Bifurcation curves

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... 

These equations represent a parametric curve with parameter y. From a set of values of the parameter y it is possible to draw a curve in the xq — yo plane, so we obtain a bifurcation curve as a function of parameter y. This curve with a cusp point can be considered as the border that dividing the plane xq — yo into domains with one and three equilibrium states respectively. 

Eq.(50) shows the variation of the equilibrium dimensionless temperature as a function of the maximum value of the dimensionless coolant flow rate X6max- Plotting XQmax versus X3e a bifurcation curve can be obtained, from which it is possible to determine the value of xsmax which gives a different behavior of the reactor in steady state. It is interesting to note that Eq.(50) is equal to Eq.(47) when we make the substitutions of Eq.(49) into Eq.(47). 

Our method shows that there can be no more than two intersections of a bifurcation curve with a line of constant U for, if A(A) denote (1 + A)2/A, the two values of A that satisfy the quadratic A = /z/ are the only two values possible. Furthermore, because their product is 1, a scale that is logarithmic in A will make the bifurcation curve symmetric about the line A = 1. As U —> 0, A = (if- 00 and one A goes to zero as 1/A, whereas the other goes to infinity as A. In fact, the A are asymptotically filU and U//jl. The curve in Fig. 15 is plotted with the [/-axis vertical, and there is a symmetric part (not shown) below the A-axis. At a = 201 and /x =. 005, it is clearly a mushroom about to become an isola. 

FIGURE 27 The bifurcation curve in three dimensions surrounded by phase-planes. 

FIGURE 4 Relation between Figures 3,5 and 8. The steady-state reaction-rate surface has bifurcation curves scribed on its surface which can be projected onto the parameter plane below or along lines of constant a to give side view. 

FIGURE 8 Detailed bifurcation diagram for variations in the reactant partial pressures (a, and a2) when y, = 0.001 and y2 = 0.002. The middle diagram is the superposition of the four surrounding bifurcation curves. The points K, L, M and N correspond to double zero eigenvalues and the points G and H are metacritical Hopf points. 

The point S of figure 8 at which the Hopf bifurcation curve crosses the boundary of the multiplicity region is not a double zero degeneracy, for the upper steady state (i.e. that with the larger 0b) is undergoing the Hopf bifurcation at the same time as the lower steady-state undergoes a saddle-node bufurcation, i.e. the conditions trJ = 0 and detJ = 0 apply at different points. It does, however, serve to show the four combinations of the two most common... 

In the 1/1 entrainment region each side of the resonance horn terminates at points C and D respectively. These points are codimension-two bifurcations and correspond to double +1 multipliers. As the saddle-node curve at the right horn boundary rises from zero amplitude towards point D, one multiplier remains at unity (the criterion for a saddle-node bifurcation) as the other free-multiplier of the saddle-node increases until it is also equal to unity upon arrival at point D. The same thing occurs for the left boundary of the resonance horn. The arc CD is also a saddle-node bifurcation curve but is different from those on the sides of the resonance horn. As arc CD is crossed from below, the period 1 saddle combines not with its companion stable node, but with the unstable node that was in the centre of the phase locked torus. As the pair collides, the invariant circle is lost and only the stable node remains. Exactly the same scenario is observed for the 1/2 resonance horn as well. 

Several codimension-two bifurcations have already been mentioned. Although they occur in restricted subspaces of parameter space and would therefore be difficult to locate experimentally, their usefulness lies in their role as centres for critical behaviour. Emanating from each local codimen-sion-two point will be two or more of the above codimension-one bifurcation curves. Their usefulness in studying dynamics is akin to that of the triple point in thermodynamic phase equilibria in which boundaries between three different phases come together at a point in a two-parameter diagram. Because some of these codimension-two points have been studied and classified analytically, finding one can provide clues about what other codimension-one bifurcation curves to expect near by and thus aids in the continuation of all of the bifurcation curves in the excitation diagram. 

The excitation diagram was found to contain saddle-node, Hopf, period doubling, and homoclinic bifurcations for the stroboscopic map. In addition, many of these co-dimension one bifurcation curves were found to meet at the following co-dimension two bifurcation points Bogdanov points (double +1 multipliers), points with double -1 multipliers, points with multipliers at li and H, metacritical period-doubling points, and saddle-node cusp points. 

The bifurcation curve relating y and a is the projection of the level curve F(a, y) = 0 onto the a-y plane, where z = 0. Our 3D plot of the surface z = F(a,y) in adiabNisosurf contains this level curve marked in black on the surface, as well as a second, isolated plot of it below the surface on the ground plane see Figure 3.6. 

Our next code and plot draws only the bottom curve of Figure 3.8 by itself from the inputs a, f3, and 7. It uses adiabNisoauxf 1 to find the actual bifurcation points of the curve, depicts them by the dotted vertical lines in the figure window, and then plots the bifurcation curve itself. This plot gives the most practical output in our view. 

Bifurcation curve graphical method output as a thin line bisection output superimposed as a thick line Figure 3.11... 

Nonadiabatic CSTR surface and bifurcation curve for Kc Figure 3.14... 

Our CD also contains the MATLAB function m file runNadiabNisokccurve. m. A call of runNadiabNisokccurve (180000,1,15,1,100,. 001, .7,1.2), for example, plots only the bifurcation curve with respect to Kc in Figure 3.16, i.e., it repeats the bottom plot in Figure 3.14. 

Nonadiabatic CSTR bifurcation curve for Kc as the bifurcation parameter... 

As mentioned above, the Kc bifurcation curves have an inverted letter-5 shape. We refer to the conclusions of Section 3.2, and to Chapter 7 for an analysis of the physical meaning of the differing shapes of the Kc and a parameter bifurcation curves when applied to industrial processes and reactors. 

In our CSTR example the constants Kc and a have opposite physical effects. If a increases, the flow rate q decreases and thus the rate of reaction increases, as does the heat of reaction. On the other hand, if Kc increases, then the heat removal by heat transfer to the cooling jacket increases, reducing the rate of reaction and the production of heat. Note that the search directions in our respective lowhighkc and lowhighal sub-programs point in opposite directions for the 5-shaped a bifurcation curves and for the inverted 5-shaped Kc bifurcation curves. 

Nonadiabatic, nonisothermal CSTR bifurcation curve for a Figure 3.18... 

Again we plot the bifurcation curve with respect to a as was done in Figure 3.17 separately. We use the command runNadiabNisoalcurve (1,12,1,0,400,0.0001,0,3.8 10 8) for Figure 3.18, for example, where the m file runNadiabNisoalcurve. m has been taken from the CD. This gives us the following plot. 

The two branches of the top bifurcation curve in Figure 3.22 will rejoin for some large-magnitude but physically impossible negative value of Kc far to the left of our window s edge. Negative values for Kc are impossible, since this would physically mean that heat is transferred from the cold part to the hot part. As depicted in Figure 3.22, the bifurcation curves look like an incomplete isola. 

Output Plot of up to three steady state solutions yO, ymid, yl in Y on the 7. bifurcation curve. 

Bifurcation curve with multiple steady states marked by + Figure 3.26... 

Bifurcation curve for y values in terms of the feedback controller gain K... 

The auxiliary function pelletrunfwd.m is used in pelletetacurve.m to evaluate r/ repeatedly for an interval of

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Fivefold bifurcating curve for ry, runtime 290 seconds Figure 5.26... 

Another fivefold bifurcating curve for 77 runtime 340 seconds... 

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