A10 Two simultaneous differential equations

Finally, we consider the more complex problem of two simultaneous (autonomous) differential equations dy/ dt = f( x, y), dy/ dt = g( x, y), the solutions of which pass through given points. In particular, let us study the non-linear problem:

Clearly, this system of equations has 2 fixed points, where dx/ dt = dy/ dt = 0, at (0,0), which is a saddleback, and at (1,1), which is a centre. By linearization, the stable and unstable invariant manifolds local to the origin are given respectively by:

That is already enough information with which to sketch by hand the essential nature of the solutions. Our concern, however, is to compute the solutions, and to display them graphically. We shall require two windows: one to show the phase diagram the ( x, y) trajectories passing through given points; the other to show the time dependence x( t) and y( t) of a chosen trajectory. This calls for the multiple document interface technique introduced in A5. Therefore, we must include the header file mdi.h as well as screen.h.

By analogy with the last section, we define a class to encapsulate the RK increment functions:

class TRKTwoDim {public:      typedef void(TRKTwoDim: :*trkf)(func, func,                  double&, double&, double,...