Figure 8: Trajectories for the Riccati equation associated with the Airy equation.
It is clear from Figure that the left half-plane t < 0, corresponding to the region of the xt-plane in Figure where trajectories seem to be non-oscillatory, is more amenable to analysis, because solutions appear to funnel together or spray apart. In fact, this is a mirror image (about the u-axis) of the phase plane analyzed by John Hubbard in . Since all of this deals with question 2, we shall defer further discussion until we have dealt with question 1: the behavior of the solutions for t > 0. The oscillatory solution for t>0 shown in Figure suggests that we try the Prüfer transformation.
We redraw Figure as Figure , adding isoclines of slopes -1, 0, +1. Portions of these three isoclines act as fences to form funnels and antifunnels.
Figure 9: Trajectories for the Riccati equation , with isoclines of slopes -1, 0, +1.