Fold bifurcation

A theoretical framework for considering how the behavior of dynamical systems change as some parameter of the system is altered. Poincare first apphed the term bifurcation for the splitting of asymptotic states of a dynamical system. A bifurcation is a period-doubling, -quadrupling, etc., that precede the onset of chaos and represent the sudden appearance of a qualitatively different behavior as some parameter is varied. Bifurcations come in four basic varieties flip bifurcations, fold bifurcations, pitchfork bifurcations, and transcritical bifurcations. In principle, bifurcation theory allows one to understand qualitative changes of a system change to, or from, an equilibrium, periodic, or chaotic state. 

If we increase / and 7 to 1.2 and 25, respectively, we find a much larger interval of values with multiple pellet efficiencies. The curve shows a nascent 5-fold bifurcation kink in the lower part of the middle branch and it requires us to solve nearly 2.7 times as many BVPs in 285 seconds. 

Figure 6.19(a) shows the main bifurcation branch, which leads to the appearance of the CS. Particularly, at 7/ = 0.02762 (LP point in the figure), a pair of period-3 orbits appears via a fold bifurcation. The branch with the smaller period leads then to the period-doubling bifurcation at rj = 0.02783... 

Fold bifurcation (/) = 0.02762), PD — period-doubling bifurcation (jj = 0.02783). T denotes the actual period of the orbit, (b) The three-1>and chaotic satldle (CS), which is createt tlirough the periori-doubling cascade of pcriod-3 orbit (broken retl line) at r) = 0.0280. 

The solution of Eqs (16)-(21) has physical significance if Da exceeds a critical value. This is the turning point of the Da - za,2 map, which represents a fold bifurcation of the mass balance equations (Fig. 3b). Here two steady states solutions are bom. The upper state (high za,2 and low-conversion) is closed-loop unstable. The instability, which can be proven on steady state considerations only, is independent of the d5mamics. Therefore, the fold point represents a feasibility and stability boundary. When zp =0 the coordinates of the fold point are given by ... 

For fixed Da either zero or two solutions exist (Fig. 6b). The behaviour is similar to the one presented in section 3.1.2, the same considerations being valid. In particular, the feasibility and stability boundary, representing a fold bifurcation, is given by ... 

A saddle-node periodic orbit (fold bifurcation) = 0, l2 e) 0. 

Fig. 14.2.8. A saddle-node (fold) bifurcation of periodic orbits in...

The dominant practice in Quantum chemistry is optimization. If the geometry optimization, for instance through analytic gradients, leads to symmetry-broken conformations, we publish and do not examine the departure from symmetry, the way it goes. This is a pity since symmetry breaking is a catastrophe (in the sense of Thom s theory) and the critical region deserves attention. There are trivial problems (the planar three-fold symmetry conformation of NH3 is a saddle point between the two pyramidal equilibrium conformations). Other processes appear as bifurcations for instance in the electron transfer... 

A reversible, direct fluoroimmunosensor for human serum albumin (HS A) measurement has been described by Bright et al.(m> Antibody Fab fragments are first immobilized on small quartz plates by hinge-region thiols, and then dansylated. The immunosensor is formed by attaching the quartz plates with bound Fab to the distal end of a bifurcated fiber-optic probe, which transmits both the excitation and emission. Binding of ffSA to the immunosensor results in a three- to five-fold enhancement of dansyl fluorescence. The sensor can be reused up to 50 times, with a detection limit of about 1.8 x 10-8 M, and a somewhat limited dynamic range. 

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. 

Taken together, the plots from Figs. 14 and 15 are very informative. For example, when going along the two stable branches of the steady-state rate (Fig. 16) far from bifurcations, we can observe no less than a four-fold difference in the eigenvalues of X. This corresponds to the fact that, in numerical experiments, the difference in relaxation times is also observed in the case when the steady state is unique. Values of X for the stable branches also differ for the lower branch, 1, the absolute value of X is much smaller than for the upper branch 3. A similar difference is also observed in the times to achieve steady states. When going along branch 1, this time amounts to... 

Con A contains no standard a-helices, however, there is one loop which contains one or two hydrogen bonds of the a-helical type (residues 80-85), but the carbonyl oxygens appear to tip outward forming bifurcated hydrogen bonds to solvent water molecules. Also, it should be noted that the first 40 amino acids, which have a large percentage of polar side chains, contribute all the atoms which bind directly to the Mn2+ and Ca2+ ions. This includes the loop which folds around this double ion site (amino acids 10-23) and, in the absence of these ions, would be completely solvated. 

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