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   	SA18 Chaos Engine   
      
   ( 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   
      
   [continued in next message]   
      
   --- SoupGate-Win32 v1.05   
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