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 curren 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 plusmn a, which are sta...

y y y 3 y y 2 a For 0 the only real root of y y 2 0 is y 0 Because y y y 2 0 for y 0 And y y y ...

For the differential equation dy/dt = ay - y3, Show that 0 is a bifurcation point of

Question:

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.