Consider the equationdy/dt = ay y2 = y(a y). (iv)(a) Again

Chapter 2, Problem 27

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Consider the equationdy/dt = ay y2 = y(a y). (iv)(a) Again consider the cases a < 0, a = 0, and a > 0. In each case find the critical points,draw the phase line, and determine whether each critical point is asymptotically stable,semistable, or unstable.(b) In each case sketch several solutions of Eq. (iv) in the ty-plane.(c) Draw the bifurcation diagram for Eq. (iv). Observe that for Eq. (iv) there are thesame number of critical points for a < 0 and a > 0 but that their stability has changed.For a < 0 the equilibrium solution y = 0 is asymptotically stable and y = a is unstable,while for a > 0 the situation is reversed. Thus there has been an exchange of stability as apasses through the bifurcation point a = 0. This type of bifurcation is called a transcriticalbifurcation.