A certain type of system has progressive states as represented by
points on the XY coordinate plane. If totally random, then points
would range evenly over the plane, but these systems have strong
attractors that are often beautiful and /or interesting.

Check out a sample of 12 from a library of 512 :
https://postimg.cc/gallery/JB8TtTj

Note capture of special program on system 93 :
https://postimg.cc/F7pdpQd1

Email casagiannoni@optonline.net to get the free special Windows
program.

* * * DETAILED DESCRIPTIVE PAPER * * *

( 1 ) Preliminary Notes

SA18 software includes, the program executable SA18 Chaos Engine.exe,
and the library binary data file sa18lib.bin containing the required
information library where 88 bytes are used to store the unique
generating, locating and scaling information for each of the many
semi-chaotic systems. Also required are the five Visual Basic v6
auxiliary run time files, Asycfilt.dll, Msvbvm.dll, Oleaut32.dll,
Olepro32.dll and Stdole2.tlb . These five must be either with the
executable, or more properly at C:\Windows\System** with other DLLs.

The software is provided "as is" for free distribution, without any
warranty or condition of any kind, express or implied, and with the
firm understanding that the user assumes all responsibility for any
consequences of the use of the software.

( 2 ) Introduction and Background :

The Chaos Engine, has evolved from a study of a unique form of
mathematically defined systems of semi-chaos. Each state of these
systems is represented by a point on the XY coordinate plane.
Subsequent states or points, are mapped via application of 18 ordered
coefficients from two 9 element, 3x3 matrices, Aij and Bij, specific
to each unique system, according to the following algorithms
recursively :

X new ( X, Y ) =

A00 + A01 Y + A02 Y^2
+ A10 X + A11 X Y + A12 X Y^2
+ A20 X^2 + A21 X^2 Y + A22 X^2 Y^2

Y new ( X, Y ) =

B00 + B01 Y + B02 Y^2
+ B10 X + B11 X Y + B12 X Y^2
+ B20 X^2 + B21 X^2 Y + B22 X^2 Y^2

Matrix coefficients are additively applied to every possible product
combination of the current X and Y state coordinates in powers 0, 1
and 2, thus defining each subsequent system state.

It was discovered that if the 18 matrix coefficients were chosen at
random from an approximate interval a bit wider than -1 to +1, then
about one in every several hundred so defined systems would exhibit
behavior that was stable or bounded, non-degenerative and
non-periodic.

This semi chaotic behavior would result in evolving points for each
subsequent state of the system, defining a progressive image
where locations in the image were clearly attractive of most
systems states ( i.e. - the system, though fundamentally chaotic in
nature, nevertheless "prefers" certain states of attraction).

Visually, it was observed that these attractors tended to have
pleasing and interesting qualities, especially if the spectral colors
are used to indicate orbital accelerations in various image areas.

A computer developed a library of images by the random process
selection of sets of matrix coefficients and rejecting systems that
lacked the desired weak chaotic behavior. Each acceptable system was
stored as the 2 x (3 x 3) = 18 matrix coefficients together with 2
locating and 2 scaling parameters, requiring 88 bytes for each image
in the library file sa18lib.bin of images.

The unique matrices can bethought of as a kind of mathematical code
for the correspondingattractor images. The Chaos Engine enables the
user to view the 2 x (3x3) = 18 matrix coefficients while the image is
evolving, and allows for the dynamic "tweaking" of any selected
coefficient and the observed effect on the dynamic image.

Given even the crude precision of the chaos engine tweaking tools,
there is still likely estimated to be a vast number indeed of
different "viable" possible images !

( 3 ) About the Colors :

The color assigned to pixel points representing each system state, is
keyed to the acceleration at that point in the progressive development
of the attractor. It is the magnitude of the change in vector
displacements, between the vector of the preceding point to the
current point, and the vector from the current point to the subsequent
point. In a qualitative sense, it is the magnitude of the "jerk" felt
at each point if one was "riding" the points around the developing
image. Normal Spectral colors are used from Blue representing the
minimal accelerations, increasing through Cyan, Green, Yellow, and up
to Red representing maximum accelerations. Excursions beyond either
extreme are represented by a progression to Magenta. The program
samples the early development of system states to define a mean and
standard deviation of accelerations. Normalized scaling from full
Magenta below Blue up to full Magenta above Red is indicative of from
-2 to +2 standard deviations.

( 4 ) Periodic Random Orbit Perturbation :

On occasion, an otherwise well behaved attractor will suddenly fall
into a repeating sequence, sometimes only involving a limited number
of system states. Image number 275 from the original library is a good
example. Click the developed image to see the limited states. The
cause of this periodic degeneracy is not well understood, but the
round off error of the floating point math describing the system
states does impose a finite limit to the possible number of system
states within the domain of each attractor, and periodic degeneracy
can be the ultimate consequence. If the attractor is especially
"tight", as indeed is the case in some of the more interesting and
beautiful figures, then this periodic degeneracy can sometimes
overtake the attractor causing further development to cease. To offset
this tendency, code has been introduced to periodically perturb a
point (1 every 2^15 = 32768 points) in both the X and Y directions, by
random amounts selected from the interval form -.0025 to +.0025. This
is often just what such a figure needs to keep moving. This feature is
selectable in the chaos engine (click the label : ON shown green, or
OFF shown red).

( 5 ) System and Program Information :

The SA18 Chaos Engine is a 32 bit Windows application requiring an
appropriate version of Windows. Use a display at the maximum normal
resolution. Start the program SA18 Chaos Engine.exe by any of the
usual methods , e.g. - double clicking SA18 Chaos Engine.exe in the
Windows Explorer or permanently installing a shortcut with the program
icon (recommended).

( 6 ) Using the Chaos Engine :

On starting the Chaos Engine a semi-chaotic system is selected at
random from the library ( currently 512 ) and the attractor displayed
using spectral colors ranging from Magenta/Blue to Red/Magenta, for
tranquilly and violently chaotic regions of the attractor
respectively.

The sizing and positioning buttons [Bigger], [Smaller], [Taller],
[Wider], [Up], [Down], [Left], [Right] all do what they say when
clicked. Left and Right Clicking are for Large and Small adjustments
respectively. [Taller] / [Wider] change the aspect ratio of the image
without changing the overall size. All of these controls do nothing to
the character of the images.

Images are selected from the library using the vertical scroll slider
and the selected image number is indicated above the top end of the
slider. Any of the 18 matrix coefficients as Aij (left) and Bij
(right) displayed at the top may be selected for "tweaking" by left
clicking the number. The selected coefficient will appear in a
different color than the rest. The coefficient will be rounded off to
six decimal places when tweaked up or down using the [Add] or [Sub.]
buttons respectively.

Six levels of additive or subtractive adjustments are possible
according to the following table :
Action Added or Subtracted Amount

Left Click 0.1
Right Click 0.01
Shift - Left Click 0.001
Shift - Right Click 0.0001
Ctrl - Left Click 0.00001
Ctrl - Right Click 0.000001

Immediately on tweaking a coefficient, the image clears and redraws
using the altered coefficient, allowing the user to observe the effect
on the image. On occasion, the tweaked coefficient will render the
system unstable or unbounded and the green "OK" indicator will
intermittently or continuously change to a red "OUT !". At this point
the user can recover to the previous stable state by reversing the
offending action using the [Add] or [Sub.] buttons appropriately. In
any case, clicking the image number will return all coefficients to
the library values and is therefore a sure way to recover.

An altered image can be stored, replacing the starting image in the
library by holding both the Ctrl and Shift keys while clicking [Save]
(wait for beep). All previous points in a developing image can be
deleted by clicking either the "OK" indicator or the display area.
This is often a good way to detect the previously mentioned periodic
degeneracy.

Exit the program by clicking [Exit].

* * * * * *