Supercritical pitchfork bifurcation

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. 

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). 

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. 

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. 

Show that a supercritical pitchfork bifurcation occurs at the origin in the system... 

After dividing through by x and neglecting higher-order terms, get ii+2- x /(> 0. Hence there is a pair of fixed points with x = -yj 6 ii + 2) for H slightly greater than —2. Thus a supercritical pitchfork bifurcation occurs at... 

Much of this picture is familiar. The origin is globally stable for r < 1. At r = 1 the origin loses stability by a supercritical pitchfork bifurcation, and a symmetric pair... 

It is interesting to note that, in this example as well as the two examples that follow, as is gradually increased, the trivial equilibrium point first goes through a supercritical pitchfork bifurcation at Q = cob and then a subcritical pitchfork bifurcation at Q = co where co is the solution of bo = 0. These bifurcations in the amplitude equation correspond to Hopf bifurcations of the original system s trivial equilibrium point. 

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

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. 

Patterns in fluids) Ahlers (1989) gives a fascinating review of experiments on one-dimensional patterns in fluid systems. In many cases, the patterns first emerge via supercritical or subcritical pitchfork bifurcations from a spatially uniform state. Near the bifurcation, the dynamics of the amplitude of the patterns are given approximately by tA — e4 - gA in the supercritical case, orx4 = e4-gA - kA in the subcritical case. Here Aft) is the amplitude, T is a typical time scale, and e is a small dimensionless parameter that measures the distance from the bifurcation. The parameter g > 0 in the supercritical case, whereas g < 0 and k > 0 in the subcritical case. (In this context, the equation iA = eA - gA is often called the Landau equation.)... 

For 0.33 < X < 0.53, the OFT is a pitchfork bifurcation, and tbe reoriented state is a D state [see the filled circles in Fig. 13(a)], This state loses its stability through a supercritical Hopf bifurcation to an O state [curve 1 in... 

Bearing in mind that the reference solution exists for all values of p, two subcases are to be distinguished if go > 0, the bifurcation is supercritical as the new solution exists for p > 0 whereas if go < 0 the new solution arises for /X < 0 and the bifurcation is said to be subcritical. In both cases we have a pitchfork bifurcation (see Figures la, b). 

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