home
***
CD-ROM
|
disk
|
FTP
|
other
***
search
/
Club Amiga de Montreal - CAM
/
CAM_CD_1.iso
/
files
/
100.lha
/
CHAOS-KIT.DOC
< prev
next >
Wrap
Text File
|
1986-11-21
|
7KB
|
159 lines
CHAOS-KIT
I. Introduction
Here is a set of four systems illustrating many of the remarkable
features of Non-Linear Dynamics (NLD). The CHAOS-KIT program was written
in AmigaBasic and compiled with Absoft AC/Basic. Use it as you wish but
please give due credit to the author. I invite comments, suggestions,
etc.; preferably by Easyplex; I also regularly attend the Science Forum
on CIS (GO SCIENCE).
It will take time for you to get to know these systems; in fact
you can spend endless hours exploring them. That is what this program
is for. I hope you enjoy these beautiful images as much as I have!
Dan Davis [CIS 71420,2332]
New York, January 1988
II. Using the Program
CHAOS-KIT must be run from the CLI by typing its name at the
prompt. The first screen presents a listing of the four systems and
prompts you to select one of them.
Once you have chosen a system, a screen appears giving the formulas
which drive the system. NONMATHEMATICIANS: DON'T PANIC! I give the math-
ematical details for those who can make use of them; they add about as
much to the enjoyment of this program as the ability to read music adds
to the enjoyment of a symphony.
All four systems work by iterating the equations; that is, by
solving them repeatedly, using the solutions at each iteration as the
starting point for the next iteration. You may choose the number of
iterations; the more iterations, the more points will be plotted and
the longer the program will run.
Each system is controlled by parameters in the equations. The
first three systems have one parameter each and the last has three.
You may choose values for these parameters. This is an important means
of controlling the output of the system.
In the first three systems you may also choose the starting point.
More about this below.
Default values have been provided throughout. As a beginner you
will probably go with the default values, but as you gain familiarity
you'll want to choose your own values. That is the way to use the prog-
ram. Explore to your heart's content, the possibilities are endless.
To choose your own values at any option, type in the number you
want and press the RETURN key. To choose the default value just hit
RETURN.
Once the values are chosen, the program begins creating the
image. You will see the two coordinate axes with tick-marks which
mark the value Scalefactor. To interrupt the program, press the space
bar; you will return to the parameter-choice screen. If you decide
to explore one of the other systems, press `m'; this will get you back
to the menu screen where you began. If you've had enough, press `q';
this quits the program.
A window will appear on screen giving the values for the options.
You can move this window around using its drag bar. If you don't want
to see it, click its `back' gadget (upper right-hand corner). If you
did this and decide you want it back, press `t'.
WARNING: I have put some anti-crash measures in the program
but it's not 100% crash-proof. For some choices of the parameters an
overflow is possible which will shut the program down. Click the mouse
as directed and start over. So far that's the worst that's happened to
me. If you meet the guru, try increasing the stack size (at the CLI
prompt type STACK 16000). If the program gives trouble let me know.
III. Notes on the Systems and NLD.
Non-linear systems have been around for a long time, but until
recently they were avoided by theoreticians since they do not in general
admit closed-form solutions. Only after the development of the computer,
with its ability to produce numerical solutions on a large scale, was it
practical to study non-linear systems in a general way. Over the last
fifteen years a vast amount has been learned. Many natural phenomena
are governed by NLD and new tools are now available for studying them.
A good layman's introduction to the subject is James Gleick's book
`Chaos', which is especially strong on the historical and human aspects
of the subject.
An important feature of NLD is the use of experimental methods,
which are not favored in classical mathematics. The experiments take
the form of computer simulations. With this program you can carry out
experiments of this type.
The first two systems were lifted from the book `Chaos', edited
by A.V. Holden. Both are from the field of population biology. The
third system was described by A.K. Dewdney in his column in `Scientific
American' for August 1987. The Lorenz Attractor is one of the most
famous objects in NLD and appears in almost every general reference.
The first three systems operate this way: the x0 and y0 chosen
by the `starting point' option are iterated 100 times (or however many
you select with the `iterations' option). All points have the same
color. Then the x0 and y0 are incremented by .01 and another set of
100 iterations (or whatever) are run from this new starting point, with
a new color. This is done thirty times in all.
1. The delayed logistic system.
Investigate this system by varying the parameter `a'. At a=2
the attracting point opens out into a limit cycle. This is an example
of a `Hopf bifurcation'. Move the starting point around for some nice
effects. Be careful, you'll get an overflow if `a' gets much above 2.27.
2. The predator-prey system.
The default value of `a' produces a network-like figure. This
is evidence for a `strange attractor' in the equations on which this
system is based. Play with the parameters.
3. The Henon system
Don't be misled by the the dull behavior at the default. When you
set theta away from zero, things start to happen. Multiples of 15 degrees
are interesting. Once you find a nice value of theta, play around with
the starting point.
4. The Lorenz Attractor
This was discovered twenty-five years ago by Ed Lorenz, professor
of theoretical meteorology at MIT. It was the first `strange attractor'
to be studied in depth. The program displays the motion of a point in
three-dimensional space projected on one of the coordinate planes which
you may choose. The point is moving on a mathematical object called a
strange attractor which is neither two-dimensional nor three-dimensional.
It is a `fractal object' with fractal dimension slightly greater than 2.
The point moves forever without crossing its path (although the two-dimen-
sional projection does cross itself) and without ever closing the path.
Play with the parameters and see if you can visualize this truly strange
object.
I hope you enjoy this program. I'd be delighted to exchange
ideas and information with anyone.
--Dan