Pitchfork bifurcation

This route should already be familiar to us from our discussion of the logistic map in chapter 4, Prom that chapter, we recall that the Feigenbaum route calls for a sequence of period-doubling bifurcations pitchfork bifurcations versus the Hopf bifurcations of the Landau-Hopf route) such that if subharmonic bifurcations are observed at Reynolds numbers TZi and 7 2, another can be expected at TZ determined by... 

A theoretical framework for considering how the behavior of dynamical systems change as some parameter of the system is altered. Poincare first apphed the term bifurcation for the splitting of asymptotic states of a dynamical system. A bifurcation is a period-doubling, -quadrupling, etc., that precede the onset of chaos and represent the sudden appearance of a qualitatively different behavior as some parameter is varied. Bifurcations come in four basic varieties flip bifurcations, fold bifurcations, pitchfork bifurcations, and transcritical bifurcations. In principle, bifurcation theory allows one to understand qualitative changes of a system change to, or from, an equilibrium, periodic, or chaotic state. 

Period-2 Limit Cycle At the second critical value, a = Q2 = 3, itself becomes unstable and is replaced by a stable attracting period-2 limit cycle, x 2) - This new bifurcation - called the pitchfork bifurcation - is shown schematically in figure 4.4 below. 

Fig. 4.4 A schematic representation of the pitchfork bifurcation a stable fixed point bifurcates into a period-2 limit cycle plus an unstable fixed point.

Period-n Limit Cycles As a increases, the system undergoes an infinite sequence of successive period-doublings via pitchfork bifurcations. In general,... 

Fig. 4.6 An illustration of some parameters describing pitchfork bifurcations used to define universaJity see text.

Note the near-horizontal and slightly disconnected black line at y yc = 1.13 in Figure 3.21. For the chosen value of a = 1.23706 106 = io6 092389299 the bifurcation diagram becomes an imperfect pitchfork. A perfect pitchfork bifurcation diagram occurs when a is slightly decreased, so that the corresponding black horizontal line in Figure 3.19... 

We note that split bifurcation regions and pitchfork bifurcation such as depicted in Figures 3.20 and 3.21 for Kc were never encountered by us for any fixed a in terms of yc. 

Pitchfork bifurcation of Yrd versus Kc for the closed-loop SISO system with simple... 

The bifurcation diagram represents an imperfect pitchfork diagram, where the middle steady state persists over the entire range of Kc, even for negative values of Kc, i.e., even for positive feed back control, which would destabilize the system. 

The static bifurcation characteristics of the resulting closed loop system have also been discussed in the previous section and we have seen that the bifurcation diagram of the reactor dense-phase dimensionless temperature, namely a plot of Yrd versus the controller gain Kc is a pitchfork. Such bifurcations are generally structurally unstable when any of the system parameters are altered, even very slightly. 

Changing Kc increases the slope of the heat removal line because its slope is 1 + K If a bifurcation diagram is drawn for this nonadiabatic case with Kc as the bifurcation parameter and the jacket cooling temperature is the temperature of the middle steady state ym, we obtain a pitchfork type bifurcation diagram as shown in Figure 9 (A-2). 

For positive values of the control parameter , stationary, spatially periodic solutions y/s(x) = y/x(x I 2n/q) of (53) may be found with and without forcing. However, in the case of a vanishing forcing amplitude (a = 0) in (53), this equation has a i//-symmetry and one has a pitchfork bifurcation from the trivial solution l/r = 0 to finite amplitude periodic solutions as indicated in Fig. 19. In the unforced case, however, periodic solutions of (53) are unstable for any wave number q against infinitesimal perturbations that induce coarsening processes [114, 121],... 

There are two very different types of pitchfork bifurcation. The simpler type is called supercritical, and will be discussed first. 

The reason for the term pitchfork becomes clear when we plot the bifurcation diagram (Figure 3.4.2). Actually, pitchfork trifurcation might be a better word ... 

Equations similar to = -x+ /Jtanhx arise in statistical mechanical models of magnets and neural networks (see Exercise 3.6.7 and Palmer 1989). Show that this equation undergoes a supercritical pitchfork bifurcation as P is varied. Then give a numerically accurate plot of the fixed points for each p. 

In real physical systems, such an explosive instability is usually opposed by the stabilizing influence of higher-order terms. Assuming that the system is still symmetric under x — x, the first stabilizing term must be x . Thus the canonical example of a system with a subcritical pitchfork bifurcation is... 

As usual in bifurcation theory, there are several other names for the bifurcations discussed here. The supercritical pitchfork is sometimes called a forward bifurcation, and is closely related to a continuous or second-order phase transition in sta-... 

We now see that a supercritical pitchfork bifurcation occurs at 7 = 1. It s left to you to check the stability of the fixed points, using linear stability analysis or graphical methods (Exercise 3.5.2). 

What is the symmetry of the governing equation Clearly the left and right halves of the hoop are physically equivalent—this is reflected by the invariance of (I) and (2) under the change of variables 0 -> -0. As we mentioned in Section 3.4, pitchfork bifurcations are to be expected in situations where such a symmetry exists. 

As we mentioned earlier, pitchfork bifurcations are common in problems that have a symmetry. For example, in the problem of the bead on a rotating hoop (Section 3.5), there was a perfect symmetry between the left and right sides of the hoop. But in many real-world circumstances, the symmetry is only approximate—an imperfection leads to a slight difference between left and right. We now want to see what happens when such imperfections are present. 

If /i = 0, we have the normal form for a supercritical pitchfork bifurcation, and there s a perfect symmetry between x and -x. But this symmetry is broken when h 0 for this reason we refer to h as an imperfection parameter. 

In the following exercises, sketch all the qualitatively different vector fields that occur as r is varied. Show that a pitchfork bifurcation occurs at a critical value of r (to be determined) and classify the bifurcation as supercritical or subcritical. Finally, sketch the bifurcation diagram of x vs. r. 

The next exercises are designed to test your ability to distinguish among the various types of bifurcations—it s easy to confuse them In each case, find the values of r at which bifurcations occur, and classify those as saddle-node, transcritical, supercritical pitchfork, or subcritical pitchfork. Finally, sketch the bifurcation diagram of fixed points X vs. r. 

Subcritical pitchfork) Consider the system x = rx + x -x which exhibits a subcritical pitchfork bifurcation. 

Nondimensionalizing the subcritical pitchfork) The first-order system u- au + bu - cm where b,c> 0, has a subcritical pitchfork bifurcation at a-Q. Show that this equation can be rewritten as... 

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