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The Riccati Transformation

The Riccati substitution

transforms the linear time-dependent system (2) into a first-order nonlinear ordinary differential equation in the single variable u:

Equation (15) is known as the Riccati equation associated with the system (2). The Riccati equation associated with the system (8) associated with the Airy equation is

A picture representing the slope field, and several trajectories, of equation (16) is shown in Figure 8.

Figure 8: Trajectories for the Riccati equation associated with the Airy equation.

It is clear from Figure 8 that the left half-plane t < 0, corresponding to the region of the xt-plane in Figure 1 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 1 suggests that we try the Prüfer transformation.

We redraw Figure 8 as Figure 9, 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.

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