Can I make a force field that is defined with differential equations? Can I make a force field in the xy plane that is defined with differential equations as:

x' = f_1(x,y) (speed in the x-direction is dependent on x,y coordinates)

y' = f_2(x,y) (speed in the y-direction is dependent on x,y coordinates)

So for example we can see a rigid body (such as wind) or liquid flow (such as a whirlpool) as a result of the force field.

Example is a phasediagram:

That is described with the following differential equations:

x' = x(2-x-y)

y' = y(-1+x)



http://systems-sciences.uni-graz.at/etextbook/sw2/ph_plane_analysis.html
As the documentation says,


  You can use a Texture force field to create an arbitrarily complicated force field, which force in the three directions is color coded. Red is coding for the X-axis, green for the Y-axis and blue for the Z-axis (like the color of the coordinate axes in the 3D View). A value of 0.5 means no force, a value larger than 0.5 acceleration in negative axis direction (like -Z), a value smaller than 0.5 acceleration in positive axis direction (like +Z).


You can create the texture corresponding to your equations using Mathematica or similar packages. Or you could write your own program to output the texture.