as broken curves, are the loci of stationary-state degeneracies, corresponding to the boundaries for isola and mushroom patterns. The curve A lies completely within the parameter regions for multiple stationary states.

Fig. 8.10. Stationary-state patterns showing multiplicity and Hopf bifurcation points distinguished by the curve A in Fig. 8.9 stable states are indicated by solid curves, unstable states by broken curves and Hopf bifurcation points by solid circles.

Fig. 8.12. The loci DH, and DH2 corresponding to degenerate Hopf bifurcation points at which the stability of the emerging limit cycle is changing. Again, these are shown relative to the loci for stationary-state multiplicity (broken curves).

In chapter 12 we discussed a model for a surface-catalysed reaction which displayed multiple stationary states. By adding an extra variable, in the form of a catalyst poison which simply takes place in a reversible but competitive adsorption process, oscillatory behaviour is induced. Hudson and Rossler have used similar principles to suggest a route to designer chaos which might be applicable to families of chemical systems. They took a two-variable scheme which displays a Hopf bifurcation and, thus, a periodic (limit cycle) response. To this is added a third variable whose role is to switch the system between oscillatory and non-oscillatory phases. 

For e and k of order unity, the full three-variable model may still display multiple stationary-states, Hopf bifurcations and sustained oscillations. 

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. 

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