Turning-point bifurcation

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... 

Figure 26.1 shows the mole fraction of H2 just above the surface vs. the surface temperature for a mixture of 10% H2 in air at various pressures. At atmospheric pressure (Fig. 26.1a), the mole fraction of H2 is almost insensitive to surface temperature until a turning point, called an ignition (/i), is reached, where the system jumps from an unreactive state to a reactive one. As the surface temperature decreases from high values, the H2 mole fraction increases, and a Hopf bifurcation (HB) point is first found at 980 K, outside the multiplicity regime. The solution branch between the HBi and the extinction is locally unstable (dashed curve). 

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 full 1 — ass versus rres bifurcation diagram is shown in Fig. 6.7(e) and reveals an S-shaped curve. Between the two turning points is a region of... 

We should also consider the behaviour along the top of the isola, on the part of the branch lying at longer residence times than the Hopf point. For Tres > t s, and with k2 still in the above range, the uppermost stationary state is unstable and is not surrounded by a stable limit cycle. The system cannot sit on this part of the branch, so it must eventually move to the only stable state, that of no conversion. Thus we fall off the top of the isola not at the long residence time turning point, but earlier as we pass the Hopf bifurcation point. 

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... 

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). 

The double-zero eigenvalue points, such as M, represent the coalescence of Hopf bifurcation and stationary-state turning points. As mentioned above, they thus represent the points at which the Hopf bifurcation loci begin and end. They also have other significance. Such points correspond to the beginning or end of loci of homoclinic orbits. For the present model, with the given choices of k1 and k2, there are two curves of homoclinic orbit points, one connecting M to N, the other K to L, as shown schematically in Fig. 12.7. 

When the forcing amplitude is very small and the midpoint of the forcing oscillation scans the autonomous bifurcation diagram, the qualitative response of the forced system for all frequencies can be deduced from the autonomous system characteristics. As the amplitude of the forcing becomes larger, one cannot predict a priori what will occur for a particular system. For this example, the most complicated phenomenon possible is a turning point bifurcation on a branch of periodic solutions where two limit cycles, one stable and one unstable, collide and disappear. This will appear as a pinch on the graph of the map [Fig. 1(d)],... 

FIGURE 5 Subharmonic saddle-node bifurcations, (a). The subharmonic period 3 isola for the surface model (o/o0 = 1.4, o0 = 0.001). One coordinate of the fixed points of the third iterate of the stroboscopic map is plotted vs. the varying frequency ratio oi/eio. Six such points (S, N) exist simultaneously, three of them (N) lying on the stable node period 3 and three (S) on the saddle period 3. Notice the two triple turning point bifurcations at o>/o>o = 2.9965 and 3.0286. In (b) the PFM of these trajectories on the isola Is also plotted, (c) shows another saddle-node bifurcation occurring (for a>o = 3) as this time the forcing amplitude is increased to o/o0 = I.6SS. 

FIGURE I Steady-state bifurcation diagrams for variations in the reactant partial pressures, (a) Partial two-parameter bifurcation diagram representing the projection of turning points, Hopf bifurcation points, and apparent triple points. (b)-(Q One parameter sections of the steady-state ffe surface. The vertical axes are the steady-state (k and range from 0 to I. The horizontal axes correspond to the appropriate axis of the two parameter diagram (a). Steady-states are stable or unstable for solid or dashed curves respectively and periodic branches are denoted by pairs of chained curves which represent the minimum and maximum values of ffe on the limit cycle. The periodic branches all terminate in Hopf bifurcations or, when a saddle is present, homoclinic (infinite period) bifurcations. (b)-(e) a, = 0.017, 0.019, 0.021, 0.025 (f)-(i) oti = 0.031, 0.028, 0.024, 0.022. 

FIGURE 3 Three-dimensional view of the steady-state reaction rate surface when -y, = 0.001, and y-i = 0.002. The inset shows a isola for a section of constant at where H and H are Hopf bifurcations and T and T are turning points. 

FIGU RE 10 Illustration of the disappearance of a limit cycle via a turning point on a periodic branch near a subcritical Hopf bifurcation, (a) A stable limit cycle surrounding an unstable focus (b) the unstable focus undergoes a subcritical Hopf bifurcation and leaves an inner unstable limit cycle surrounding a stable focus (c) the two limit cycles combine into a metastable configuration and disappear altogether as the parameter is further increased. 

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