Subcritical pitchfork bifurcation

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... 

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... 

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.)... 

Figure 6.11 shows the bifurcation diagram of the amplitude equation, (6.40), where the damping coefficient, c, is taken as the control parameter. The trivial solution (i.e., the equilibrium point of the original system) undergoes a subcritical pitchfork bifurcation [69] at approximately Cd = 2.32 x lO Nms/rad. This value agrees with (6.50). It can be shown that this bifurcation corresponds to a... 

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. 

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. 

Like pitchfork bifurcations, Hopf bifurcations come in both super- and subcritical varieties. The subcritical case is always much more dramatic, and potentially dangerous in engineering applications. After the bifurcation, the trajectories must jump to a distant attractor, which may be a fixed point, another limit cycle, infinity, or—in... 

If the graph of f is concave down near x, the cobweb tends to produce a small, stable 2-cycle close to the fixed point. B ut like pitchfork bifurcations, flip bifurcations can also be subcritical, in which case the 2-cycle exists below the bifurcation and is unstable—see Exercise 10.3.11. 

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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