Consider the equation dy/dt = a y2 . (ii) (a) Find all of the critical points for Eq

Chapter 2, Problem 25

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Consider the equation dy/dt = a y2 . (ii) (a) Find all of the critical points for Eq. (ii). Observe that there are no critical points if a < 0, one critical point if a = 0, and two critical points if a > 0. (b) Draw the phase line in each case and determine whether each critical point is asymptotically stable, semistable, or unstable. (c) In each case sketch several solutions of Eq. (ii) in the ty-plane. (d) If we plot the location of the critical points as a function of a in the ay-plane, we obtain Figure 2.5.10. This is called the bifurcation diagram for Eq. (ii). The bifurcation at a = 0 is called a saddlenode bifurcation. This name is more natural in the context of second order systems, which are discussed in Chapter 9. 2 1 1 2 2 1 1 2 3 4 Unstable Asymptotically stable y a FIGURE 2.5.10 Bifurcation diagram for y = a y2.