For the differential equation dy/dt = ay - y3,
Show that 0 is a bifurcation point of the parameter a as follows
(a) Show that if a ‰¤ 0 there is only one equilibrium point at 0 and it is stable.
(b) Show that if a > 0 there are three equilibrium points: 0, which is unstable, and ± ˆša, which are stable.
(c) Then draw a bifurcation diagram for this equation. That is, plot the equilibrium points (as solid dots for stable equilibria and open dots for unstable equilibria) as a function of a, as in Fig. 2.5.11 for Example 3. Figure 2.5.13 shows values already plotted for a = -2 and a = +2; when you fill it in for other values of a, you should have a graph that looks like a pitchfork. Consequently, a = 0 is called a pitchfork bifurcation; when the pitchfork branches at a = 0, the equilibrium at y = 0 loses its stability.

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