V1.5                  - Lyapunovia Users' Guide -                    Page 1




                                Welcome to



                             -----------------
                            | LYAPUNOVIA V1.5 |
                             -----------------



                       Copyright 1992/93 Jesper Juul






                                 Contents:
                                 ---------


         Introduction to Lyapunovia.............................2

         Installation...........................................4

         Getting started........................................5

         Program functions......................................6

         Tips & tricks..........................................9

         In-depth explanations.................................10

         Contacting the author, The serious part...............12

         Notes.................................................13

         Footnotes.............................................15




                      "Why do I have to keep reading
                         these technical manuals?"

                               -Roger Waters














V1.5                  - Lyapunovia Users' Guide -                    Page 2


                        Introduction to Lyapunovia
                        ==========================


-Tired of zooming endlessly around the Mandelbrot set?


-Irritated  by  little  graphic  thingys  daftly  named  "Sea of solitude",
"Dragons mouth", "Mountain of magic", or even worse?


-Bored by a 1000 dull fractal  programs  all  the  same,  except  for small
twists concerning the "cache modes" of the "68030"?


Well, this is no cure. This is Lyapunovia V1.5.
              --               ---------------


But what then, is Lyapunovia?
-----------------------------

-To be brief, Lyapunovia makes pictures. 

-To be  more elaborate,  Lyapunovia IS a fractal program, and it does allow
you to zoom.... But it is  NOT a  Mandelbrot program.  (Everybody should be
screaming  with  relief  at  this  point.)  The  great  thing  here  is the
variations of the images; from cute candy-like patterns to ragged  and torn
metal.   Lyapunovia   pictures   contain  depth  and  strange  interacting,
ever-changing shapes with NO names.

-Lyapunovia is Shareware.


And what is Lyapunovia V1.5?
----------------------------

When I originally  released  Lyapunovia,  I  used  the  traditional "stingy
programmer" shareware  concept: I  released a  _good enough_ program to the
public, but kept all the fancy  functions such  as AGA  and FPU  support to
myself and the registered users. This worked out just fine.
But after a while, a feeling of sillyness came to me: Why spend hours, days
and weeks in social isolation for the benefit of so few people?
I thought for a long time and decided  to release  _the full_  program this
time. This  makes _me_  feel good  anyway, and I hope that _you_, the user,
will appreciate it.

From the original V1.0, a lot  of things  has happened;  the user interface
has  improved,  many  functions  have  been speeded up, support for various
chips and 24-bit output has been added. Most of the  changes were suggested
by various users.
I've tried  to make  a program that takes advantage of AGA graphics and the
68040 internal FPU on a 4000, but still runs on an old '500.


(The mathematically disinterested  should  skip  the  following explanatory
part and go directly to "Installation" and "Getting started".)





V1.5                  - Lyapunovia Users' Guide -                    Page 3


The mathematical way
--------------------
Where Mandelbrot  graphics (the  ones you've  seen a  100 times before) are
renditions of the "Mandelbrot  Set",  Lyapunovia  renders  "Lyapunov Space"
(unsurprisingly   named   after   the  russian  matematician  Aleksandr  M.
Lyapunov).  If  the  Mandelbrot  set  is   the  "most   complex  object  in
mathematics",  Lyapunov  Space  must  be  the  juiciest,  spiciest and most
outrageous object ever found within numbers.

The specific formulas  used  to  produce  these  breathtaking  picures were
thought up  by  Mario Markus of the Max Planck institute for Nutrition. And
it all reached my mind by means of the "Mathematical Recreations" column in
the September, 1991 issue of "Scientific American".

If you've  ever spent  some time with the Mandelbrot set, zooming, changing
colors, etc... You'll be  well off  to understanding  how Lyapunovia works.
Picture  a  square  with  coordinates,  going horisontally from 2.0 to 4.0,
vertically from 2.0 to 4.0:

              (X1)       (X2)
               2.0        4.0
               --------------
      (Y1) 2.0 |            |
               |    The     |
               |  Picture   |
      (Y2) 4.0 |            |
               --------------

That's what you see  when running  the program.  Lyapunovia enables  you to
zoom in,  to watch  in closer detail whatever part of the screen appeals to
you. This is like the Mandelbrot set.

What makes Lyapunov Space perhaps so much more fascinating, is  the ability
to switch  between an  endless amount of different domains, each possessing
an individual "personality" of sorts. And  things are  very strange, jagged
and torn. Very appealing to any decently deranged imagination.

Like  the  Mandelbrot  set,  Lyapunov  Space  is  a  map of chaos, meaning:
Lyapunovia calculates the "Lyapunov exponent" of each pixel;  an indication
of whether  the formula  is order  or chaos at the given X and Y-positions.
Chaos is mapped as black, order is mapped with the  highest colors  for the
orderliest function.  The basic  formula is  equivalent to the one used for
making "Feigenbaum trees"; x=rx(1-x).  The Lyapunov  exponent is calculated
like this:


;A SMALL PROGRAM FOR DETERMINING THE LYAPUNOV EXPONENT

X=0.5     ;JUST AN INITIAL VALUE
TOTAL=0
ITERATIONS=50
R=3.5     ;OR SOMETING ELSE BETWEEN 2 AND 4.

FOR I=1 TO ITERATIONS
 X=RX(1-X)
 TOTAL=TOTAL+LOG(ABS(R-2RX))/LOG(2)
NEXT I

TOTAL=TOTAL/ITERATIONS



V1.5                  - Lyapunovia Users' Guide -                    Page 4


The  TOTAL  variable  now  holds  the  Lyapunov  Exponent  for  the formula
X=RX(1-X).
Now you should be asking yourself:  How  does  one  plot  a two-dimensional
picture when  the formula only has one parameter? Right: What we do is, for
each iteration, to  replace  R  with  either  the  X  coordinate  or  the Y
coordinate  of  the  pixel  we're  going  to  plot.  The sequence parameter
determines how X & Y should alternate: An "AB" sequence does X,  then Y, X,
then Y and so on... "AAB" does X,X,Y,X,X,Y... and so on and so forth.

Naturally,  there  are  various  ways  to  rewrite  the formula to speed up
calculations as I've done,  but this  should give  you an  idea of  what is
going on.

Reading on Lyapunov Space:
--------------------------
Scientific American,  September 1991.  A.K. Dewdney: "Leaping into Lyapunov
Space".

Generally on fractals & chaos:
------------------------------
Brian H. Kaye: A Random Walk Through Fractal Dimension. VCH, 1989.
H.O. Peitgen, P.H. Richter: The Beauty of Fractals. Springer, 1986.


--------------------------------------------------------------------------


                           Installing Lyapunovia
                           =====================


Lyapunovia  uses  Nico  Francois'  ReqTools  library  for  file-requesters,
information boxes, palette requesters and the like, therefore:

Before running,  REQTOOLS.LIBRARY (version 2.0 or above) must be present in
your LIBS: directory. In the "Install" drawer of  this distribution, you'll
find instructions as how do this.

Also  make  sure  to  have  the mathtrans.library in the LIBS: directory as
well. (It comes with the computer, so don't worry.)


Program versions
----------------
Three different program versions are included in this distribution:
"Lyapunovia V1.5 68000"  - for unaccelerated A500/A1000/A2000.
"Lyapunovia V1.5 68020+" - for accelerated Amigas, A1200,3000,4000.
"Lyapunovia V1.5 FPU"    - for Amigas with a floating point unit.
All three versions will run under 1.3-3.0, and will automatically adjust to
your computer.


Libraries, screens and memory allocation
----------------------------------------
Whenever the  program is unable to open a library or a screen, or unable to
allocate some required memory, it will instantly exit  and clean everything
up. An alert (the flashing thing) will then inform you of the error.
If using  WB3.0+ on  a non-AGA  machine, trying to load a 256-color picture
will result in a "unable to open screen"-error.
Using 1.2-2.1, the picture just won't  be loaded,  instead calculations are
started based on the coordinates.

V1.5                  - Lyapunovia Users' Guide -                    Page 5


                              Getting started
                              ===============

To quickly get a feel of what Lyapunovia does, do the following:
Install  Lyapunovia.  Double-click  on  the  icon.  The  title-screen  will
appear, press "OK" to continue.

(In the following, letters within {}  indicate  that  the  function  can be
activated by press "right amiga" and that key.)

The program will start making a very simple Lyapunovia picture. Note that I
generally programmed Lyapunovia to  automatically  begin  calculating every
time a  parameter has been changed. You can always stop the program via the
menus or by pressing right amiga+"." (the latter being much faster).


Do the following:
-----------------

Stop the program. {.} Try choosing a new screenmode:
-On 1.2/1.3, select # of colors etc.. from the menus.
-On 2.0-3.0, press {M} to get a screenmode requester.

Let the picture finish.

That was easy enough, but you've  probably noticed  that the  picture isn't
very interesting.  So lets try loading a coordinate file: A coordinate file
is simply a little  file describing  a certain  "place" in  Lyapunov Space.
When  you   load  a   coordinate  file,   Lyapunovia  automatically  starts
calculating it in the current screenmode, with the current palette.
Load a coordinate file, say "Cross" or "ABBBBA  24 -  3". {D}  (You'll find
the files in the "Coordinates" drawer of this distribution.)

While the  picture is  being calculated,  you can experiment with different
palettes; with menus or by pressing  right-amiga and  1,2,3, or  4. -Or try
bringing up the palette requester. {P}

Once the  picture's done, try loading some palette files. In the "Palettes"
drawer, you'll find a lot of different palettes:
-On a 256-color AGA screen, try "256 - Shiny" or "256 R&G".
-On a 32-color screen, try "32 - lift up my nights".

Try selecting a cycle speed of 1 {Z}, then try the "cycle" function. {Y}

If you like the picture, try  saving it,  using "Save  IFF". {S}  From then
on, you  can print  it, edit  it or  whatever you want to do. At some later
time, you can also load the  picture again,  change the  colors or  zoom in
somewhere that looks interesting - this is the next point:

Zooming is  very simple: You can select the area you want to see in further
detail either from corner to corner {R} or from center  to corner  {N}. Try
it; a requester will appear, asking if you're sure. If you were, the screen
is cleared and the image you've selected starts being drawn.


To more fully appreciate what Lyapunovia has to offer, this is how  to make
some of my favorite pictures: (Load the palette, then the coordinate file.)





V1.5                  - Lyapunovia Users' Guide -                    Page 6


Coordinate file               Palette (AGA / non-AGA)
------------------------------------------------------
Flash; Curve II               256 Cycle - II / 32 Cycle - III
5A5B 50, ibd 11               256 - Shiny / #1
5A5B 50, 'In-between, detail' #4 / #4
5A5B 50, 'In-between'         #4 / #4
5A5B 50, ibd something        256 - R&G / 32 - Light up my nights
6A6B 30, Agurk                #4 / #4
ABABA 50, Radiant             256 - R&G / 32 Cycle, Bilal II
ABABA 50, Spider              256 - Lift my days / 32 - Lift my Days
5A5B 50, ibd 5                256 - Shiny / #4
5A5B 50, The Big Parable      256 - Yellowie Eyes / 32 - Greenscale
AB 100, 'Indian Eyes"         #4 / #4
6A6B 30, 'CrissCross'         256 Cycle - III / 32 Cycle - III
5A5B 50, 'Loneliest Object'   #4 / #4
Flash; 6AAB, 'Cross'          256 Cycle - Seventies  / 32 Cycle - II

(#1 means standard palette 1; right amiga + 1.)

If you  like a  picture, you  san always save it {S}, and then reload it at
some later time for deeper zooming.

If you want to produce a 24-bit file, just load  the coordinate  file, stop
{.}, press  {F}. Select  dimensions, select  a palette file, then an output
file name.
To make sure that the color transitions are absolutely smooth, you may want
to expand  a palette  file first  {X}: Select the palette to expand, select
the # of colors to expand to (just say 4096), then the file name of the new
palette. To avoid confusion, it can be a good idea to name the palettes the
way I've done: Thus expand "256 - Shiny" to "4096 - Shiny".


A general rule is  that you  should zoom  on the  border between  black and
color. That's where the "action" usually is.


--------------------------------------------------------------------------


                             Program functions
                             =================

     (NOTE: letters  within {}  indicate that the function can be activated
     by press "right amiga" and that key.)

     (In order  to  gain  speed,  Lyapunovia  will  only  check  for menus,
     pressed keys  etc. every now and then. This is unnoticable on pictures
     with a low "iteration" value; with large values (larger than, say, 500
     on  an  unaccelerated  Amiga),  there  will be a noticeable delay from
     selecting a menu till the program reacts. Don't worry, your Amiga will
     remember what you did!)




Menu 1: PROGRAM
---------------------------------
This concerns basic program and file options.

Stop:             Stops drawing the picture. {.}


V1.5                  - Lyapunovia Users' Guide -                    Page 7


Go:               Though   changing   almost  any  parameter  automatically
                  restarts  calculations,  you  sometimes  need  a separate
                  function to do this. This is it. {G}

Save IFF:         Saves  the  current  picture  as a standard IFF file. {S}
                  (Note 2.)

Load IFF:         Loads  a  picture   saved   from   Lyapunovia,  including
                  coordinates etc... This enables you to continue exploring
                  where you left off... {L}

Save Prefs:       Saves     the     current     screen      settings     as
                  "S:Lyapunovia.prefs".  This  file is automatically loaded
                  when the program is started.

Load Prefs:       Loads the preference settings.

About:            A bit of information about Lyapunovia. {A}

Priority:         Sets the task priority of Lyapunovia. Generally speaking,
                  "1" gives  Lyapunovia priority  over most other programs,
                  "0" splits the CPU time, and  "-1" makes  the program run
                  quietly in the background.

Quit:             Quits Lyapunovia. {Q}


Menu 2: SCREEN
--------------
This concerns screen resolution and colors.

Palette:

 Edit:            Edit palette. {P}

 Palette 1-4:     Four different standard palettes. {1,2,3,4}

 Cycle:           Performs a standard color cycle on the current picture. I
                  realise  that  this  is  probably  the  oldest  and least
                  original  effect  on  the  Amiga,  but  boy! it does look
                  great. Anyway, press mouse button or any key to stop. {Y}

 Cycle speed:     Some people asked for  a faster  cycle, some  asked for a
                  slower. This  function allows you to select the delay (in
                  screen frames) between each step of a cycle. {Z}

 Save:            Saves the current palette in a standard format. {V}

 Load:            Loads a new palette. Note that it is  possible to  load a
                  palette from any standard IFF file.
                  Loading a  palette with a different number of colors than
                  the the current will delete the  current image.  A prompt
                  appears, asking you to confirm. {O}

 Rotate:          When you  perform a  cycle, sometimes  you'll notice some
                  unexpectedly good-looking color-combination that you want
                  to  keep.  This  function  moves through a standard cycle
                  step by  step,  allowing  you  to  save  the good-looking
                  colors when they appear.

Title bar:        Toggles the title bar. {T}

V1.5                  - Lyapunovia Users' Guide -                    Page 8




Screenmode:       Users with  WB2.0+ can  change screen resolution and # of
                  colors using a  single  menu.  This  replaces  the below-
                  mentioned  menus.  Lyapunovia  supports  ALL  resolutions
                  supported by the system software. If you seem  to be able
                  to a  select a mode that causes strange flickering or the
                  like, please note that this is due to a faulty preference
                  setting, not a faulty program!
                  As  standard,   the  program   assumes  the  screen  size
                  specified  as  "Text  overscan"  in  the  preferences. To
                  produce  pictures  bigger  than  the  screen,  enter  the
                  desired dimensions in the  "width" and  "height" gadgets.
                  If  the  screen  requires  more  chip ram than avaliable,
                  you'll be returned  to  the  previous  resolution.  On an
                  oversize  picture,  the  screen will automatically scroll
                  when you move your pointer around. (Note 1.)

Hires:            Turns  high  resolution  on/off.  (640  pixels  per line,
                  rather than 320.) * Under WB1.3 only!

Interlace:        Turns interlace  mode on/off. (PAL: 512 lines rather than
                  256. NTSC: 400 lines  rather  than  200.)  *  Under WB1.3
                  only!

2-32 Colors:      Selects the  number of colours on screen. (2, 4, 8, 16 or
                  32.) (Note 1.) * Under WB1.3 only!


Menu 3: IMAGE
-------------
For moving around in Lyapunov Space.

Coordinates:      Alters the specifications of the current  image. See "In-
                  depth explanations" for elaboration.

Zoom in:

 Center/corner:   To  zoom  with  the  mouse.  Point  at  the center of the
                  "place" you want to view, hold left mouse button and move
                  mouse  to  the  corner  of  the  square.  Release button.
                  Lyapunovia will ask you if you're sure. {N}

 Corner/corner:   Same as above, except  that  you  select  from  corner to
                  corner. {R}

Zoom out:

 All the way:     Returns you to the initial coordinates.

 2 times:         Zooms out two times. 

 4 times:         Zooms out four times.

Save:             This  saves  the  current  image  setting;  the  settings
                  specified under  the  "Coordinates  menu":  What sequence
                  you're using, and where you're zooming, etc.. {E}

Load:             This loads  such a  file. You  can also  load them from a
                  Lyapunovia IFF file. {D}


V1.5                  - Lyapunovia Users' Guide -                    Page 9


Precision:

 Half:            Half precision is a special "preview" mode to quickly see
                  what a  picture will  look like.  Faster, especially on a
                  68000 system. This mode has to be selected  manually. Not
                  very useful, really.

 Single:          Single precision is the default precision. Precise enough
                  for most purposes. Switching back to this mode  has to be
                  done manually.

 Double:          Double  precision  is  for extra deep zooming. Lyapunovia
                  will  automatically  switch  to  this  mode  when  it  is
                  required.

24-bit output:
                  This is  what a  lot of people have been asking for for a
                  long time. It's fairly simple; to  produce a  picture you
                  just enter  the dimensions  (in pixels), select a palette
                  and an output file name, and away you go. The  picture is
                  always  drawn  based  on the current coordinates. Maximum
                  dimensions are  65000 x  65000 pixels.  Note: The picture
                  on the  screen doesn't  have to be fully drawn before you
                  do the 24-bit output; the normal  display should  be seen
                  as  a  navigator  to  help  you  produce  the best 24-bit
                  pictures  possible,  24-bit  output   is  independent  of
                  current screen resolution etc.. (Note 4".) {F}

24-bit palette expansion:
                  Since a lot of people don't have AGA, and since even with
                  AGA it can be hard to manipulate colormaps  with over 256
                  colors, this is a small tool to help you: What it does is
                  that it takes a previously existing palette file and then
                  "expands it", smoothing any jumpy color-differences along
                  the way, allowing you to get smooth color  transitions in
                  your 24-bit  output. (Incidentally: This can also be used
                  to turn old 16/32 color palettes  into 256-color palettes
                  for use on AGA machines.) {X}


--------------------------------------------------------------------------


                               Tips & Tricks
                               =============

Cycle
-----
For optimum  results, use  a palette  where color 1 is nearly equivalent to
the last color. That way, cycling is less "jumpy". The  "32 color  cycle I"
palette is a good example thereof, so is "256 - shiny".


Where do i zoom?
----------------
Well, in the end you should let your own tastes decide:

Do you  like the  soft &  smooth quality  of "Candy" with palette #1? -Then
focus on pictures with  a low  "iteration" value;  don't zoom  to deep, but
experiment with "color-divide" and palettes instead.


V1.5                  - Lyapunovia Users' Guide -                   Page 10


If  you  prefer  the  spaced-out  metallic  of  "AAB 100, Indian eyes" with
palette #4, then stick to fairly high iteration  values and  always zoom on
the edges between black and color, in places that seem layered.

For  the  rather  moving,  "loneliest mathematical object in the universe"-
feeling of "5a5b 50, zoom1", zoom  on the  little islands  of color  in the
blackness.

Note  that  all  images  change appearance radically when changing palette.
Try the "X-ray" palettes!

Sometimes, but seldomly, you may experience  the feeling  that the pictures
you get tend towards the repetitive.
The  official  advice  is  this:  Zoom  where  you don't usually zoom! That
usually does the job. After having played with Lyapunov Space  for nearly a
year, I still get a lot of surprises. The possibilities are endless.....


--------------------------------------------------------------------------


                   In-depth explanation of a few points:
                   =====================================


Coordinates, the short explanation:
-----------------------------------

If you're  not that  interested in  mathematics, you  only need to know one
thing: The "coordinates" menu decides what your picture should look like:
"Sequence" make the picture ...... different.
"Color-divide" changes the brightness of the picture.
"Iterations": The bigger the number, the  stranger the  picture, the slower
the program.
"Plot repeat": Repeats the palette.


Coordinates, the in-depth explanation:
--------------------------------------

Sequence
--------
Selects the  general formula  you want to look at. Not as complicated as it
sounds: Lyapunovia works by  -  at  each  point  on  the  screen  - running
through  a  basic  formula  as  many  times  as  you  specify  in the field
"Iterations".
In this field, only the letters "A" or "B" are legal. "A"  runs through the
formula  using  the  X  coordinate  of  the  current  point, "B" uses the Y
coordinate. To give an example: A  sequence  of  "ABABB"  and  7 iterations
will repeat the formula like this:

A,B,A,B,B,A,B            -The "sequence", repeated.
X Y X Y Y X Y            -Its' X or Y partner.
1 2 3 4 5 6 7            -The current iteration.

That is to say that sequences are wrap-around.






V1.5                  - Lyapunovia Users' Guide -                   Page 11


Color-divide
------------
Determines  the  general  brightness  of  the  picture.  When using default
palette #1, a high  color-divide makes  the picture  light, a  low one will
make the  picture dark. I can't tell you what color-divide should be set to
at all times -  rather  you  should  experiment  from  picture  to picture,
adjusting this  number along with the palette. Nevertheless, a good rule is
to stay between 1500 and 5000.


Iterations
----------
This determines how many times the basic formula  should be  repeated - for
each iteration, the program moves to the next step of the "sequence". Thus,
if you specify fewer iterations than the sequence is long, only part of the
sequence will be executed.
The more  iterations, the  more weird the picture... And the longer time to
do calculations. (Note 3.)


Plot repeat
-----------
With a value of 1, the palette is distributed evenly by mathematical means.
Higher  "plot   repeat"  values   will  distribute  the  palette  _several_
consecutive times. If you then alter the palette  so color  1 and  the last
color are close, and then cycle, you'll have a great effect of moving bars.
Check out the "Curve  x4 II"  with different  cycle-palettes to  see what I
mean. Do try it with a cycle speed of 1!

Only 1-4 are valid plot repeat values.


Errors
------
When  pressing  "OK"  in  coordinates,  the  program will inform you of any
faulty values... Respond accordingly. If you're lost, just  press "UNDO" or
"CANCEL".

























V1.5                  - Lyapunovia Users' Guide -                   Page 12


                           Contacting the author
                           =====================

Contacting me
-------------
First of  all, you're  always welcome to contacting me with any suggestion,
bug report (aargh!) or comment that springs into your mind. 

If you've  found somewhere  nice in  Lyapunov Space;  a great  palette or a
great picture  or some  great coordinates, please mail them to me. You will
then .... be mentioned in the next doc. file!


The serious part
----------------
And now the serious part: Since the last release of the program,  I decided
not to keep a speciel registered version of the program to myself and a few
registered users, but rather to share it with as many people as possible.


In return for that, I'd like _you_ to share something with me:

The suggested shareware fee for this program is 50 DKR,  15 DM,  US$ 10, or
the equivalent  in any  other currency. If you think I deserve it, this can
be communicated in several ways:


* By mail: Wrap the bills in some dark paper and write  me letter  and give
me a  few comments,  tell me  a bit  about your Amiga and what you're doing
with it.
* By EuroCheque, addressed  to "Jesper  Juul". Remember  to write  the card
number at the back of the cheque.
* In most countries, you can pay at the post office via my Giro account:
Girobank Denmark, 1-205-3207.
* But no national cheques, not even Bank-signed ones.

If you don't feel like sending money, you can mail me something else:

*  Records,  tapes  and  CDs!  (Keywords:  Pixies, Stravinsky, Jethro Tull,
Living Colour.)
* Good books! (Keywords: Paul Auster, Jonathan Culler, Douglas Adams, Jorge
Luis Borges.)
* Or some (preferably good) software you've written.
* Or  if you  make something like t-shirts or posters or videos or demos or
anything from the program, I'd like a copy.
* Or whatever you think.

* No  matter  what  you  do,  be  sure  to  mail  me  some  comments and/or
suggestions!


Anyway, this is my address:


                  Jesper Juul
                  Guldsmedgade 34, 1.
                  DK-8000 Aarhus C.
                  Denmark

E-mail:           norjj@stud.hum.aau.dk


V1.5                  - Lyapunovia Users' Guide -                   Page 13


                                  Notes:
                                  ======

The future
----------
There are (still) a lot of things I might want to add to this program.

-Speed.
-Movie mode, letting you save animation-files.
-A color dithering mode.
-Extended rendering choices.
-More mathematical formulas to choose from.
-Whatever you can think of....... AREXX interface and SCSI support?


An  animation  mode  is  the  next  project,  after  that I'll probably try
looking at some other formulas.



                              Lighter notes:
                              ==============

     "It's too bad they don't do this covernote thing anymore,  'cause it's
     not a bad thing to say something, even if it doesn't amount to much."
          -Ian Anderson


The hard road to excess
-----------------------
Lyapunovia  was  written  in  100%  pure  assembler. It was written late at
night, early in  the  morning,  while  cooking  Spaghetti  Bolognese, while
brushing my  teeth, while  I was supposed to be doing something else, while
half asleep, while everybody  else was  watching TV,  while I  had too many
drinks the  night before, while I didn't really want to, while under almost
divine inspiration, while reading, while making  phonecalls, while watching
the tiny  little bit  of blue  sky I can see from my room if I push my head
flat against the window and look directly up,  while singing  loudly, while
thinking  'bout  the  night  before,  while  drying  my  hair, while eating
breakfast, while snapping my fingers, and while I was in small gaps between
everything  else  that  my  life  contains.  And  now  it's here. Makes you
wonder....


Slogans
-------

The official slogan of this manual is, as you know, this:

     "Why do I have to keep reading these technical manuals?"
          -Roger Waters.


The official slogan of this program was suggested by Iivo Vehvilainen:

     "Way off in some indistinguishable distance  was a  stunning peak that
     overarched  the  sky,  climbed  and  climbed and spread out in flowing
     aigrettes, agglomerates and archimandrites."
          -Douglas Adams: Mostly Harmless.



V1.5                  - Lyapunovia Users' Guide -                   Page 14


The official slogan of my entire life is this:

     "Though they may have won all the battles, we had all the good songs."
          -Tom Lehrer.


The slogans, by the way, are self-contained and have no slogans.


Laws
----

The official law governing  the production  of this  program is Hofstadters
law:

     "Things always take longer time to finish than you expect, even if you
     take Hofstadters law into account."
          -Douglas R. Hofstadter.


Copyrights
----------

Lyapunovia is  1992-93 Jesper Juul.

ReqTools is  Nico Francois.

The rest of the copyrights belong to whoever owns them.


Thanks go to:
-------------

Nico Francois for ReqTools.
Jan van den Baard for GadgetEd.
Stefan Borberg for LhA.
Olaf "Olsen" Barthel for Term.
Fred Fish for his library.

Mads Christiansen, Mads Rydahl  and  Jens  Christian  Hansen  for different
things.
(The  former  Mads  especially  for  showing  me  the  article  that got me
started, the latter Mads for designing disklabels.)

Kaare Heinsen, Phill Coxon, Jesper Skov, and Jes Srensen for beta-testing.
(Special thanks to Jesper for "answering questions".)

Lotte, Kim, Annette,  Anne,  Henrik,  Lilo,  Line,  Marianne,  Hanne, Lars,
Rasmus, and Trine for endurance.

And finally, the great mathematical minds behind it all:
A. K. Dewdney, Mario Markus, Benno Hess, and Aleksandr M. Lyapunov.










V1.5                  - Lyapunovia Users' Guide -                   Page 15


                                 Footnotes
                                 =========


NOTE 1:

On Amigas without Fastmem, selecting 16 colors in hires mode (256 colors, a
1280 pixel  screen on  AGA machines)  will significantly  slow down program
execution.  But  also  notice  that  this  is only true when the Lyapunovia
screen is  being  displayed.  Therefore,  bring  Workbench  to  front while
Lyapunovia is  calculating; the  screen will  automatically be re-displayed
when the picture has finished.


NOTE 2:

When  saving  an  IFF   file,  Lyapunovia   will  include   the  coordinate
parameters,  thus  enabling  you  to  load  the picture some other time and
continue zooming.  However:  If  you  process  the  picture  (in  a drawing
program, for example), that information will be lost, so keep a copy of the
orignal file. (For the technically interested: The information  is saved as
a TEXT chunk, which is probably the best solution.) 


NOTE 3:

The  "iteration"  parameter  is  a  source of frequent misunderstanding for
experienced Mandelbrot users.  Lyapunovia  is  very  different  in  that an
increased number  of iterations has nothing to do with increased precision;
rather a low iteration count leads to smooth  curvy pictures,  a high count
leads to ragged and chaotic pictures. Precision is constant.


NOTE 4:

You should  of course  use the program version that suits your system best.
The 68020+ version will be faster than the normal version,  the FPU version
will  _generally_  be  faster  than  the  68020+  version. However: On some
systems, single precision calculations will  actually  be  faster  with the
68020+  than   with  the  FPU  version.  And  also:  Single  precision  FPU
calculations are a bit less precise.
The  program  attemps  to  automatically  adjust  the  level  of  precision
according to the zoom depth. With FPUs, the program may not be that good at
guessing the needed precision.  With an  FPU, my  best advice  is to select
"double precsion" at all times.
