Study the relationship between the values of the parameter b in the differential equation dy /dt = y2 + by + I and the equilibrium points of the equation and their stability.
(a) Show that for lbl > 2 there are two equilibrium points; for |b| = 2, one; and for |b| < 2, none.
(b) Determine the bifurcation points for b -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.