Tangent bifurcation

Figure C3.6.6 The figure shows tire coordinate, for < 0, of tire family of trajectories intersecting tire Poincare surface at cq = 8.5 as a function of bifurcation parameter k 2- As tire ordinate k 2 decreases, tire first subhannonic cascade is visible between k 2 0.1, tire value of tire first subhannonic bifurcation to k 2 0.083, tire subhannonic limit of tire first cascade. Periodic orbits tliat arise by tire tangent bifurcation mechanism associated witli type-I intennittency (see tire text for references) can also be seen for values of k 2 smaller tlian tliis subhannonic limit. The left side of tire figure ends at k 2 = 0.072, tire value corresponding to tire chaotic attractor shown in figure C3.6.1(a). Otlier regions of chaos can also be seen.

Figure 7.8 shows a plot of the iterative map /2(p) for rule R2 as a function of p for four different values of p p = 1 (top curve), p > Pc, P = Pc and p < Pc, where Pc 0.5347. Notice that all four curves have zero first and second derivatives at the origin. This ensures the existence of some critical value Pc such that for all p < Pc, p t + 1) < p t) and thus that limt->oo p t) = 0. In fact, for all 0 < p < Pc the origin is the only stable fixed point. At p = Pc, another stable fixed point ps 0.373 appears via a tangent bifurcation. For values of p greater than Pc, /2 undergoes a... 

One can also gain an insight into the bifurcation for instance, assume that we introduce a parameter A that permits displacing the curve graphical interpretation for a bifurcation of the first kind. 

Figure 11. Same as Fig. 9 for the tangent bifurcation scenario for nonsymmetric XYZ. 

In the mapping (4.9) with v = a = p = 0 and x < 0, this main tangent bifurcation, which replaces the antipitchfork bifurcation, occurs at pi, = 3(—2/4) /3, which is shifted from zero to positive values when k does not vanish. 

The bottom of the exit channels is at -3194 cm-1 if the origin corresponds to the saddle of the Karplus-Porter surface. The pair of tangent bifurcations occur at E = 1670 cm 1, which is followed by the subcritical antipitchfork bifurcation at Ea = 2633 cm 1. The bifurcation scenario is thus similar to the CO2 system, and we may expect a three-branch Smale horseshoe in this system as well. 

The varying behavior of the multiple steady states in Figure 3.2 is called bifurcation. The bifurcation points for the parameter s are determined by the tangent lines with extreme slopes s and s as depicted by the dashed and dotted lines in Figure 3.2. For any s < s < s there are three steady states, while for any s > s or for any s < s there is only one steady-state solution of the system. And of course, for s = s and s = s there are precisely two steady states. 

Figure 4.33 illustrates the PSPS and bifurcation behavior of a simple batch reactive distillation process. Qualitatively, the surface of potential singular points is shaped in the form of a hyperbola due to the boiling sequence of the involved components. Along the left-hand part of the PSPS, the stable node branch and the saddle point branch 1 coming from the water vertex, meet each other at the kinetic tangent pinch point x = (0.0246, 0.7462) at the critical Damkohler number Da = 0.414. The right-hand part of the PSPS is the saddle point branch 2, which runs from pure THF to the binary azeotrope between THF and water. 

When intersections disappear or new intersections appear, these intersections are tangent. Thus, we suggest that tangency signals bifurcation in the connections among NHIMs. Moreover, we expect that the tangency of intersections gives birth to a transition of chaos. 

Bexp(0/(1 + 0/y)) and the heat removal line (x 6 — 9c). It has been shown that bifurcations or critical conditions exist when the heat generation and heat removal curves are tangent to each other, such that the following pair of equations is satisfied ... 

Now imagine we start decreasing the parameter r. The line r-x slides down and the fixed points approach each other. At some critical value r = r, the line becomes tangent to the curve and the fixed points coalesce in a saddle-node bifurcation (Figure 3.1.6b). For r below this critical value, the line lies below the curve and there are no fixed points (Figure 3.1.6c). 

Now we use a microscope to zoom in on the behavior near the bifurcation. As r varies, we see a parabola intersecting the x-axis, then becoming tangent to it, and then failing to intersect. This is exactly the scenario in the prototypical Figure 3.1.1. 

The critical case occurs when the horizontal line is just tangent to either the local minimum or maximum of the cubic then we have a saddle-node bifurcation. To find the values of h at which this bifurcation occurs, note that the cubic has a local maximum when - (rx - x ) = r-3x = 0. Hence... 

One can show analytically that the value of r at the tangent bifurcation is 1 =3.8284.. . (Myrberg 1958). This beautiful result is often mentioned in... 

We should not be surprised to see ghosts—they always occur near saddle-node bifurcations (Sections 4.3 and 8.1) and indeed, a tangent bifurcation is just a saddle-node bifurcation by another name. But the new wrinkle is that the orbit returns to the ghostly 3-cycle repeatedly, with intermittent bouts of chaos between visits. Accordingly, this phenomenon is known as intermittency (Pomeau and Manneville 1980). 

We commented at the end of Section 10.2 that a copy of the orbit diagram appears in miniature in the period-3 window. The explanation has to do with hills and valleys again. Just after the stable 3-cycle is created in the tangent bifurcation, the slope at the black dots in Figure 10.4.1 is close to -i-l. As we increase r, the hills rise and the valleys sink. The slope of / (x) at the black dots decreases steadily from -1-1 and eventually reaches -1. When this occurs, a flip bifurcation causes... 

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