Unusual bifurcations

Stability, Bifurcations, Limit Cycles Some aspects of this subject involve the solution of nonlinear equations other aspects involve the integration of ordinaiy differential equations apphcations include chaos and fractals as well as unusual operation of some chemical engineering eqmpment. Ref. 176 gives an excellent introduction to the subject and the details needed to apply the methods. Ref. 66 gives more details of the algorithms. A concise survey with some chemical engineering examples is given in Ref. 91. Bifurcation results are closely connected with stabihty of the steady states, which is essentially a transient phenomenon. 

Slow transitions produced by enzyme isomerizations. This behavior can lead to a type of cooperativity that is generally associated with ligand-induced conformational changes . A number of enzymes are also known to undergo slow oligomerization reactions, and these enzymes may display unusual kinetic properties. If this is observed, it is advisable to determine the time course of enzyme activation or inactivation following enzyme dilution. See Cooperativity Bifurcation Theory Lag Time... 

Holmes 1983) states that when the above transversal homoclinic intersection occurs, that there is a structurally stable invariant Cantor set like the one for the Horseshoe map. It has also been shown by Holmes (1982) that this invariant set contains a countable, dense set of saddles of all periods, an uncountable set of non-periodic trajectories and a dense orbit. If nothing else is clear from the above, it is at least certain that homoclinic bifurcations for maps are accompanied by some very unusual phase portraits. Even if homoclinic bifurcations are not necessarily accompanied by the formation of stable chaotic attractors, they lend themselves to extremely long chaotic like transients before settling down to a periodic motion. Because there are large numbers of saddles present, their stable manifolds divide up the phase plane into tiny stability regions and extreme sensitivity to perturbations is expected. 

More recently the term bifurcated has also been applied to the configuration 5, proposed for interactions between water and dimethyl phosphate or formate ions [68]. An unusual bifurcated configuration 6 is reported in 1,3-dimethyl uracil [69]. 

The hydrogen-bond structure of cytidinium nitrate [CYTIDN] (Fig. 17.33) contains the unusual bifurcated/three-center bond configuration, with strong bonds to the nitro oxygens (2.01 A and 1.94 A) and weak bonds (2.74 A and 2.58 A) to the hydroxyl oxygen 0(3 ). There is evidence of a three-center intramolecular bond ( ) with C(6)-H as donor and 0(4 ),0(5 ) as acceptors, similar to that observed in 0(2 )-methylcytidine (Fig. 17.36), and there is another three-center bond with the intramolecular 0(2 ) - H 0(3 ) as minor component ( ). The cytosine moiety is protonated at N(3). 

The hydrogen bonding in 2,-deoxycytidine HCl [DOCYTC] centers around the four-coordinated chloride ion (Fig. 17.40). It contains an unusual bifurcated bond... 

Three-center (bifurcated) hydrogen bonding between adjacent base pairs explains unusual properties of poly(dA) poly(dT). The stability of the B-form of this duplex (and of its ribo analog) formed by two homopolymers might be a consequence of the extreme propeller twist (Box 20.1) reported for the central portions (the A-tracts) of the double helical dodecamer d(CGCAAAAAAGCG) [702] and d(CGCAAATTTGCG) [703]. In both crystal structures the DNA molecules show the same unusual three-center bonds linking adjacent base pairs. 

The examples in Fig. 5-7 deserve some comment, since they are newer proposals of inlarge molecules can show chelation (often with multiple H bonds, as in this example) (f) exhibits a large, internally bonded ion (k) is a very unusual case which may not be a H bond at all—as proposed, it places the hydrogen atom in one tetrahedral face where it may have some similarity to a bifurcated H bond. 

Klatskin, G. Adenocarcinoma of the hepatic duct and its bifurcation within the porta hepatis. An unusual tumor with distinctive clinical and pathological features. Amer. J. Med. 1965 38 241 -256... 

Unusual bifurcations) In discussing the normal form of the saddle-node bi-... 

We start with the two-body Smoluchowski model (2BSM) the details of the formulation (matrix and starting vector) are discussed in Section II.C. A stochastic system made of two spherical rotators in a diffusive (Smoluchowski) regime has been used recently to interpret typical bifurcation phenomena of supercooled organic liquids [40]. In that work it was shown that the presence of a slow body coupled to the solute causes unusual decay behavior that is strongly dependent on the rank of the interaction potential. 

Hightower SE, Corcoran RC et al (2005) Unusual, bifurcated photorcactivity of a rhenium(I) carbonyl complex of triethynylphosphine. Inoig Chem 44 9601-9603... 

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