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 LYAP(6X)                                                           LYAP(6X)




 NAME
      lyap - display an array of Lyapunov exponents graphically

 SYNOPSIS
      lyap [-BLps][-W width][-H height][-o filename][-a n ] [-b n ] [-w n ]
              [-h n ] [-i xstart] [-M n ] [-R p ] [-S n ] [-D n ] [-F
              string][-f string][-r n ] [-O n ] [-C n ] [-c n ] [-m n ] [-x
              xpos] [-y ypos]

 DESCRIPTION
      lyap generates and graphically displays an array of Lyapunov exponents
      for a variety of iterated periodically forced non-linear maps of the
      unit interval.

 OPTIONS
      -C n    Specifies the minimum color index to be used for negative
              exponents

      -D n    Specifies the "dwell" or number of iterations over which to
              average in order to calculate the Lyapunov exponent. Default
              is 400.

      -B      Causes the stop, go, spin and quit buttons to be displayed.

      -H n    Specifies the height of the window. Default is 256.

      -L      Indicates use log(x) + log(y) rather than log(xy).

      -M r    Specifies the real value to compare exponent values to for
              indexing into a color wheel. The default value is 1.0.

      -O n    Specifies the minimum color index to be used for positive
              exponents

      -R p    Specifies pseudo-random forcing with probability p of using
              parameter value 'a'.

      -S n    Specifies the "settle" or number of iterations prior to the
              beginning of the calculation of the Lyapunov exponent. Default
              is 200.

      -W n    Specifies the width of the window. Default is 256.

      -a r    Specifies the real value to use as the minimum parameter value
              of the horizontal axis. Default is 3.0 for the logistic map.

      -b n    Specifies the real value to use as the minimum parameter value
              of the vertical axis. Default is 3.0 for the logistic map.

      -c n    Selects one of six different color wheels to use. The default
              color wheel is a rainbow palette.



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 LYAP(6X)                                                           LYAP(6X)




      -F 10101010
              Specifies the "Function" forcing function to use. The example
              above would alternate between iterating the circle and
              logistic maps. An argument of "-F 2323" would alternate
              between left and right logistic maps. The default is to only
              use the single specified map (see the description of -m).

      -f abbabaab
              Specifies the forcing function to use. The default is to
              alternate between the "a" parameter and the "b" parameter.

      -h r    Specifies the real value to be used as the range over which
              the vertical parameter values vary. The default is 1.0.

      -i r    Specifies the real value of the initial condition to use.
              Default is 0.05.

      -m n    Selects between available non-linear maps of the unit
              interval. A value of 0 specifies the logistic map. A value of
              1, the circle map. A value of 2, the left-logistic. A value of
              3, the right-logistic. A value of 4, the double-logistic. The
              default is 0, the logistic map.

      -o filename
              Specifies the output filename to be used. If the -o option is
              given, this file will automatically be written out at the
              completion of the drawing.  If it is not specified, a default
              filename of lyap.out is used and only written if the 'f' or
              'F' keys are pressed during a run. The format of the output
              file is PPM for color and PGM for monochrom. The parameters
              used to calculate the picture are included as comments at the
              beginning of the output file.

      -p      Switches color indices for negative and positive exponents.
              Generally, causes negative exponents to be displayed in more
              detail while darkening and narrowing the color range for
              positive exponents. This can be toggled during runtime by
              pressing the 'p' key.

      -r n    Specifies the maximum rgb value to be used. Default is 35000.

      -s n    Specifies the length of the color wheel spin.

      -u      Produces a usage message.

      -v      Prints out the various values to be used and exits.

      -w r    Specifies the real value to be used as the range over which
              the horizontal parameter values vary. The default is 1.0.





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 LYAP(6X)                                                           LYAP(6X)




      -x n    Specifies the x screen coordinate of the window (default is
              256).

      -y n    Specifies the y screen coordinate of the window (default is
              256).



 NOTES
      During display, pressing any mouse button allows you to select the
      area to be investigated with the mouse. The upper left hand corner of
      the desired area is the location of the cursor when the button is
      pressed. The lower right hand corner is specified by the cursor when
      the button is released.


      Use of the keys bBeEfFkKjJmnrRsSwWxXqQ indicates:

           (<) Halve dwell value.
           (>) Double dwell value.
           ([) Halve settle value.
           (]) Double settle value.
           (B or b) Toggle button display on/off
           (E or e) Recalculate the indices into the color wheel using a
      different method
           (F or f) Save current screen to ouput file (not yet implemented)
           (H or h or ?) Display brief help message
           (i) Decrement the interval between stripes for the striped color
      map.
           (I) Increment the interval between stripes for the striped color
      map.
           (K) Decrease value exponents are compared against by 0.05.
           (J) Increase value exponents are compared against by 0.05.
           (M) Decrease value exponents are compared against by 0.005.
           (N) Increase value exponents are compared against by 0.005.
           (m) Increment the map index, changing the map to be iterated.
           (P or p) Toggle positive/negative exponent display.
           (r) Redraw the window using previously calculated exponents.
           (R) Redraw the window using the newly set dwell and/or settle
      values.
           (S) Spin the color wheel
           (s) Halve the length of the spin and spin the color wheel
           (u) Go up to the window just prior to the most recent zoom.
           (U) Go all the way up to the original window.
           (V or v) Display values of various parameters currently in use
           (W or w) Use next color map.
           (X or x) Clear window
           (Q or q) quit






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 LYAP(6X)                                                           LYAP(6X)




 AUTHOR
              Ronald Joe Record
           The Santa Cruz Operation
                P.O. Box 1900
             Santa Cruz, CA 95061
                  rr@sco.com



 ACKNOWLEDGEMENTS
      The algorithm was taken from the September 1991 Scientific American
      article by A. K. Dewdney who gives credit to Mario Markus of the Max
      Planck Institute for its creation. Additional information and ideas
      were gleaned from the discussion on alt.fractals involving Stephen
      Hall, Ed Kubaitis, Dave Platt and Baback Moghaddam. Assistance with
      colormaps and spinning color wheels and X was gleaned from Hiram
      Clawson. Rubber banding code was adapted from an existing Mandelbrot
      program written by Stacey Campbell.




































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