*626080007
*235090028
*327120084
*438090121
*138090179
*737080288
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*62608
You have a choice of
monochrome or color
graphics.  Monochrome
graphics show more
detail` but are fuzzy
and slow.*
To get the most out of
monochrome graphics on
a color monitor` turn
the color all the way
down or switch off the
color if your monitor
has such a switch.*
You may exit this
program by pressing
the Escape key or
Control-C.

At any time` you can
press the ? key to get
help and information.
*23509
You are selecting a portion of
the graph to enlarge. Using the
arrow keys or           /\
place the     Control-   K
box over             < H  U >
the area you            J
wish to enlarge.         \/*
Because drawing the graph may
take some time` the program
will beep when it finishes
plotting the graph.
You can toggle this sound at
any menu by hitting Control-S.*
To change the size of the frame`
press < or >.

To toggle the distance by which
the frame moves` press Space.

When the frame is sized and
positioned over the area you
wish to enlarge` press return.*
The new graph will be drawn over
the old` starting from the left`
so you will usually be able to
see how things are going by
comparing what is being drawn
with what is visible in the
frame.*
Because each new graph is
calculated from scratch in order
to bring out details` the new
graph may differ a bit from what
you expected.  Sometimes` if
you changed the parameters too
much` the new graph may even be
blank!*
This happens because the
accuracy of the graph depends
upon the value of the first
number you give it.  Changing
that number changes the shape
of the graph` and as you blow
up ever-smaller parts of the
graph` small changes in shape
become very large.*
If a blank graph is drawn you
will probably have to start over
because there is no way to go
back one screen.*
So just experiment and you'll
get the hang of it.
With patience` you can dive
deep` deep into that fractal
landscape.  Who knows what
you'll find?
*32712
You can either look at
time-series analyses of
this graph` or enlarge
a section of it for
more detailed study.*
What you are looking at
is a graph of the stable
population size reached
by the population at
different rates of
growth.*
This graph provides
a way of looking at
the overall behavior of
the equation.  Blowing
up smaller and smaller
sections of it lets you
see detail far too tiny
to be visible on the
initial graph.*
A time-series plots the
changing population size
from season to season.
So it will let you
watch the equilibrium
settle in over time.*
When viewing a time-
series` you can also
count how many seasons
it takes the graph to
repeat itself.  This is
called the period of the
equation at one growth
rate` and cannot be
observed directly after
the initial graph has
been blown up.
*43809
You are looking at a graph of the
stable population levels reached
at different rates of growth` with
the growth rates increasing from
left to right.  From here` you can
select a particular growth rate`
and examine seasonal growth at
that rate.*
Use the arrows or Control-H and
Control-U to move the cursor line.

To toggle the distance by which
the line moves` press Space.

When the line is positioned where
you want it` press Return.*
The graph will then be replaced
by a time-series analysis.  You
can see the population rise and
fall and settle at a stable--
equilibrium level.*
At higher growth rates` this
level will split` and the population
will alternate between two levels on
alternate years.*
At even higher growth rates` the
level will continue to split`
finally breaking up into random
noise.

But look closer . . .
Even here` order lurks in the
numbers if you can just see it.*
The number displayed with the
time-series represents the growth
rate.  Like all the numbers in
these graphs` its actual value
is not very meaningful` but this
number tells us a lot about the
equation.  For example` chaotic
behavior sets in at around 3.6`
called the point of accumulation.*
It is also possible` with a bit of
mathematical savvy and a lot of
patience` to use this number to
predict period doublings.  This
is important` since it allows
predictions to be made about
something that is otherwise
random in appearance.*
This is part of why Chaos is
important.  Imagine an engineer
working on aircraft` car engines
or artificial hearts.  Using
Chaos` a mathematician might be
able to tell the engineer where
turbulance will crop up and how
bad it will be.
*13809
Chaos Explorer needs two bits of
information from you.  The first
is the number of times to run the
equation before plotting anything.
The second tells the program
how many points to plot once it
has run it to that point.*
Chaos Explorer uses an equation
that models the behavior of a
population over time.  The
equation is given a growth rate
and a current population size
and computes the population size
for the next year or generation.*
If the growth rate is too low
then the population` like that of
the giant panda` will dwindle.
And as the population size gets
larger` it restricts itself just
like any real population where
food and other resources are
scarce.*
The result is a model that
behaves like several real
populations` and that does
some pretty odd things.*
Chaos Explorer shows you this
behavior in two different ways`
so that you can see how the
population does at a given rate
of growth and also see its
overall behavior.*
The graph of overall behavior
is what the two numbers are for.
The graph will be plotted with
the growth rate increasing from
left to right and population
size increasing from bottom to
top.*
In order to compress all the
behavior of the population at
many growth rates onto one graph`
this graph will plot the
population reached after many
generations (the equilibrium
point) for each rate of growth.*
This is where the two numbers
come in.  The first determines
how many generations to wait before
plotting.  The larger this number`
the more likely the populatin is to
settle down to its equilibrium.
Thus` the more accurate--but time-
consuming--the graph.*
As you blow up ever-smaller chunks
of the graph` this number must get
very large to keep the graph
honest` because you are enlarging
the whole equation` including the
parts that reach equilibrium very
slowly` thereby distorting the
graph.*
The second number determines how
many points to plot.  If there is
a stable population size` then only
one point will show up at that
equilibrium` and if the population
never settles down then many`
seemingly random` points
will appear.*
This number need not be huge--on
the first graph even a tiny number
like eight will give a crude idea
of the shape of things.*
As you blow up smaller sections of
the graph however` this number also
must rise because only the points
falling within the current section
can be plotted--the rest are simply
ignored.*
For decent results on the initial
graph` try values of about 20 for
both parameters.  On later graphs`
you will be asked for new numbers.
This does not change the equation`
only the number of generations to
wait before plotting and the number
of generations to plot.*
The reason Chaos Explorer asks for
these numbers each time it blows up
the graph is that you may want to
use very much larger numbers when
blowing up tiny sections of the
graph` but you just don't need such
large numbers when viewing the
whole thing.*
And you wouldn't want to wait for
the initial graph to plot waiting
a thousand or more generations.
That would produce accuracy far
beyond that needed or useful.*
These numbers could be preset` but
that would take all the fun out of
things.  How is the computer to
know whether you want a  S-L-O-W`
high-quality graph or a quick` crude
one unless you tell it?  So just
experiment and use what you like.
After all YOU'RE the explorer here!
*73708
You can calculate a new graph or
load one that has been previously
saved to disk.  Saving a graph to
disk allows you to move its image
to another program or to load it
again later.*
When you save a file to disk`
two files are created with the
name you supply.  One file` with
'.Dat' added to the name is a
data file.  The other file` with
'.Pic' added to the name` is a
memory dump containing the image
from the screen.*
This file can be loaded into
other programs like any other
graphics file.  A graphics utility
may therefore be used to make a
printout of the graph.*
The other file contains
data used by Chaos Explorer.
Basically` it tells the program
exactly what part of the graph is
stored in the image file of the
same name.*
So when you load a file` you pick
up exactly where you were when you
saved that file.  This lets you
stop your exploration when it's
time to wash the dog and continue
another day.*
When you save a graph` the two
files will be created on the
current directory so there must
be at least ten bytes free on that
directory.  To save the graph on
a different directory` just use
the complete pathname.*
For example` if you enter
/SomeDir/MyGraph as a filename`
the correctly displayed graph
will be saved in the files
MyGraph.Pic and MyGraph.Dat on
the directory SomeDir.
~
