Bifurcated interactions

The aim of this chapter is to present the current understanding of the dynamics seen in Figures 1 and 2 within the context of symmetric bifurcation theory. It will be shown, through a computational bifurcation analysis of reaction-diffusion equations, that the the dynamics landscape in Figure 2 is organized around a parameter point at which a Hopf bifurcation interacts with... 

The recent simulations of Pearson [44] on a fed autocatalator [104], on the other hand stages interactions between Turing and saddle-node bifurcations or even situations where a Hopf bifurcation interacts with the two previous ones. Then a plethora of scenarios may result. However, the Turing bifurcations are genuine but they occur on the stable branch of an isola of uniform states and may exist beyond the limits of the isola. 

One may also observe a transition to a type of defect-mediated turbulence in this Turing system (see figure C3.6.12 (b). Here the defects divide the system into domains of spots and stripes. The defects move erratically and lead to a turbulent state characterized by exponential decay of correlations [59]. Turing bifurcations can interact with the Hopf bifurcations discussed above to give rise to very complicated spatio-temporal patterns [63, 64]. 

Weak intramolecular interactions between sulfur or selenium and nitrogen are a recurrent phenomenon in large biomolecules. They may occur in the same residue or between neighbours of a peptide chain. The formation of four- or five-membered rings of the types 15.1 and 15.2, respectively, is most common. A feature that is unique to proteins is the participation of sulfur atoms in bifurcated N S N contacts. 

In 1990, Baumeister et al. [127] described the crystal and molecular structure of 4-ethoxy-3 -(4-ethoxyphenyliminomethyl)-4 -(4-methoxy-benzoy-loxy)azobenzene. The molecules have a bifurcated shape. The phenyliminom-ethyl branch is bent markedly from the nearly linear three ring fragment, but is almost coplanar with the azobenzene moiety. They found that the molecular conformation is affected by an intramolecular interaction of the carboxylic and azomethine groups. The crystal packing was described in terms of a sheet structure with interdigitating rows of molecules. 

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. 

The approach used in these studies follows idezus from bifurcation theory. We consider the structure of solution families with a single evolving parameter with all others held fixed. The lateral size of the element of the melt/crystal interface appears 2LS one of these parameters and, in this context, the evolution of interfacial patterns are addressed for specific sizes of this element. Our approach is to examine families of cell shapes with increasing growth rate with respect to the form of the cells and to nonlinear interactions between adjacent shape families which may affect pattern formation. 

INTERACTION OF SHALLOW CELLS CELLULAR DYNAMICS Evolution of Shallow Cells The Role of Codimension Two Bifurcations. The importance of nonlinear interactions between spatially resonant structures is... 

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 ... 

The infinite potential barrier, shown schematically in figure 10 corresponds to a superselection rule that operates below the critical temperature [133]. Above the critical temperature the quantum-mechanical superposition principle applies, but below that temperature the system behaves classically. The system bifurcates spontaneously at the critical point. The bifurcation, like second-order phase transformation is caused by some interaction that becomes dominant at that point. In the case of chemical reactions the interaction leads to the rearrangement of chemical bonds. The essential difference between chemical reaction and second-order phase transition is therefore epitomized by the formation of chemically different species rather than different states of aggregation, when the symmetry is spontaneously broken at a critical point. 

Results of in vitro studies suggest an interaction between calcium ions and cyanide in cardiovascular effects (Allen and Smith 1985 Robinson et al. 1985a). It has been demonstrated that exposure to cyanide in metabolically depleted ferret papillary muscle eventually results in elevated intracellular calcium levels, but only after a substantial contracture develops (Allen and Smith 1985). The authors proposed that intracellular calcium may precipitate cell damage and arrhythmias. The mechanism by which calcium levels are raised was not determined. Franchini and Krieger (1993) produced selective denervation of the aortic and carotid bifurcation areas, and confirmed the carotid body chemoreceptor origin of cardiovascular, respiratory and certain behavioral responses to cyanide in rats. Bradycardia and hyperventilation induced by cyanide are typical responses evoked by carotid body chemoreceptor stimulation (Franchini and Krieger 1993). 

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