Loci, Hopf bifurcation

Fig. 3.6. The dependence of local stability and character of the stationary-state solution on the parameters /i amd ku. (a) The locus of Hopf bifurcation points with tr(J) = 0 beneath...

The second significant difference between the predictions and the actual results is that oscillations survive beyond the time. This arises because the pseudo-stationary state has focal character just after the second Hopf bifurcation (i.e. the slowly varying eigenvalues i1>2 are complex conjugates with now negative real parts) so there is a damped oscillatory return to the locus. In Fig. 3.10(a) this can be seen after t 3966, whilst t = 3891. 

Fig. 4.3. The ft-K parameter plane showing loci of changes in local stability or character for the model with the exponential approximation (a) full plane (b) enlargement of region near origin showing, in particular, the locus of Hopf bifurcation (change from stable to unstable focus) and the locus corresponding to the maximum in the ass(/r) curves (broken line).

The locus of these Hopf bifurcation points is also shown in Fig. 4.3 and can be seen to be another closed loop emanating from the origin. It lies in the region between the loci for changes between nodal and focal character, so the condition tr(J) separates stable focus from unstable focus. The curve has a maximum at... 

There are then, also, two values of the dimensionless concentration of the reactant, /if and /if with /if > /if say, on this locus. For our example these are /if = 0.05797 and /if = 0.2070. In between these solutions, the stationary state is unstable. For any other particular system with a different value of k, the appropriate Hopf bifurcation points can be calculated in a similar way, as given in Table 4.3, or read off Fig. 4.3. However, if k is small, we can also estimate /if and /if directly by using an approximate, but quite accurate,... 

We may also note from this last result that the lower Hopf bifurcation point /i f lies at a slightly higher value of /i than the maximum in the ass locus (which occurs at /i = k). 

We have already determined the following information about the behaviour of the pool chemical model with the exponential approximation. There is a unique stationary-state solution for ass, the concentration of the intermediate A, and 0SS, the temperature rise, for any given combination of the experimental conditions /r and k. If the dimensionless reaction rate constant k is larger than the value e-2, then the stationary state is always stable. If heat transfer is more efficient, so that k Hopf bifurcation points along the stationary-state locus as /r varies (Fig. 4.4). If these bifurcation points are /r and /z (with the stationary state... 

Fig. 4.8. The h k parameter plane showing changes in local stability and character for y = 0.1 (a) full parameter plane (b) enlargement for small /r and k showing locus of Hopf bifurcation (transition from stable to unstable focus and (as broken lines) the loci for the maximum and...

Equation (4.74) has distinct real roots provided y < . Hopf bifurcation cannot occur if the activation energy E becomes too small compared with the thermal energy RT i.e. if E < 4RTa. This is the same condition on y as that for the existence of the maximum and minimum in the ass locus. In fact, the Hopf bifurcation points always occur for p values between the maximum and minimum, i.e. on the part of the locus where ass is decreasing, as shown in Fig. 4.8(b) where the loci of turning points are shown as broken lines. 

The locus described by these equations in the K-fx parameter plane is reproduced in Fig. 5.1, which also shows a typical stationary-state bifurcation diagram for fixed k. Hopf bifurcations occur at two values of the precursor reactant concentration /if, 2, with /if < /if, for any given k less than a... 

Fig. 5.1. (a) A typical parameter plane showing a locus of Hopf bifurcation points. For any given value of the parameter k on the ordinate we may construct a horizontal (broken line) the Hopf bifurcation points, pi and pi, are then located as shown. The corresponding stationary-state loci, shown in (b) and (c), have unstable solutions between pi and pi. ... 

This corresponds to the maximum in the locus of Hopf bifurcations in... 

Fig. 5.3. Locus of Hopf bifurcation points in K-fi parameter plane for thermokinetic model with the full Arrhenius temperature dependence and y = 0.21. The nature of the Hopf bifurcation point and, hence, the stability of the emerging limit cycle changes along this locus at k = 2.77 x 10 3. Supercritical bifurcations are denoted by the solid curve, subcritical bifurcations occur along the broken segment, i.e. at the upper bifurcation point for the lowest k. The stationary-state solution is unstable and surrounded by a stable limit cycle for all parameter values within the enclosed region. Oscillatory behaviour also occurs in the small shaded region below the Hopf curve, where the stable stationary state is surrounded by both an unstable and...

Fig. 5.6. Typical tc-g parameter plane for the thermokinetic model with the full Arrhenius temperature depedence a region of stationary-state instability lies within the locus of Hopf bifurcation points (solid curve). Also shown, as broken lines, are the loci corresponding to the maximum and minimum in the g(a, 0) = 0 nullcline (see text).

Fig. 8.5. The locus of Hopf bifurcation points t s-k2 described by eqn (8.42) for cubic autocatalysis with decay but no catalyst inflow. Also shown are the loci of the extinction points TreS, marking the ends of the isola, given by eqn (8.27). The three curves meet at the common...

Fig. 8.9. The locus A of double-zero eigenvalue degeneracies of the Hopf bifurcation for cubic autocatalysis with decay. Also shown, as broken curves, are the loci of stationary-state degeneracies, corresponding to the boundaries for isola and mushroom patterns. The curve A lies completely within the parameter regions for multiple stationary states.

The condition tr(J) = det(J) = 0 corresponds to a Hopf bifurcation point moving exactly onto the saddle-node turning point (ignition or extinction point) on the stationary-state locus. Above the curve A the system may have two Hopf bifurcations, or it may have none as we will see in the next subsection. Below A there are two points at which tr (J) = 0, but only one of... 

Fig. 8.11. The locus H of degenerate Hopf bifurcation points described by the transversality condition (merging of two Hopf points), eqn (8.51). Below this curve, the stationary-state locus exhibits Hopf bifurcation (dynamic instability) at some residence times above it, the system does...

Fig. 10.3. The locus of Hopf bifurcation points indicating the conditions for loss of local stability for the spatially uniform stationary-state solution. Inside this region the system may show spatially uniform time-dependent oscillations.

Fig. 10.7. Representation of conditions in the K-fi parameter plane for instability of the uniform stationary state with respect to spatial perturbations for a system with f = 10. Also shown (broken curve) is the Hopf bifurcation locus, within which the uniform state is itself unstable. The two loci cross at some point on their upper shores.

In the previous section we have taken care to keep well away from parameter values /i and k for which the uniform stationary state is unstable to Hopf bifurcations. Thus, instabilities have been induced solely by the inequality of the diffusivities. We now wish to look at a different problem and ask whether diffusion processes can have a stabilizing effect. We will be interested in conditions where the uniform state shows time-dependent periodic oscillations, i.e. for which /i and k lie inside the Hopf locus. We wish to see whether, as an alternative to uniform oscillations, the system can move on to a time-independent, stable, but spatially non-uniform, pattern. In fact the... 

We have specified that the conditions of interest here are those lying within the Hopf bifurcation locus, so tr(U) will be greater than zero. This means that the eigenvalues appropriate to this uniform state must have positive real parts. The system is unstable to uniform perturbations. We must exclude this n = 0 mode from any perturbations in the remainder of this section. 

With p = 0.019, the traverse cuts both Hopf curves, so the stationary-state locus has four Hopf bifurcation points, as shown in Fig. 12.6(c), each one supercritical. There are two separate ranges of the partial pressure of R over which a stable limit cycle and hence sustained oscillations occur. 

Next, consider the case with p = 0.02014. The traverse across Fig. 12.6(a) as r is varied now also cuts the region of multi stability. It passes above the cusp point C (see Fig. 12.5), giving rise to two turning points in the stationary-state locus, but below the double-zero eigenvalue point M. There are still four intersections with the Hopf curve, so there are four points of Hopf bifurcation. The Hopf point at highest r is now a subcritical bifurcation. The dependence of the reaction rate on r for this system is shown in Fig. 12.6(d). 

In order to model the oscillatory waveform and to predict the p-T locus for the (Hopf) bifurcation from oscillatory ignition to steady flame accurately, it is in fact necessary to include more reaction steps. Johnson et al. [45] examined the 35 reaction Baldwin-Walker scheme and obtained a number of reduced mechanisms from this in order to identify a minimal model capable of semi-quantitative p-T limit prediction and also of producing the complex, mixed-mode waveforms observed experimentally. The minimal scheme depends on the rate coefficient data used, with an updated set beyond that used by Chinnick et al. allowing reduction to a 10-step scheme. It is of particular interest, however, that not even the 35 reaction mechanism can predict complex oscillations unless the non-isothermal character of the reaction is included explicitly. (In computer integrations it is easy to examine the isothermal system by setting the reaction enthalpies equal to zero this allows us, in effect, to examine the behaviour supported by the chemical feedback processes in this system in isolation... 

Our results are summarized in the stability diagram in Figure 8.3.4. The boundary between the two regions is given by the Hopf bifurcation locus b = 3a/5 - 25/a. ... 

Figure 12 Constraint diagram showing parameter values of Turing bifurcation locus (left line) and Hopf bifurcation locus (right line). The Turing instability occurs for parameter values in the middle region. (Reprinted with permission from Ref. 34.)...

Modulated waves in the full system correspond to limit-cycle behavior for and C in the reduced system and the bifurcation to modulated waves corresponds to a Hopf bifurcation from a steady state ( 1, Ci) to n limit cycle. An expression for the locus of Hopf bifurcations is obtained as follows. The stability matrix for Equations (11) is ... 

The first equation gives the value of at the Hopf bifurcation, in terms of ai. The second then gives 02 for the Hopf bifurcation in terms of and a. This Hopf locus is plotted in Figure 10 and is the boundary of the region of modulated waves. 

These requirements specify two loci one of them, labelled DH l in Fig. 8.12, emanates from the points / = 0, k2 = 9/256, as located in 8.3.6. This curve cuts through the parameter space for isola and mushroom patterns, but always lies below the curve A. (In fact it intersects A at the common point P0 = i(33/2 - 5), k2 = rg(3 - /3)4(1 -, /3)2 where the locus H also crosses.) In the vicinity of DH x, the stationary-state curve has only one Hopf point. This changes from a subcritical bifurcation (unstable limit cycle emerging) for conditions to the right of the curve to supercritical (stable limit cycle emerging) to the left. 

Stable, limit cycle. The latter occurs in the Salnikov case and the modified bifurcation diagram is shown in Fig. 5.11(b). The stable limit cycle born at the lower Hopf point overshoots the upper Hopf point but is extinguished by colliding with the unstable limit cycle born at the upper Hopf point which also grows in amplitude as )jl is increased. Over a, typically narrow range, then there are two limit cycles, one unstable and one stable around the (stable) steady-state point. If we start with the system at some large value of /r, so we settle onto the steady-state locus, and then decrease the parameter, we will first swap to oscillations at the Hopf point /r - At this point there is a stable limit cycle available as the system departs from the now unstable steady-state, but this stable limit cycle is not born at this point and so already has a relatively large amplitude. We would expect to... 

The MTW locus emerges from the Hopf locus at the codimension-two point where the Hopf frequency equals the rotation frequency. This point is easily found. We define 0 2 to be the Hopf frequency, that is = V5et, where Det is the determinant of the stability matrix (15) at the bifurcation. Then ... 

---