Figure 10: Example of upper and lower fences
There are two important observations about fences: (1) Almost any curve is a fence of some sort, at least in appropriate regions; and (2) you do not need to solve the differential equation to check whether a curve is a fence.
Example. Does the equation have any monotone increasing solutions? (This problem is important when tring to understand the lowest eigenfunction for certain Schrodinger equations.)
Solution. As Figure 11 shows, some solutions increase until they hit the curve , then decrease to become asymptotic to x=-1. Are any "fenced away" from the line x=-1? Yes: The curve is an upper fence when t>2, since when t>2. Thus all solutions that pass through points with t>2 and increase monotonically toward the asymptote x=-1, but remain below this fence. Note that this property is pretty delicate: there are no such solutions for .
Figure 11: Monotone increasing solutions of
Fences are especially interesting when two are disjoint and close together. This can happen in two ways; If the upper fence is above the lower fence, together they form a funnel; if below it, they form an antifunnel (Figure 12). As the drawing suggests, once a solution enters a funnel, it can never leave it. Especially when the boundaries of a funnel are close together, this can allow a very precise description of the ultimate fate of the whole classes of solutions.
Figure 12: Example of funnels and antifunnels.
Example. Show that all solutions of (sketched in Figure 13) with satisfy for sufficiently large t.
Figure 13: Solutions of with a funnel and antifunnel
Solution. The curves and are isoclines of slope 0 and -1 respectively. The former is a lower fence, and the latter an upper fence when . Superimposing these curves over the solution curves in Figure 13, we see that any solution in the region x<-1, t>2 will enter this funnel (compare to Figure 9).
Even more interesting are the antifunnels. A first result is that in every antifunnel, there is at least one solution. Often there is exactly one solution, for instance if the antifunnel narrows and in the antifunnel. Then you can characterize a particular solution by its long-term behavior. Often this behavior is exceptional, separating packages of solutions that all behave in the same way. In most examples of real interest, this is the most important information we can hope for. It often answers such questions as: What amount of hunting will lead to the extinction of a species? What amount of heat will make the boiler explode?
Example. Show that the differential equation of Figure 13 has exactly one solution asymptotic to as .
Solution. We superimpose on the graph of solutions of isoclines of slopes 0 and 1 in Figure 13), and use portions as fences to define an antifunnel. The isoclines and are an upper and a lower fence, respectively, for . Thus, the region between them is a narrowing antifunnel. Since there, the uniqueness conclusion follows.
Students have no conceptual difficulties with fences, funnels, and antifunnels. On the other hand, the algebraic manipulation of inequalities needed to use them effectively gives many students a lot of trouble. For the qualtitative analysis of differential equations, mastery of inequalities becomes a fundumental skill. Identifying funnels and antifunnels furnishes an endless collection of motivated exercises.