Bifurcating

Figure Al.2.10. Birth of local modes in a bifurcation. In (a), before the bifiircation there are stable anhamionic symmetric and antisymmetric stretch modes, as in figure Al.2.6. At a critical value of the energy and polyad number, one of the modes, in this example the symmetric stretch, becomes unstable and new stable local modes are bom in a bifurcation the system is shown shortly after the bifiircation in (b), where the new modes have moved away from the unstable syimnetric stretch. In (c), the new modes clearly have taken the character of the anliamionic local modes.

Golubitsky M and Schaeffer D G 1985 Singularities and Groups in Bifurcation Theory vol 1 (New York Springer)... 

This Is a didactic Introduction to some of the techniques of bifurcation theory discussed In this article. 

This survey compares algebraic methods with more standard approaches, and the bifurcation approach in this article. Bunker P R 1979 Moleoular Symmetry and Speotrosoopy (New York Academic)... 

Figure A3.14.3. Example bifurcation diagrams, showing dependence of steady-state concentration in an open system on some experimental parameter such as residence time (inverse flow rate) (a) monotonic dependence (b) bistability (c) tristability (d) isola and (e) musliroom.

Ezra G S 1996 Periodic orbit analysis of molecular vibrational spectra-spectral patterns and dynamical bifurcations in Fermi resonant systems J. Chem. Phys. 104 26... 

B3.5.7.3 BIFURCATION OF THE REACTION PATH AND VALLEY-RIDGE INFLECTION POINTS... 

Valtazanos P and Ruedenberg K 1986 Bifurcations and transition states Theor. Chim. Acta 69 281... 

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 C3.6.5 The first two periodic orbits in the main subhannonic sequence are shown projected onto the (c, C2) plane. This sequence arises from a Hopf bifurcation of the stable fixed point for the parameters given in the text. The arrows indicate the direction of motion, (a) The limit cycle or period-1 orbit at k 2 = 0.11. (b) The first subhannonic or period-2 orbit at k 2 = 0.095.

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.

The quasiperiodic route to chaos is historically important. It arises from a succession of Hopf birfurcations. As already noted, a single Hopf bifurcation results in a limit cycle. The next Hopf bifurcation produces a phase flow tliat can be represented on tire surface of a toms (douglmut). This flow is associated witli two frequencies if tire ratio of tliese frequencies is irrational tlien tire toms surface is densely covered by tire phase trajectory, whereas if... 

Consider the analogue of such a bifurcation in a spatially distributed system and imagine tuning a bifurcation... 

This complex Ginzburg-Landau equation describes the space and time variations of the amplitude A on long distance and time scales detennined by the parameter distance from the Hopf bifurcation point. The parameters a and (5 can be detennined from a knowledge of the parameter set p and the diffusion coefficients of the reaction-diffusion equation. For example, for the FitzHugh-Nagumo equation we have a = (D - P... 

Now we may state tlie well known conditions for a Turing bifurcation. If A <0 and A22 < 0 we say species A j is tlie activator and speciesis tlie inliibitor. Then, for a Turing bifurcation to occur we must have detB = 0,Tt B> 0 and A 7)2 + > 0. The (unique) wavenumber at tlie bifurcation is... 

Furtliennore, since tlie bifurcation must occur from a stable homogeneous steady state we must have D ID < 1 i.e. tlie diffusion coefficient of tlie inhibitor is greater tlian tliat of tlie activator. The critical diffusion ratio at tlie bifurcation is... 

Consequently, when D /Dj exceeds the critical value, close to the bifurcation one expects to see the appearance of chemical patterns with characteristic lengtli i= In / k. Beyond the bifurcation point a band of wave numbers is unstable and the nature of the pattern selected (spots, stripes, etc.) depends on the nonlinearity and requires a more detailed analysis. Chemical Turing patterns were observed in the chlorite-iodide-malonic acid (CIMA) system in a gel reactor [M, 59 and 60]. Figure C3.6.12(a) shows an experimental CIMA Turing spot pattern [59]. 

Figure C3.6.12 a) Turing spot pattern in the CIMA reaction. (A) Tio-temporal turbulence near the Turing bifurcation. Reproduced by pennission from Ouyang and Swinney [59].

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]. 

The method will, however, fail badly if the Gaussian form is not a good approximation. For example, looking at the dynamics shown in Figure 2, a problem arises when a barrier causes the wavepacket to bifurcate. Under these circumstances it is necessary to use a superposition of functions. As will be seen later, this is always the case when non-adiabatic effects are present. 

Abstract. A model of the conformational transitions of the nucleic acid molecule during the water adsorption-desorption cycle is proposed. The nucleic acid-water system is considered as an open system. The model describes the transitions between three main conformations of wet nucleic acid samples A-, B- and unordered forms. The analysis of kinetic equations shows the non-trivial bifurcation behaviour of the system which leads to the multistability. This fact allows one to explain the hysteresis phenomena observed experimentally in the nucleic acid-water system. The problem of self-organization in the nucleic acid-water system is of great importance for revealing physical mechanisms of the functioning of nucleic acids and for many specific practical fields. 

Changing the constants in the SCF equations can be done by using a dilferent basis set. Since a particular basis set is often chosen for a desired accuracy and speed, this is not generally the most practical solution to a convergence problem. Plots of results vs. constant values are the bifurcation diagrams that are found in many explanations of chaos theory. 

J. P. Knapp and M. F. Doherty, "Minimum Entrainer Flows for Extractive Distillation. A Bifurcation Theoretic Approach," submitted to AIChE J. (1993). 

Kuhicek, M., and M. Marek. Computational Methods in Bifurcation Theory and Dissipative Structures, Springer-Verlag, Berhn (1983). 

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. 

Generally it was found that resolution R is practically the same for isoeluotropic mixtures methanol and acetonitrile with water. The dependencies were obtained between capacity factors for derivatives of 3-chloro-l,4-naphtoquinone at their retention with methanol and acetonitrile. Previous prediction of RP-HPLC behaviour of the compounds was made by ChromDream softwai e. Some complications ai e observed at weak acetonitrile eluent with 40 % w content when for some substances the existence of peak bifurcation. 

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 contour plot is given in fig. 43. As remarked by Miller [1983], the existence of more than one transition states and, therefore, the bifurcation of the reaction path, is a rather common event. This implies that at least one transverse vibration, q in the case at hand, turns into a double-well potential. The instanton analysis of this PES has been carried out by Benderskii et al. [1991b]. The... 

Fig. 44. Bifurcational diagram for the potential (6.13) in the (Qo, P) plane. Domains (i), (ii) and (iii) correspond to Arrhenius dependence (stepwise thermally activated transfer), two-dimensional instanton and one-dimensional instanton (concerted transfer), respectively.

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