Items tagged with fractal fractal Tagged Items Feed

Hi everyone,

at the suggestion of Carl I am making my question a post.

History:

Newbies often get fascinated with the power tower: x^x^x^...

The generalized power tower z^z^z^... is a special case of the Euler sequence:

z_{n+1}=c^z_n, for c\in C.

Like the Mandelbrot set: z_{n+1}=z^2-c, the power tower, often called infinite exponential, also has a general periodic map. It is included below and is taken from Daniel Geisler's site:
www.tetration.org

Shel and Thron proved that the infinite exponential conveges whenever c belongs to the red region, called today Shell-Thron region.

Definition of Julia Sets for the iterated exponential map:

Also like the Mandelbrot set, the infinite exponential admits Juila Sets. The Julia Sets of the infinite exponential however, are defined differently from the Julia Sets of the Mandelbrot set. They are defined to be  the closure of the periodic repellers of the Euler sequence . They are Cantor Bouquets.  Geisler's colored map then is a general map of how the corresponding Julia set behaves roughly, with c taken from the map. 

We can then introduce small cuts which go from the interior of the Shell-Thron region to the exterior, crossing at various angles, and this will tells us how the infinite exponential evolves. Generally speaking, each time one crosses the Shell-Thron boundary, one wittnesses what's called a Knaster explosion, wherein the exponential explodes into p subregions, where p is the pre-period of the multiplier.


When the parameter c exits the Shell-Thron region at angles of 2*Pi and Pi from the real axis (cuts right and left, p=1, 2), the infinite exponential either transitions from converging to a single feature to exploding into multiple indecomposable contiua (p=1), or it breaks into a period 2 bifurcation (p=2), which itself, also may explode into continua.

When it exits at angles 2*Pi/p, where p>2 is the preperiod of the multiplier, then the infinite exponential evolves from converging to a single feature, to exploding into  p major regions, called Fatou regions, each one having its own attractor, displaying a p-furcation.


In all cases,  the Knaster explosions may introduce the presence of indecomposable continua, as some Fatou regions end up covering entire parts of the complex plane after each transition. In the animations, Knaster explosions occur whenever the background is red. There may be more than one explosion present in the evolution of the power tower. 

Cantor Bouquets are strange creatures. They are essentially quantum sets, and no point of them is actually visible. The probability a point is visible varies directly with the area of the corresponding bouquet "finger" which is rendered in the area of interest. Devaney uses these "fingers"  to obtain "iteneraries" of the iterated exponential map.

Points "close"  to the fingers-hairs o the Cantor Bouquets escape towards complex infinity at (final) angles 2*Pi/p.

The "hairs" of a Cantor Bouquet are C^\infty curves, hence they can be termed easthetically pretty. Only hairs from the main Cantor bouquet for c=e^{1/e} are globally convex/concave. Every other bouquet may contain hairs which change curvature "unpredictably".

Inlcuded files:

1) Euler.mw

2) EulerAnimation.mw

3) EulerAnimation2_noaxes.mw

 

Code for static fractal with given parameters in 1). Parameters can be changed in the constant section (Try p=2,4 or Pi)

Code for morphing animations through cuts in the Shell-Thron region in 2). Parameters d1-d2, N

Code for zooming animations in 3). Parameters M, M1-M2,

Limitations:

Will show attractors of broken Fatou basins only up to pre-period p=5. No big deal though. If you want to increase the preperiod past p=5, you need to add the relavnt tests in procedure ppc(c), otherwise the new attractors are not calculated and the plot ends up red.

Colors are assigned a bit more dispersed, using ln(m). You can also use m in that place. It depends on which area you are in.

Basins of attraction and Fatou regions change appearance under different epsilons. If you want different shapes for the basins, you can try using the Manhattan metric in proc Jhf, instead of |z-lim|.

Included below are the main map of tetration by Daniel Geisler, a short excerpt of the Shell-Thron region which shows pre-periodic cuts and  6 morhing transitions, for p\in {1,2,3,4,5} and one which exits at the angle of 2 radians (p=Pi).

References:

More than 300. To be published with my thesis. Patches, problems and code corrections in the original question page:

http://www.mapleprimes.com/questions/203593-Maple-13-Fast-Maple-18-Crawling

Hi. I want to generate a Julia set, and teh first instruction in the demo, applications fractal is

GenerateComplex(a,b,c,d,e)

I have Maple13 and I can not find such instruction. Could you tell me where can I get it, or how to define it?

 

Thanks

It is a relatively recent innovation that complex-number computations can be done in the evalhf environment. When combined with plots:-densityplot, this makes escape-time fractals in the complex domain very easy to plot. This fractal is based on the Collatz problem. This Wikipedia article has a high-resolution picture of this fractal. I've switched the real and imaginary axes and reversed the direction of the real axis purely for asthetic reasons.

 

Collatz:= proc(b,a)  #Axes switched
local z:= -a+b*I, k;  #real part negated
     for k to 31 while abs(Im(z)) < 1 do
          z:= (1+4*z-(1+2*z)*cos(Pi*z))/4
     end do;
     k #escape time
end proc:

#Test evalhf'ability:

evalhf(Collatz(0,1));

32.

plotsetup(
     jpeg, plotoutput= "C:/Users/Carl/desktop/Collatz.jpg",
     plotoptions="height= 1024, width= 1024, quality= 95"
);

 

CodeTools:-Usage(
     plots:-densityplot(
          Collatz,
          -1..1, # imaginary range
          -0.5..4.5, #negative of real range
          colorstyle= HUE, grid= [1024, 1024], style= patchnogrid,
          labels= [Im,-Re], labelfont= [TIMES, BOLD, 14],
          axes= boxed,
          caption= cat("      Happy New Year ",                  

                StringTools:-FormatTime("%Y")),
          captionfont= [HELVETICA, BOLDOBLIQUE, 18]
     )
);

memory used=24.08MiB, alloc change=24.00MiB, cpu time=7.78s, real time=7.79s

 

Download Collatz_fractal.mw

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

restart;
a:= 2.75:  b:= 7.:  N:= 2^4:
X:= t-> add(sin(b^k*t)/a^k, k= 0..N):
Y:= t-> add(cos(b^k*t)/a^k, k= 0..N):
plot(
   [X, Y, 0..2*Pi], numpoints= 2^12
  ,scaling= constrained
  ,axes= none
  ,color= cyan
  ,caption= cat("Happy New Year ", StringTools:-FormatTime("%Y"))

Using techniques previously used for generating color images of logistic maps and complex argument, attached is a first draft of a new Mandelbrot set fractal image applet.

A key motive behind this is the need for a faster fractal generator than is currently available on the Application Center as the older Fractal Fun! and Mandelbrot Mania with Maple entries. Those older apps warn against being run with too high a resolution for the final image, as it would take too long. In fact, even at a modest size such as 800x800 the plain black and white images can take up to 40 seconds to generate on a fast Intel i7 machine when running those older applications.

The attached worksheet can produce the basic 800x800 black and white image in approximately 0.5 seconds on the same machine. I used 64bit Maple 15.01 on Windows 7 for the timings. The attached implementration uses the Maple Compiler to attain that speed, but should fall back to Maple's quick evalhf mode in the case that the Compiler is not properly configured or enabled.

The other main difference is that this new version is more interactive: using sliders and other Components. It also inlines the image directly (using a Label), instead of as a (slow and resource intensive) density plot.

Run the Code Edit region, to begin. Make sure your GUI window is shown large enough for you to see the sides of the GUI Table conveniently.

The update image appearing in the worksheet is stored in a file, the name of which is currently set to whatever the following evaluates to in your Maple,

cat(kernelopts('homedir'),"/mandelbrot.jpg"):

You can copy the current image file aside in your OS while experimenting with the applet, if you want to save it at any step. See the start of the Code Edit region, to change this filename setting.

Here's the attachment. Comments are welcome, as I'd like to make corrections before submitting to the Application Center. Some examples of images (reduced in size for inclusion here) created with the applet are below.

NewtonBlackArea.mw

I have been working with Newton-Raphson fractals for some time.  Like others it was necessary to deal with the "black areas" many times, so I performed some additional analysis and present some of these results here.  This will allow others to stop coloring these areas black and allow visualization of the structure inside these areas.  It will also help demonstrate...

I have the fractcal of a snowflake

I have a pbs: Suppose 3 vector A, B,C with noted [A:B]:=(list)[ A,B,-A,-B] (i.e a paralelogram);

An endomorphism T such that T(.) where (.)=A,B or C are B,C, and (list)[C,-A,-B,-B]
 and T([x:y])=[T([x]):T([y])] where x,y = A,B or C.

Problem: draw T^{n}[A:B] , i.e its boundary is a fractal figure when n is big.

Hi,

I am designing an antenna based on Peano-Gosper fractal. I have limited math skills and the most difficult part for me is deriving the generator for this fractal that takes two points (beginning and end) of initiator and generate 8 points PG curve consisting of 7 segments. Could you please suggest where I can find such generator written in Maple?

Thank you very much Vadim

How could I go about creating the fractal fern for which the iteration process is as described here...

http://en.wikipedia.org/wiki/Iterated_function_system

Under the section - Example: a fractal "fern"

 

Its mostly the probability part that would be the biggest problem for me.

While visiting a cathedral in Germany, Bob Schipke, a retired Harvard mathematician was astounded to find a glyph in a 13th century manuscript that looked remarkably like the Mandelbrot set.  This led to a remarkable voyage of discovery that was publicised in a

Page 1 of 1