Consider the equation dy dt = a y2. (29) a. Find all of the critical points for equation

Chapter 2, Problem 24

(choose chapter or problem)

Consider the equation dy dt = a y2. (29) a. Find all of the critical points for equation (29). Observe that there are no critical points if a < 0, one critical point if a = 0, and two critical points if a > 0. G b. Draw the phase line in each case and determine whether each critical point is asymptotically stable, semistable, or unstable. G c. In each case sketch several solutions of equation (29) in the ty-plane. Note: 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 equation (29). The bifurcation at a = 0 is called a saddle -- node 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.