Points 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... 

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]. 

At the point 0 another bifurcation point (of the first kind) is encountered SU -> (SU) — U, resulting in an unstable singular point. Thus, there will be a jump from A to B and the operation will be established on the upper stable cycle. 

The two jumps at the bifurcation points form a kind of hysteresis loop, which B. van der Pol and E. Appleton, who discovered the phenomenon, call oscillation hysteresis. ... 

Some electrode potentials that are characteristic of corrosion can be regarded as the bifurcation points, and thus the abmpt change of corrosion state can be understood as a kind of bifurcation phenomenon. Figure 2 is... 

Considering the similarity between Figs. 1 and 2, the electrode potential E and the anodic dissolution current J in Fig. 2 correspond to the control parameter ft and the physical variable x in Fig. 1, respectively. Then it can be said that the equilibrium solution of J changes the value from J - 0 to J > 0 at the critical pitting potential pit. Therefore the critical pitting potential corresponds to the bifurcation point. From these points of view, corrosion should be classified as one of the nonequilibrium and nonlinear phenomena in complex systems, similar to other phenomena such as chaos. 

The next step should clarify why the unstable growth of the variable x occurs through a stable state at the bifurcation point. To determine the stability of the bifurcation point, it is necessary to examine the linear stability of the steady-state solution. For Eq. (1), the steady-state solution at the bifurcation point is given as jc0 = 0. So, let us examine whether the solution is stable for a small fluctuation c(/). Substituting Jt = b + Ax(f) into Eq. (1), and neglecting the higher order of smallness, it follows that... 

The linear instability theory of the behavior of a system near the bifurcation point can be successfully applied to many self-organization problems, such as thermal convection in hydrodynamics4 and crystal growth in solution.5 In these theories, various initial fluctuations play important roles. Occasionally the fluctuations arise from the thermal motion of atoms or molecules. If a system reaches an unstable mode over... 

As shown in Fig. 21, in this case, the entire system is composed of an open vessel with a flat bottom, containing a thin layer of liquid. Steady heat conduction from the flat bottom to the upper hquid/air interface is maintained by heating the bottom constantly. Then as the temperature of the heat plate is increased, after the critical temperature is passed, the liquid suddenly starts to move to form steady convection cells. Therefore in this case, the critical temperature is assumed to be a bifurcation point. The important point is the existence of the standard state defined by the nonzero heat flux without any fluctuations. Below the critical temperature, even though some disturbances cause the liquid to fluctuate, the fluctuations receive only small energy from the heat flux, so that they cannot develop, and continuously decay to zero. Above the critical temperature, on the other hand, the energy received by the fluctuations increases steeply, so that they grow with time this is the origin of the convection cell. From this example, it can be said that the pattern formation requires both a certain nonzero flux and complementary fluctuations of physical quantities. 

The numbers on Fig. 7 refer to the number of unstable modes (corresponding to eigenvalues of the linearized form of (8) with positive real part) for shapes along segments of the families. Only the planar shape up the bifurcation point with the (lAe)>family and a portion of the (lAe)-family are stable to disturbances with the symmetry imposed by this sample size. 

The stability of the (lAe)-family is lost at a Hopf bifurcation point denoted by the open circle (o) on Fig. 7, where the real parts of a complex conjugate pair of eigenvalues change sign. No stable time-periodic solutions were found near this point, indicating that the time-periodic states evolve sub-critically in P and are unstable. Haug (1986) predicted Hopf bifurcations for codimension two bifurcations of the form shown in Fig. 7. but did not compute the stability of the time-periodic states. 

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. 

Since the solution changes stability at the turning points, these are Important to the understanding of the overall reactor behavior. In principle, the Infiated system may be used to determine the bifurcation points and switch solution branches by Increasing the arclength parameter, s, and... 

Fig. 20 Reaction pathways in the reduction of methyl (a) and /-butyl chloride (b) by NO". , reactant and products , transition states. In (a) and (b), the full line is the mass-weighted IRC path from the reactant to the product states the dashed line is a ridge separating the Sn2 and ET valleys and the dotted-dashed line is the mass-weighted IRC path from the Sn2 product state to the ET product state (homolytic dissociation). The dotted line in (a) represents the col separating the reactant and the SN2 product valleys. The dotted line in (b) represents the steepest descent path from the bifurcation point, B, to the Sn2 product. In (a), B is the point of the col separating the reactant and the SN2 product valleys where the ridge separating the SN2 and ET valleys starts.

In parallel with the studies described above, which concern perfectly deterministic equations of evolution, it appeared necessary to complete the theory by studying the spontaneous fluctuations. Near equilibrium, any deviation is rapidly damped but near a bifurcation point, a fluctuation may may lead the system across the barrier. The fluctuation is then stabilized, or even amplified this is the origin of the phenomenon which Prigogine liked calling creation of order through fluctuations. More specifically, one witnesses in this way a step toward self-organization. 

From a mathematical point of view, the onset of sustained oscillations generally corresponds to the passage through a Hopf bifurcation point [19] For a critical value of a control parameter, the steady state becomes unstable as a focus. Before the bifurcation point, the system displays damped oscillations and eventually reaches the steady state, which is a stable focus. Beyond the bifurcation point, a stable solution arises in the form of a small-amplitude limit cycle surrounding the unstable steady state [15, 17]. By reason of their stability or regularity, most biological rhythms correspond to oscillations of the limit cycle type rather than to Lotka-Volterra oscillations. Such is the case for the periodic phenomena in biochemical and cellular systems discussed in this chapter. The phase plane analysis of two-variable models indicates that the oscillatory dynamics of neurons also corresponds to the evolution toward a limit cycle [20]. A similar evolution is predicted [21] by models for predator-prey interactions in ecology. 

Fig. 26.1a). At first, multistage ignitions and extinctions occur followed by a relaxation (long period) mode [7]. Oscillations die a few degrees below the ignition temperature at a saddle-loop infinite-period homoclinic orbit bifurcation point. This is an example where both ignition and extinction are oscillatory. 

This notion of bifurcation point, connected with those of instability and fluctuations, is the basis of this branch of science of self-organization in out-of-equilibrium systems. The story begins with Alan Turing, who, in search of the chemical basis of... 

However at a critical distance from equilibrium, the system must choose between two possible pathways, represented by the bifurcation point Ac. The continuation of the initial pathway, indicated by a broken hne, indicates the region of instability. The concentration of the species A and the value of A assume quite different values, and the more so, the further from equilibrium. An important point is that the choice between the two branching directions is casual, with 50 50 probability of either. The critical point Ac has particular importance because beyond it, the system can assume an organized structure. Here the term self-organization is introduced as a consequence of the dissipative structures, dissipative in the sense that it results from an exchange of matter and energy between system and environment (we are considering open systems). 

Figure 5.21 Bifurcation far from equilibrium, (a) Primary bifurcation is the distance from equilibrium, at which the thermodynamic branching of minimal entropy production becomes unstable. The bifurcation point or critical point corresponds to the concentration (b) Complete diagram of bifurcations. As the non-linear reaction moves away from equilibrium, the number of possible states increases enormously. (Adapted, with permission, from Coveney and Highfield, 1990).

Fig. 11. Bifurcation diagram for an even wave number and

In Fig. 21 we have drawn the bifurcation diagram of the fundamental steady-state solutions for three values of p [ Kxn is plotted versus UK) as the bifurcation parameter]. There is a subcritical region in the upper or lower branch, depending on the relative height of the peaks in Fig. 20c. The asymptotes K and K" of these branches correspond to half-period solutions of infinite length. When p 2 the asymptote K merges with the w-axis therefore situation 2 above can be viewed as a particular case of situation 3 above, in which the bifurcation point moves to infinity. 

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