Back to Basics: Return to Sanity?

So, all those renderings that I posted for the attractor fractals before were "wrong." They are something, and they might be interesting, but they are not renderings of the actual attractor set. After clearing up the math with Professor Taylor, I produced the following renders of the set that is created by this function f(x, y):

f(xn, yn)=(yn-1*xn-1*(1-xn-1), yn-1)

or

vect f(vect v)
{
vect result;
result.x=v.y*v.x*(1-v.x);
result.y=v.y;
return result;
}

Points are collected and drawn starting at a certain value (at which point we hope that the point is approaching the set), such as 995-1000 for the wide shots and up to 1,999,995-2,000,000 for the focus shots. The increment of y on this shot is (ranging from 3.0-4.0) is 0.0001.
Here is a zoom on the portion of the graph where y is between 5.575 and 5.5759. The increment was at 0.0001.
The detail is not high enough to make out the structures clearly, so I decreased the increment to 0.000001 to achieve the next few shots and set the n limit to 2,000,000. It took several minutes to calculate the points before they were rendered at this level of detail.

There is still instability in the system, even at this high level of iteration, which blows my mind. You can tell this due to the peppering of seemingly uniformly distributed points that clouds the structure of the image. If the system were converging faster, there would be more empty space like in the original image.