The explanation of d=atan(d) is pretty simple... first of all you need to know how movements work:
The formulas you enter specify the *source* pixel in function of the *target*. When AVS applies the movement, it looks up the source pixels for every target pixel and copies the colour from source to target.
So when you specify d=d*2, you're saying that pixels halfway to the edge (d=0.5) should get the colour that is at the edge (d=0.5*2=1). This means that the image will be zoomed *out*: the area (-1...1) gets squised into (-0.5...0.5).
Now if this makes sense, you'll understand why atan(d) makes a tunnel movement. Open this in a new window:
The red line is the plot of f(x)=x, a 45° angled line. This is 'no movement', every point gets projected onto itself.
The blue curve is an approximation of the plot of f(x)=atan(x). I exaggerated the curving to make it easier to illustrate.
We take a point at a certain distance (the leftmost fat ellipse on the bottom axis) and we're going to apply the movement d=atan(d). We trace its location on the graph upwards (gray line). We need to draw a horizontal line and intersect with the blue curve. This is where this point would end up after one step of movement. Again, this is not how AVS works, but this is how it works intuitively.
We trace down again and find the target distance as the second point from the left on the axis. You can see the point moved ******ds.
Now we keep applying this: trace up, trace horizontal, intersect with blue, trace down to find out where it will end after several frames (and repeated movement). You can see that because the distance between the red line and the blue line increases, the point will move ******ds faster and faster.
What we're doing is actually a kind of source map now. AVS looks at the graph 90° rotated. You can understand that, if the blue curve doesn't cover the entire vertical range, that some points on the red line might not have a target horizontally of them. This is why a source-map can leave holes in the image.
Now what this example doesn't show is that there can be a huge difference is resulting movement from two slightly different curves, because the shape of the movement is the result of repeatedly following the points, rather than just once.