SA18 Attractors
( 1 ) Preliminary Notes :
A complete understanding of the material in the following three
sections ( 2, 3 and 4 ), is not necessary for using the Chaos Engine ,
(program file, SA18 Chaos Engine.exe) and working with the many stored
images).
SA18 software consists of the following eight files : the previously
mentioned executable for the Chaos Engine; the 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 stored images, the five auxiliary
files SVBVM60.DLL, OLEAUT32.DLL, OLEPRO32.DLL, ASYCFILT.DLL and
STDOLE2.TLB, and this Word document file SA18.doc. The software is
provided as the zip file SA18.ZIP.
The software was developed in Visual Basic v6.0, and 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 chaos. Each state of these systems
is defined 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 algorithm :
X new ( X, Y ) = A00 + A01Y + A02Y2 + A10X + A11XY +
A12XY2 + A20X2 + A21X2Y + A22X2Y2
Y new ( X, Y ) = B00 + B01Y + B02Y2 + B10X + B11XY +
B12XY2 + B20X2 + B21X2Y + B22X2Y2
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 weakly
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 was assigned the task of developing 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 18 matrix coefficients together
with scaling, locating and dimensional parameters, requiring 88 bytes
for each image in the library file of images. The unique matrices can
be thought of as a kind of mathematical code for the corresponding
attractor images. The Chaos Engine enables the user to view the 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 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. 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. Up-to-date versions of following files
must be in the Windows System subfolder, with other DLL files :
MSVBVM60.DLL, OLEAUT32.DLL, OLEPRO32.DLL, ASYCFILT.DLL, and
STDOLE2.TLB.
For the Chaos Engine, using the highest screen image resolution that
will permit a color depth of at least 64K (16 bit ) and will display
the developing images in a reasonable time, will produce the best
viewing.
For the Chaos Engine, the program file SA18 Chaos Engine.EXE and the
Library Image file SA18LIB.BIN should be placed in the same folder
location. Start the program
SA18 Chaos Engine.EXE in Windows by any of the usual methods , e.g. -
double clicking SA18 Chaos Engine.EXE in the Windows Explorer, using
the Run command, or permanently installing a shortcut with the program
icon (recommended).
( 6 ) Using the Chaos Engine :
On starting the Chaos Engine an image is selected at random from the
library and 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 (using
the [Add] or [Sub] buttons) 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].
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]
Steve Giannoni
casagiannoni@optonline.net
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