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Never be the 1st ...
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 to try something like the new Windows 10, until all the bugs and glitches have been worked out. Just solid good sense. Probably best to wait about a year ! Comments, ?? |
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Never be the 1st ...
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