Tue Sep 30

MAIN POINTS
Pitchfork bifurcation is a third type of bifurcation that arises in problems with a symmetry. The supercritical pitchfork bifurcation has the form x'=rx-x^3. This equation has a single, central stable fixed point for all values of r ≤ 0, and three fixed points for positive r. In this case, the two symmetric fixed points on either side are stable, and the central fixed point is unstable. Plotting this on a bifurcation diagram shows a pitchfork shape. Example 3.4.2 shows how this idea conveys onto a diagram of Potential. As x increases past 0, two pits form on either side of the origin. The subcritical bifurcation has two symmetric FP's that are unstable and converge towards the origin as r is increased to zero from a negative value.

3.5 discusses the overdamped bead on a rotating hoop. They find a way to simplify the second order Newton equation to a first order.

CHALLENGES
The bead/hoop example was kind of impenetrable to me. I expect that we'll go over this in class. On p.59, the book says that the detailed analysis of (3) is left in exercises... perhaps it would be good to go over how the bifurcation diagram for that equation with the 5th degree term is derived.

REFLECTION
What are some examples of situations where the symmetry causes a pitcfork bifurcation? Are there any simple ones? Does it arise in chemistry or physics?