attractor fractals anyone?

(Update: got some decent detail on this one. n=10, f(x)=4x(1-x) read below for more info...)

Ok, so we were learning about a new sort of fractal in class today, and I used my computer to display some data for some of the functions that we were looking at.

These are all fractals of the sort where we have a function f and we calculate a point xn by plugging in a value x0 into the recursive definition xn+1=f(xn) n number of times.

The graph that you can see is made up of a series of xi values on the vertical axis (from x0-xn, for example x0-x20 or so), where the horizontal axis is the value of the starting input x0, ranging over the unit interval (0.0, 1.0). I made low-detail renderings by limiting n to 5, and I made high-detail renderings by letting n go up to 20. That means on the high-detail graphs, we have 20 plotted values for x at each x0.

The first one that was mentioned was f(x)=3/2*x for x between 0 and 1/2 and f(x)=-3/2*(x-1) for x between 1/2 and 1. Here is a low-detail rendering (n=5).
(****Note: for a larger view of these pictures, left click on them. My blogger displays shrunken "thumbs" instead of full resolution. On some of these renderings, getting the larger view is critical. -Ed)

Here is a high-detail rendering (n=20). See the other equations below for better results.


The second function that was mentioned was f(x)=2*x for x between 0 and 1/2 and f(x)=-2*(x-1) for x between 1/2 and 1. This is a low-detail rendering of the above equation (n=5).here is the most highly detailed rendering (n=20). I think this is a really awesome illustration of the infinite detail contained in such simple math.

This is another new shot. I decreased the line width and increased the increment in order to show more detail. I still am unclear about two aspects of these fractals: I don't understand exactly what an "orbit" is for one of these attractors and how it causes the detail. Also, and this is related to the first question, why is it called an attractor in the first place? if you know the answer to either of these questions, then by all means post a comment!
In the closup here, you can really see the layers upon layers of lines that seem to reflect off the "walls" of the graph over and over again.
The third one is f(x)=4x*(1-x). Here is a low detail shot of the fractal (by low detail, once again I mean that each x0-xn series is only 5 terms long, as in n=5)


This one is a slightly more detailed rendering (n=7), If you look closely at the two renderings, you can see exactly what the value n controls. If you look at any point on the horizontal axis, there are 6 curves that are crossing that spot (x0 is not drawn). If you refer to the previous shot, you can clearly see the 4 individual curves. As you increase n, you get more curves and thus more detail!

here we can see the most highly detailed rendering (n=20). I am still trying to figure out a way to zoom properly or display it so that the infinity is clear to the eye, but I think that this shot hints at the infinite detail of the fractal.

This final image is a rendering of an attractor-like equation that I discovered myself after playing around with the math for awhile.
This one is an n=10 rendering of the function f(x)= - (x-0.5)^3 for x less than 0.5 and f(x)=x^3 for x greater than 0.5.

here is a listing of the source code that renders these images. For this code to work, you need to have your opengl window set up already.

(note: code was removed, as blogger thought that less-than signs were html tags...
once i figure out how to post code, I will add the source here. -Ed)