If a nonlinear sys1em depends on a parameter le (such as a damping constant, spring constant, or chemical concentration), a critical value k0 where the qualitative behavior of the system changes is called a bifurcation point. Show that k = 0 is a bifurcalion point for the system
X' = -x y2 + 1).
Y' = y2 + k.
as follows. Illustrate each part with a phase portrait.
(a) Show that (15) has two equilibrium points for k < 0.
(b) Show that (15) has one equilibrium point for k = 0.
(c) Show that (15) has no equilibrium points for k > 0.
(d) Calculate the linearization about the equilibrium point for k = 0. Relate the phase portraits for (b) and (d).