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The following (downsized) images of Lyapunov fractals were each generated in a few seconds, in Maple 16.

 

I may make an interface for this with embedded components, or submit it in some form on the Application Center. But I thought that I'd share this version here first.

I'm just re-using the techniques in the code behind an earlier Post on Mandelbrot and Julia fractals. But I've only used one simple coloring scheme here, so far. I'll probably try the so-called burning ship escape-time fractal next.

 

 

 

 

Here below is the contents of the worksheet attached at the end of this Post.

 

 

The procedures are defined in the Startup code region of this worksheet.

 

It should run in Maple 15 and 16, but may not work in earlier versions since it relies on a properly functioning Threads:-Task.

 

The procedure `Lyapunov` can be called as

 

          Lyapunov(W, xa, xb, ya, yb, xresolution)

          Lyapunov(W, xa, xb, ya, yb, xresolution, numterms=N)

 

where those parameters are,

 

 - W, a Vector or list whose entries should be only 0 or 1

 - xa, the leftmost x-point (a float, usually greater than 2.0)

 - xb, the rightmost x-point (a float, usually less than or equal to 4.0)

 - ya, the lowest y-point (a float, usually greater than 2.0)

 - yb, the highest y-point (a float, usually less than or equal to 4.0)

 - xresolution, the width in pixels of the returned image (Array)

 - numterms=N, (optional) where positive integer N is the number of terms added for the approx. Lyapunov exponent

 

 

The speed of calculation depends on whether the Compiler  is functional and how many cores are detected. On a 4-core Intel i7 under Windows 7 the first example below had approximately the following performce in 64bit Maple 16.

 

 

Compiled

evalhf

serial (1 core)

20 seconds

240 seconds

parallel (4 cores)

5 seconds

60 seconds

 

 

 

with(ImageTools):


W:=[0,0,1,0,1]:
res1:=CodeTools:-Usage( Lyapunov(W, 2.01, 4.0, 2.01, 4.0, 500) ):

memory used=46.36MiB, alloc change=65.73MiB, cpu time=33.87s, real time=5.17s


View(res1);


W:=[1,1,1,1,1,1,0,0,0,0,0,0]:
res2:=CodeTools:-Usage( Lyapunov(W, 2.5, 3.4, 3.4, 4.0, 500) ):

memory used=30.94MiB, alloc change=0 bytes, cpu time=21.32s, real time=3.54s


View(res2);


W:=[1,0,1,0,1,1,0,1]:
res3:=CodeTools:-Usage( Lyapunov(W, 2.1, 3.7, 3.1, 4.0, 500) ):

memory used=26.18MiB, alloc change=15.09MiB, cpu time=18.44s, real time=2.95s


View(res3);


W:=[0,1]:
res4:=CodeTools:-Usage( Lyapunov(W, 2.01, 4.0, 2.01, 4.0, 500) ):

memory used=46.25MiB, alloc change=15.09MiB, cpu time=33.52s, real time=5.18s


View(res4);

 

 

Download lyapfractpost.mw

A set of three taped video interviews with famous physicist and mathematician Cornelius Lanczos (1893-1974) has been made available online by the University of Manchester.

This is a wish list for the version of Maple after whatever immediate next version is being developed.

It's about three-quarters of a year since Maple 16 came out, and if Maple follows its recent history of annual releases then development of new features of the very next release after 16 would be quite well along, and possibly even nearing completion. But that could mean that this is a good time to discuss wish lists for the version after that -- the version two releases after 16.

There are two interesting threads about definite integration on the usenet group comp.soft-sys.math.maple posted in the past few days.

The first thread is more about the wrong result coming out of Maple's int command when computing via its `MeijerG` method. Motivating examples there include

int(cos(t)/(1+t^k), t=-infinity..infinity);      # for k=10,12,14,...

The poster demonstrates using residues to obtain a terse result.

In the second thread the poster examines the ideas that the solutions involving RootOfs (as returned by the `ftoc` method) may suffer from numerical difficulties during floating-point evaluation and are unnecessarily complicated when re-expressed in terms of more standard functions (ie. allvalues, then simplification...).

For those interested, an existing `contour` method exists within `int`. It is not very strong however. One can examine its source using commands such as,

showstat(`int/definite/contour`);

showstat(`int/definite/contour/def1`);

etc.

See also Methods of contour integration on wikipedia.

Following on from the 3D plots of the Earth globes in comments to an earlier post, here's some hacky code to grab longtitude and latitude.

(nb. This code attempts to send the IP address of your primary DNS to www.geobytes.com so don't run it if you don't want that action.)

if kernelopts(platform)="windows" then
   res:=ssystem("ipconfig /all"): res:=res[2];
   StringTools:-Search("Primary Dns Suffix",res);
   ans:=StringTools:-Split(res[%+36..%+70])[1];
else
   ans:=ssystem("hostname -y"...
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