Subcritical pitchfork bifurcation

Example 4:

Equilibrium and stability behavior for this equation are shown in Figure 6 (right). There is again a loss of stability at the bifurcation point (x0, p0) = (0, 0). In contrast to the example 3 , there is no exchange of stability. Instead, the stability is lost locally at the bifurcation point. Differential equation and stationary solutions have the same symmetry as those of the previous example.

The terms 'supercritical' and 'subcritical' are occasionally used in a fashion solely oriented toward decreasing or increasing values of the parameter p and not defined when all the half-branches are unstable. Supercritical bifurcation has locally stable solution on both sides of the bifurcation. In this vein a pitchfork bifurcation at p = p0 is said to be supercritical if there is locally only one solution to p < p0 - that is, 'supercritical' bifurcates to the right and 'subcritical' to the left (the p-axis points to the right).

So far we have introduced the concepts of simple bifurcation point. Physically, an equilibrium represents a situation without 'life'. It may mean no motion of a pendulum, no reaction in a reactor, no nerve activity, no flutter of an air-foil, no laser operation, or no circadian rhythms of biological clocks. A stationary bifurcation does not give rise to such exciting phenomena; the branching solutions show no more life than the previously known solutions.

The mathematical vehicles that describe this kind of 'life' are the function x(t) or y(t). The corresponding function spaces include equilibrium as the special-case constant solutions. The type of bifurcation that connects equilibrium with periodic motion is Hopf bifurcation.

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