Nonhomogeneous systems and higher dimensional systems*

If your system is nonhomogeneous:

The problem is that this system doesn't just vary with X and Y, it also varies with t.  If you think of t as time (which it often is), however, you can use things like animation to show this.  One way to animate a graph is to make a table of "frames", which you can then double-click on and animate.  So, if we use one frame for every "tick" of the clock we can make some interesting vector fields (we let t go from 0 to 10, with a stepsize of 1 here):

While this is pretty cool, I'm not sure it is really all that useful.  (You can't, for example, use one of these graphs to trace out a trajectory.  You would have to trace it out in real-time while it is being animated...)

Notice, that we don't have to worry about this when graphing the parametric equations:  , graphing x vs. y in the plane (the "trajectory" in the XY plane).  It is just another parametric equation (though a bit ugly...)

If , then we have:

Notice how this has been "distorted" from the homogeneous case above.

So, what if you have a system of more than 2 dimensions?  Well, if it is three dimensional, you could use the 3-dimensional graphing capabilities of Mathematica to draw a vector field or trajectories.  (This may or may not be useful to you.)  Another possibility is to take the components 2 at a time and treat them like a 2-dimensional case (to tell you something at least, of how those 2 components interact).  I may put up some examples of this later if I have time.