## Sunday, August 04, 2013

### Programs from the past, Part II

I could have sworn I wrote a bit more about a program I wrote while at FAU;
specifically, just how long it took to run back then vs now (or rather, a
few years ago), but having failed to find any using Google or locally searching the entries on
the server, I ended up spending all day yesterday *reading* through
ten years of archives and *still* not finding anything, I guess I
haven't.

Well. [Deep subject. —Editor] [Shut up you! —Sean]

So I wrote a series of programs to search through the domain of the following pair of equations:

x_{i+1} = (Ay_{i} + B) x_{i} (1 - x_{i})

y_{i+1} = (Cx_{i} + D) y_{i} (1 - y_{i})

The intent was to generate a large number of `(x y)`

pairs and
plot the results for a given set of values for `A`

,
`B`

, `C`

and `D`

. Something like:

The top window shows the resulting chaotic attractor when
`A`

is 2.4375, `B`

is 1.5624, `C`

is
-0.8659 and `D`

is 4.0 as you loop through the above two
equations 15,000 times to generate 15,000 points (all between 0 and 1 for
both the `x`

and `y`

axis). And the thing about a
chaotic attractor is, the initial starting point (`x`

and _{0}`y`

) is largely immaterial (but not quite, as the
starting values must be between somewhere between 0 and 1). Any given value
for _{0}`x`

and _{0}`y`

will result in
the same figure being drawn for the given values of _{0}`A`

,
`B`

, `C`

and `D`

.

The program shown allows you to change the settings of the four
controlling variables `A`

, `B`

, `C`

and
`D`

by sliding the named boxes horizontally (the botton one
allows you to slide multiple variables at the same time—in this case,
`A`

and `C`

will slide to the left if you slide the
bottom box to the left).

That was fine, but my bosses at the time wanted another view generated—a “map” of the chaotic attractors, if you will. To generate this “map,” first you calculate how many points are generated before you settle into an attractor:

int mainloop(const double A,const double B,const double C,const double D) { double xn,xn1; double yn,yn1; int ix; int iy; int count; bool pix[DIM][DIM]; /* have we seen this (xy) pair yet? */ memset(pix,0,sizeof(pix)); for (count = 0 , xn = yn = 0.5 ; count < MAXLOOP ; count++) { xn1 = ((A * yn) + B) * xn * (1.0 - xn); yn1 = ((C * xn) + D) * yn * (1.0 - yn); xn = xn1; yn = yn1; /* exit if we're outside the bounds */ if (xn < 0.0) return MAXLOOP; if (xn >= 1.0) return MAXLOOP; if (yn < 0.0) return MAXLOOP; if (yn >= 1.0) return MAXLOOP; ix = (int)(xn * DIM); iy = (int)(yn * DIM); if (pix[ix][iy]) break; pix[ix][iy] = true; } return(count); }

The assumption—if this returns `MAXLOOP`

, there is no
attractor for the given set of values for `A`

, `B`

,
`C`

and `D`

; othersise it's an indication of how
quickly (or how “showey”) we settle into the pattern.

Then it's a matter of going through a range of values and for the map images, since they're two dimensional, I only sweep through two of the four governing variables over their range:

for (A = -4.0 ; A <= 4.0 ; A += 0.016) for (B = -4.0 ; B <= 4.0 ; B += 0.016) plot(A,B,count_as_color(mainloop(A,B,-0.8659,4.0));

(`count_as_color()`

just maps a count to a color)

And we end up with an image something like:

In this image, `A`

is the horizontal axis, going left to right
from -4.0 to 4.0, and `B`

is the veritcal axis, going, bottom to
top, from -4.0 to 4.0; `C`

and `D`

are fixed, at
-0.8659 and 4.0 respectively. The red plus in the upper right hand quadrant
is the location of the “French horn” attractor above. The black area
represents areas that have no chaotic attractor.

But my bosses didn't just want *one* map—no, they wanted a
larger set. So, I generated a series of images as this, each image having a
different value for `C`

. And while the map may not change
drastically—for instance, if we bump C up a notch to -0.8499:

The resulting chaotic attractor is completely different:

And I generated a ton of maps by looping through values of `C`

from -4.0 to 4.0—500 images in total.

Now, when I did this the first time, twenty years (or a bit more—maybe
up to twenty-two years ago) I was doing this on a state of the art Unix
workstation, an SGI workstation
(perhaps you've seen their commerical?). It cost
$30,000 and it took a *year* to generate the 500 images. Each
individual image took some *ten hours* to generate.

Fast forward to today. On a laptop that is maybe two years old now. I reran the code and was able to generate an entire image (in fact, both of the map images above) in only 8 seconds.

With four cores, that means, on a two year old laptop that might have
cost $1,500 (tops) would take *17 minutes* what it took me a
*year* to generate twenty years ago.

Oh, and about those eight seconds …

#### Update on August 5^{th}, 2013

Oops, I made a slight mistake …