Consider the two differential euuationseach having the critical points a, b, and c; suppos...

Consider the two differential euuations

each having the critical points a, b, and c; suppose that a < b < c. For one of these equations, only the critical point b is stable; for the other equation, b is the only unstable critical point. Construct phase diagrams for the two equations to determine which is which. Without attempting to solve either equation explicitly, make rough sketches of typical solution curves for each. You should see two funnels and a spout in one case, two spouts and a funnel in the other.

i

FIGURE. Solution curves for harvesting a population of alligators.

Then substitute the expressions given in (22) for x′ and y′ to obtain

Tt follows that

Then differentiation of r2 = x2 + y2 yields

so r = r(t) satisfies the differential equation

FIGURE 1. Spiral trajectories of the system in Eq. (22) with Ic = 5.

In Problem 22 we ask you to derive the solution

where fq = r(0). Thus the typical solution of Eq. (22) may be expressed in the form

If r0 = 1, then Eq. (25) gives r(t) = 1 (the unit circle). Otherwise, if r0 > 0, then Eq. (25) implies that r(t) − > 1 as 0. Hence the trajectory defined in (26) spirals in toward the unit circle if f(q) < 1 and spirals out toward this closed trajectory if 0 > ly > 1. Figure 1 shows a trajectory spiraling outward from the origin and four trajectories spiraling inward, all approaching the closed trajectory r(t) = 1. ■

Under rather general hypotheses it can be shown that there are four possibilities for a nondegenerate trajectory of the autonomous system

The tour possibilities are these:

1. (x(f), y(t)) approaches a critical point as t + ∞.

2. (x(t), y(t)) is unbounded with increasing t.

3. y(t)) is a periodic solution with a closed trajectory.

4. (x(f), y(t)) spirals toward a closed trajectory as t→ + ∞.

As a consequence, the qualitative nature of the phase plane picture of the trajectories of an autonomous system is determined largely by the locations of its critical points and by the behavior of its trajectories near its critical points. We will see in Section 7.3 that, subject to mild restrictions on the functions F and G, each isolated critical point of the system x′ = F(x, y) y′ = G(x, y) resembles qualitatively one of the examples of this section=it is cither a node (proper or improper), a saddle point, a center, or a spiral point.