Brian Edwards' fractals - finite attractors.

Brian Edwards' fractals site

Understand fractals. Inside Mandelbrot,Volterra-Lotka. Critical points, compound Mandelbrots. Convergence fractals to finite and strange attractors. Julia types, Newton etc.

I would welcome any comments, appreciation or criticism of the content of this site.
NEW. Users can now input their own functions so that the Julias and Mandelbrot can be drawn for new Z = f ( Z , c ) for any unconditional function f. See revisions.

There is basic information for the newcomer here as well as possibly new material for the more experienced. Is the work on Compound Mandelbrots new? It arose as a natural consequence of colouring the interior of a Mandelbrot by period. (How unusual is that?). The parameter maps of the Volterra -Lotka may be unusually good and where the parameters give rise to a strange attractor the convergence steps to it are shown with more accuracy than I have previously seen.

The main purpose of this site is to introduce and make available a free fractals drawing program called GENFRACT, which has drawn the site pictures.
Genfract differs from programs which were mainly designed for 'escape to infinity' fractals in that it has been specifically written to also hunt out finite attractors and show the convergence fractals. The program can use continuous potential and its ability to draw Compound Mandelbrots may be unique. The user is given full control of all aspects of the program so that it can be used as an investigative tool.
At present the scope is confined to Julia and Mandelbrot types for any user supplied unconditional function. A number of simpler functions and the traditional Inverse Julias and a few Newtons are built in.

	The program can show up to eleven finite attractors, each in their own colour and stepped to show the convergence. On the left is the Newton for Z^11 = 1 showing the eleven basins of attraction.

The program can also colour the inside of a Mandelbrot set. The colour shows the period of the attractor which lives in the various parts of the set, stepped according to the rate at which the critical point converges. To the right is the traditional set for Z = Z^2 + C .

	The Mandelbrot for Z = Z ^ -2 + C has no exterior. But there is an internal black sea of chaos in which swim myriads of little Mandelbrots whose colours reflect their periods. Without internal colouring there would be nothing to see at all ! The boundary shape with three cusps is an epicycloid.

This twin spiral is a detail from the traditional Mandelbrot. It was drawn using the 'Waves' and 'Smooth' colour settings. The fractals themselves are in fact smooth artefacts. The bands in which they are usually drawn are no more real than the contours which are drawn to represent a smooth hill.

	Are the fractals for exponential functions still regarded as difficult to draw? Here the function Z = C ^ ( Z / C ) has been entered as a user function. The program has made a very satisfactory job of drawing the Mandelbrot. Julias drawn from it are also fine.

You may click on most pictures on this site for a compressed 1024x768 version.
If you don't have the good old drawing program 'Fractint' , down-load it as frain###.zip from http://spanky.triumf.ca/pub/fractals/programs/ibmpc
Another newer good program is 'ChaosPro' available at http://www.chaospro.de
Both of these are much wider in scope and with many more features than I am offering and are free. They can't however distinguish between the basins of different attractors. Nor can they draw Compound Mandelbrots. Neither can they look for 'strange attractors', so they can't deal with Volterra - Lotkas properly.

Example outputs

A basic explanation of Julia and Mandelbrot fractals

Critical values and compound Mandelbrots

An explanation of Volterra- Lotka fractals

Description and examples of Volterra-Lotka fractals

A short description of what Genfract can do

Down-load Genfract (version 3.1, zip 291 kb)

e-mail :-brian.edwards@felicite-parmentier.freeserve.co.uk

Program update, 22 May 2005