Problem

Pitchfork Bifurcation For the differential equationshow that 0 is a bifurcation point of t...

Pitchfork Bifurcation For the differential equation

show that 0 is a bifurcation point of the parameter α as follows.

(a) Show that if α ≤ 0 there is only one equilibrium point at 0 and it is stable.
(b) Show that if α > 0 there are three equilibrium points: 0, which is unstable, and , 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 α, as in Fig. 11 for Example 3. Figure 13 shows values already plotted for α = −2 and α = +2; when you fill it in for other values of α, you should have a graph that looks like a pitchfork. Consequently, α = 0 is called a pitchfork bifurcation; when the pitchfork branches at α = 0, the equilibrium at y = 0 loses its stability.