Problem

Another Bifurcation Study the relationship between the values of the parameter b in the di...

Another Bifurcation Study the relationship between the values of the parameter b in the differential equation dy/dt = y2 + by + 1 and the equilibrium points of the equation and their stability.

(a) Show that for |b| > 2 there are two equilibrium points; for |b| = 2, one; and for |b| < 2, none.
(b) Determine the bifurcation points for b (see Problem)—the b-values at which the solutions undergo qualitative change.
(c) Sketch solutions of the differential equation for different b-values (e.g., b = −3, −2, −1, 0, 1, 2, 3) in order to observe the change that takes place at the bifurcation points.
(d) Determine which of the equilibrium points are stable.
(e) Draw the bifurcation diagram for this equation; that is, plot the equilibrium points of this equation as a function of the parameter values for −∞ < b < ∞. For this equation, the bifurcation does not fall into the pitchfork class.

Saddle-Node Bifurcation Explore this type of bifurcation for the equation dy/dt= y2 + r.

Problem

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.