XLYAP(6X)XLYAP(6X)

xlyap - display an array of Lyapunov exponents graphically

xlyap[-BLps][-W width][-H height][-o filename][-an] [-bn] [-wn] [-hn] [-i xstart] [-Mn] [-Rp] [-Sn] [-Dn] [-F string][-f string][-rn] [-On] [-Cn] [-cn] [-mn] [-x xpos] [-y ypos]

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

-random A good choice for use with xscreensaver: picks random parame- ters from a built-in list. -CnSpecifies the minimum color index to be used for negative expo- nents -DnSpecifies 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. -HnSpecifies the height of the window. Default is 256. -L Indicates use log(x) + log(y) rather than log(xy). -MrSpecifies the real value to compare exponent values to for indexing into a color wheel. The default value is 1.0. -OnSpecifies the minimum color index to be used for positive expo- nents -RpSpecifies pseudo-random forcing with probabilitypof using parameter value ’a’. -SnSpecifies the "settle" or number of iterations prior to the beginning of the calculation of the Lyapunov exponent. Default is 200. -WnSpecifies the width of the window. Default is 256. -arSpecifies the real value to use as the minimum parameter value of the horizontal axis. Default is 3.0 for the logistic map. -bnSpecifies the real value to use as the minimum parameter value of the vertical axis. Default is 3.0 for the logistic map. -cnSelects one of six different color wheels to use. The default color wheel is a rainbow palette. -F10101010Specifies 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). -fabbabaabSpecifies the forcing function to use. The default is to alter- nate between the "a" parameter and the "b" parameter. -hrSpecifies the real value to be used as the range over which the vertical parameter values vary. The default is 1.0. -irSpecifies the real value of the initial condition to use. Default is 0.05. -mnSelects between available non-linear maps of the unit interval. A value of 0 specifies the logistic map. A value of 1, the cir- cle 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. -ofilenameSpecifies the output filename to be used. If the -o option is given, this file will automatically be written out at the com- pletion 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 posi- tive exponents. This can be toggled during runtime by pressing the ’p’ key. -rnSpecifies the maximum rgb value to be used. Default is 35000. -snSpecifies the length of the color wheel spin. -u Produces a usage message. -v Prints out the various values to be used and exits. -wrSpecifies the real value to be used as the range over which the horizontal parameter values vary. The default is 1.0. -xnSpecifies the x screen coordinate of the window (default is 256). -ynSpecifies the y screen coordinate of the window (default is 256).

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 keysbBeEfFkKjJmnrRsSwWxXqQindicates: (<) 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 dif- ferent 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 val- ues. (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

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

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 Claw- son. Rubber banding code was adapted from an existing Mandelbrot pro- gram written by Stacey Campbell. Viciously hacked for xscreensaver by Jamie Zawinski, 20-Nov-97.XLYAP(6X)

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