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 [3]. 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.