Bifurcation value

If for certain values of a parameter A in the differential equation, the qualitative aspect of the solution (i.e., the phase portrait ) of the differential equation remains the same (in other words the changes are only quantitative) such values of A are called ordinary values. If however, for a certain value A = A0 this qualitative aspect changes, such a special value is called a critical or bifurcation value. 

In their subsequent works, the authors treated directly the nonlinear equations of evolution (e.g., the equations of chemical kinetics). Even though these equations cannot be solved explicitly, some powerful mathematical methods can be used to determine the nature of their solutions (rather than their analytical form). In these equations, one can generally identify a certain parameter k, which measures the strength of the external constraints that prevent the system from reaching thermodynamic equilibrium. The system then tends to a nonequilibrium stationary state. Near equilibrium, the latter state is unique and close to the former its characteristics, plotted against k, lie on a continuous curve (the thermodynamic branch). It may happen, however, that on increasing k, one reaches a critical bifurcation value k, beyond which the appearance of the... 

Whereas two bifurcation values for the glucose input rate define the domain of oscillations in yeast extracts [40], only a single bifurcation value below which oscillations occur is found in intact yeast cells [47]. This does not necessarily imply a difference in oscillatory mechanism but merely indicates that in intact cells the glucose transporter becomes saturated before the intracellular glucose input has reached the upper bifurcation value above which oscillations disappear in yeast extracts [38]. 

Stochastic simulations confirm the existence of bifurcation values of the control parameters bounding a domain in which sustained oscillations occur. The effect of noise diminishes as the number of molecules increases. Only when the maximum numbers of molecules of mRNA and protein become smaller than a few tens does noise begin to obliterate the circadian rhythm. The robustness of circadian rhythms with respect to molecular noise is enhanced when the rate of binding of the repressor molecule to the gene promoter increases [128]. Conditions that enhance the resistance of genetic oscillators to random fluctuations have been investigated [130]. 

Recall that a Hopf bifurcation is termed supercritical if its bifurcation diagram is as shown schematically in Fig. 6.2.2a. Correspondingly, in this case a stable limit cycle is born around the equilibrium, unstable hereon, only at a critical (bifurcation) value of the control parameter A = Ac. In contrast, in the subcritical case (Fig. 6.2.2b), the equilibrium is surrounded by limit cycles already for A < Ac, with an unstable limit cycle separating the stable one from the still stable equilibrium. At the bifurcation A = Ac the unstable limit cycle dies out with the equilibrium, unstable hereon, surrounded by a stable limit cycle. Thus the main feature of the subcritical case (as opposed to the supercritical one) is that a stable equilibrium and a stable limit cycle coexist in a certain parameter range, with a possibility to reach the limit cycle through a sufficiently strong perturbation of the equilibrium. 

Note that the dimensionless time rf, which gives the length of the pre-oscillatory period, will only be positive if the initial concentration /i0 exceeds the upper Hopf bifurcation value /if. If we start with a lower initial reactant concentration, so that /i0 < /if (but still with /i0 > /if), there will be no pre-oscillatory period the system will jump straight into oscillations which will persist until time rf. 

In several experiments, in particular the study by Temkin and co-workers [224] of the kinetics in ethylene oxidation, slow relaxations, i.e. the extremely slow achievement of a steady-state reaction rate, were found. As a rule, the existence of such slow relaxations is ascribed to some "side reasons rather than to the purely kinetic ("proper ) factors. The terms "proper and "side were first introduced by Temkin [225], As usual, we classify as slow "side processes variations in the chemical or phase composition of the surface under the effect of reaction media, catalyst deactivation, substance diffusion into its bulk, etc. These processes are usually considered to require significantly longer times to achieve a steady state compared with those characterizing the performance of chemical reactions. The above numerical experiment, however, shows that, when the system parameters attain their bifurcation values, the time to achieve a steady state, tr, sharply increases. 

Slow relaxations can be exemplified by the system behaviour corresponding to the adsorption mechanism (8) when the parameters % are close to their bifurcation values. 

Localization of this steady state as a point of intercept for the null dines x = 0 and y = 0 as a function of the k x value is shown in Fig. 16. At low k x this point is localized sufficiently close to the region of probable initial conditions (at k x = 0 it becomes a boundary steady state). It is the proximity of the initial conditions to the steady state outside the reaction polyhedron that accounts for the slow transition regime. Note that, besides two real-valued steady states, the system also has two complex-valued steady states. At bifurcation values of the parameters, the latter become real and appear in the reaction simplex as an unrough internal steady state. The proximity of complex-valued roots of the system to the reaction simplex also accounts for the generation of slow relaxations. 

But as has already been noted, in the neighbourhood of the bifurcation point, the closer the parameter is to the bifurcation value, the longer is the relaxation process ("critical retardation ). 

To see what is happening when 9 passes through the bifurcation value 9 = 3, we examine the stability at the second iteration. The second iteration can be thought of as a first iteration in a model where the iterative time step is 2. The fixed points are solutions of... 

For both analyses, the procedure is to use dimensionless parameters in the set of differential equations describing the model, look for the steady state, investigate the linear stability, and determine the conditions for instability. Near the bifurcation values of the parameters, which initiate an oscillatory growing solution, a perturbation analysis provides an estimate for the period of the ensuing limit cycle behavior. 

Figure 3.8 Typical kinetic curves (A, C, E) and the corresponding phase trajectories (B, D, F) of the evolution of thermodynamic rushes of intermediates Y (solid line) and Z (dash line) in scheme (3.35) with damped oscillations. Calculations are given for the cases of constant values Sq = 2, S2 = 0.5, and S3 = 5 and starting condition Ya =2, Za = t at different values of the controlling parameters R R = 1 in diagrams A and B (stationary state at Y = 10.4, Z = 0.4) R = 30 in diagrams C and D (stationary state at Y — 22, Z — 12), and R = 50 in diagrams E and F (stationary state Y = 30, Z — 20). There is a bifurcation value of the controlling parameter R = 25 for all the other external parameters.

If attention is restricted to the vicinity of the bifurcation point, then a nonlinear perturbation analysis can be developed for describing analytically the nature of the pulsating mode [111]. In effect, the difference between A and its bifurcation value is treated as a small parameter, say , and oscillatory solutions for temperature profiles are calculated as perturbations about the... 

The problem is reduced to finding the phase trajectories of the equation system (104) at the (g, 0)-plane at different y values (dimensionless reaction rate) and values of p (relationship of the rates of relaxation g and heat removal at T = Tq). Dependence of the solution on x and in the physically justified ranges of their variation (tj > I at q qi ij< 1) turns out to be relatively weak. The authors of ref 234 applied the well-known method of analysis of specific trajectories changing at the bifurcational values of parameters [237], In the general case, the system of equations (104) has four singular points. The inflammation condition has the form... 

The bifurcational values of y and f) determining the confluence line of sites and focuses are the solutions of the equations... 

The aromatic rings present the highest and the antiaromatic systems the lowest bifurcation values of ELF. The bifurcation of the ELF occurs close to 0.75, except in system where sigma delocalization exists. In this way, the aromaticity of polycyclic aromatic hydrocarbons was well predicted, and also the aromaticity of new molecules was corroborated. In the all-metal aromatic compound Al - important contributions to stability from the two tt aromatic electrons and the cr system in the plane of the molecule were observed. The isosurfaces of the total ELF, ELFOT and ELF functions are showed in the Figure 6. 

Scheme 1 Bifurcation values of ELF and ELF functions for some aromatic and antiaromatic molecules...

When r < 0, there is a quadratic minimum at the origin. At the bifurcation value r = 0, the minimum becomes a much flatter quartic. For r > 0, a local maximum appears at the origin, and a symmetric pair of minima occur to either side of it. ... 

For each of the following questions, draw the phase portrait as function of the control parameter jLt. Classify the bifurcations that occur as // varies, and find all the bifurcation values of //. 

Figure 10.6.3 schematically illustrates the meaning of S. Let A = - a ,., denote the distance between consecutive bifurcation values. Then 5 as... 

The graph in Figure 10.5.2 suggests that A = 0 at each period-doubling bifurcation value r,. Show analytically that this is correct. 

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