Bifurcation variable

It is important to note that while the selection of the bifurcation variable does not affect the maximal number of steady-state solutions, it affects the number and type of bifurcation diagrams. For example, if we selected the coolant or feed temperature as the bifurcation variable then Eq. (18) would be the universal unfolding for both the adiabatic and the cooled case and no isolas would exist [1 2 ]. 

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... 

In flame extinction studies the maximum temperature is used often as the ordinate in bifurcation curves. In the counterflowing premixed flames we consider here, the maximum temperature is attained at the symmetry plane y = 0. Hence, it is natural to introduce the temperature at the first grid point along with the reciprocal of the strain rate or the equivalence ratio as the dependent variables in the normalization condition. In this way the block tridiagonal structure of the Jacobian can be maintained. The flnal form of the governing equations we solve is given by (2.8)-(2.18), (4.6) and the normalization condition... 

One of the interesting consequences of changing specifications is the effect on the equation structure. With formulations C and D, the occurrence matrix is symmetric. But if external flows, w, and model parameters, / , are introduced as the unknown variables, the symmetry may be destroyed. One way of preserving the local symmetry is to augment the system of equations and to bifurcate the variables in terms of state and design variables (M2). 

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. 

Hopf bifurcation analysis commonly signals the onset of oscillatory behaviour. This chapter uses a particular two-variable example to illustrate the essential features of the approach and to explore the relationship to relaxation oscillations. After a careful study of this chapter the reader should be able to ... 

In order to evaluate P2 we need to consider how the governing equations for mass and energy balance themselves vary with changes in the variables. In the case of the present model this means evaluating various partial derivatives of (5.1) and (5.2) with respect to a and 0. Before proceeding, however, we should take a look at the elements of the Jacobian matrix evaluated for Hopf bifurcation conditions ... 

When the catalyst decays we have a two-variable system and hence there is the potential for Hopf bifurcations and sustained oscillations. In our flow reactor, we have the possibility of oscillation about one stationary state... 

Again we have a two-variable system, so we can look for points of Hopf bifurcation in terms of the trace and determinant of the Jacobian matrix evaluated for the stationary-state solutions. Thus we seek conditions such that... 

Local stability analysis and Hopf bifurcation for n variables... 

As described above, the first condition on the eigenvalues for a Hopf bifurcation in a three-variable scheme is that the principal pair should be purely imaginary and the third should be real and negative. For this to be the... 

Let us imagine a scenario for which a supercritical Hopf bifurcation occurs as one of the parameters, fi say, is increased. For fi < fi, the stationary state is locally stable. At fi there is a Hopf bifurcation the stationary state loses stability and a stable limit cycle emerges. The limit cycle grows as ft increases above fi. It is quite possible for there to be further bifurcations in the system if we continue to vary fi. With three variables we might expect to have period-doubling sequences or transitions to quasi-periodicity such as those seen with the forced oscillator of the previous section. Such bifurcations, however, will not be signified by any change in the local stability of the stationary state. These are bifurcations from the oscillatory solution, and so we must test the local stability of the limit cycle. We now consider how to do this. 

This concludes the mathematical preliminaries needed for our discussion of systems with three or more independent concentrations. The different dynamical responses which arise from the various bifurcations listed above will now be exemplified through a number of model schemes, each with three variables. 

In chapter 12 we discussed a model for a surface-catalysed reaction which displayed multiple stationary states. By adding an extra variable, in the form of a catalyst poison which simply takes place in a reversible but competitive adsorption process, oscillatory behaviour is induced. Hudson and Rossler have used similar principles to suggest a route to designer chaos which might be applicable to families of chemical systems. They took a two-variable scheme which displays a Hopf bifurcation and, thus, a periodic (limit cycle) response. To this is added a third variable whose role is to switch the system between oscillatory and non-oscillatory phases. 

Using the usual Hopf analysis we can readily find that the two-variable scheme, with fixed c, has a Hopf bifurcation at c = 1.102. Now let us consider the behaviour of the full three-variable scheme. 

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