ENDO

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NAME

endo - display an iterated endomorphism of the plane

SYNOPSIS

endo [-^+01234567RUVZpquv][-S settle][-n dwell][-W width][-H height][-o filename] [-8 0-3][-9 0-3][-P 1-8][-abcdefgijklm r ][-ABCDEFGIJKLM r ][-w r [ , r , r ][-h r [ , r , r ][-x r [ , r , r ][-y r [ , r , r ][-t delta][-z cdelta][-N n ][-O n ][-X n ][-Y n ][-Q n n [ , n ][-T n ][-r n ][-s n ]

DESCRIPTION

endo generates and graphically displays an iterated endomorphism of the plane. Endomorphisms are into mappings. Thus, each iterate (f(x,y),g(x,y)) may not have a unique inverse, (x,y). The basic idea is to iterate an endomorphism of the plane (R x R). Each point (x,y) is mapped to (f(x,y),g(x,y)). The new point is then mapped to (f(f(x,y),g(x,y)),g(f(x,y),g(x,y))) and so on. Each point is then colored according to how many iterates it took to get within some "delta" of a previous iterate (e.g. a periodic orbit is detected). Lyapunov exponents for a region of parameter space can be calculated and displayed in a separate window. Phase portraits can be constructed and histographic data displayed in a third window. Critical curves (curves for which the determinant of the Jacobian is zero) and their iterates may be displayed in a fourth window. Finally, pre-iterates of the critical curves may be displayed in a fifth window. By varying parameter(s), the phase portrait window can be used to display bifurcation diagrams. The "attract velocity" window can also be used to display basins of attraction. A variety of methods of display and colormaps can be selected. For instance, bifurcation diagrams can be displayed in 3 dimensions (2 spatial and one parametric), 2 spatial dimensions, 1 spatial and 1 parametric dimension, or 2 dimensional with the 2 spatial dimensions overlayed on the y-axis while the parametric dimension is mapped onto the x-axis. In each case, the user can select whether to clear the window each time a parameter is incremented.

OPTIONS

-^: reverses the colormap divisions (e.g. negative Lyapunov exponents are given the portion of the colormap usually reserved for positive Lyapunov exponents).
-+: indicates display coordinate axes.
-0: selects the demonstration mode.
-1: display y coordinate bifurcation diagram only
-2: display x and y coordinate bifurcation diagrams
-3: display 3-D bifurcation diagrams
-4: display x coordinate bifurcation diagram only
-6: selects animation drawing mode (i think bifurcation diagrams look best this way)
-7: selects histogram drawing mode
-8 n: selects Lyapunov mode (0 indicates no Lyapunov sets calculated or displayed, 1 indicates display only bifurcation diagrams, 2 indicates calculate and display both bifurcation diagrams and Lyapunov exponents, 3 is the same as 2 but the subsequent initial condition is chosen to be the last point calculated).
-9 n: selects Mandelbrot mode (0 indicates no Mandelbrot/Julia sets calculated or displayed, 1 indicates display Mandelbrot sets with Lyapunov exponents, 2 indicates calculate and display Mandelbrot set in black, 3 indicates display Mandelbrot set using rate of attraction of critical point).
-n n: Specifies the "dwell" or number of iterations over which to calculate and display iterate values.
-S n: Specifies the "settle" or number of iterations prior to the beginning of the calculation of exponents or display of the iterates.
-H n: Specifies the height of the window.
-ABCDEFGIJKLM n: Specifies the 12 possible parameter values for the selected map. For instance, "-A 0.04 -B 0.056" would select values for the first and second parameters of the map.
-abcdefgijklm n: Specifies the 12 possible additional parameter values for the North-South models. These values are those to be used for the North region.
-O n: Specifies the minimum color index to be used for positive exponents
-W n: Specifies the width of the window.
-x r [ , r , r ]: Specifies the real value to use as the minimum value of the horizontal axis. First value is for the Attractor/Lyapunov window. Second (optional) value is for the Trajectory window. Third (optional) value is for the Critical Curve window. Values are a comma-separated list.
-y r [ , r , r ]: Specifies the real value to use as the minimum value of the vertical axis. First value is for the Attractor/Lyapunov window. Second (optional) value is for the Trajectory window. Third (optional) value is for the Critical Curve window. Values are a comma-separated list.
-P n: Turns on critical curve detection and display. 1 indicates display of critical curves only, 2 indicates display of critical and precritical curves, 3 indicates display of precritical curves only, 4 indicates display of pre-images of the diagonal only, 5 indicates the attractors are symmetrically divided by the diagonal, 6 indicates display of pre-images of the origin, 7 indicates display of pre-images of the diagonal with the pre-iterates of the origin colored differently, 8 indicates display of pre-iterates of the critical curve and the origin with the pre-iterates of the origin colored differently.
-h r [ , r , r ]: Specifies the real value to be used as the range over which the vertical axes values vary. First value is for the Attractor/Lyapunov window. Second (optional) value is for the Trajectory window. Third (optional) value is for the Critical Curve window. Values are a comma-separated list.
-R: Turns on selection of random initial conditions. A new initial condition is selected for each horizontal slice through parameter space when calculating Lyapunov exponents.
-U r: Specifies the real value to be used as the minimum x-axis of the trajectory window.
-Q n n [ , n ]: Specifies which parameter indices to vary during calculation of Lyapunov exponents. For instance, "-Q 0 2" varies parameters A and C while calculating Lyapunov exponents for each pair of parameter values. The third (optional) value specifies the method of traversing parameter space when constructing bifurcation diagrams. By default, parameters A and B are varied and the parameter region is traversed row by row. A value of 1 indicates to just compute the diagonal. A value of 2 computes a row then a column. A value of 3 computes a column then a row. A value of 4 says use a mesh and refine it until all values are calculated.
-T n: Selects between available non-linear maps of the plane. A value of 0 specifies the Gaertner map; a value of 1, the Goodwin map; a value of 2, the Guckenheimer map; a value of 3, the "double circle" map; a value of 4, the "double sine hump" map; a value of 5 selects a variant of the double circle map, while a value of 6 selects the Standard map. Values of 7-9 select the either the Chichilnisky North-South models or the Gardini figures depending on how the program was compiled, 10 selects the "rotor" map, 11 the "twist and flip" map, 12 the "Dorband double logistic" map, 13 the untwisted double logistic, 14 the Lorenz map, 15 the Volterra-Lotka equations, 16 the familiar Julia and Mandelbrot sets, and 17 the Brusselator. There is no default map. If none is specified, a menu is displayed to select from.
-X r: Specifies the real value to be used as the initial x value of the trajectory.
-Y r: Specifies the real value to be used as the initial y value of the trajectory.
-Z r: Specifies the real value to be used as the minimum y-axis of the trajectory window.
-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 endo.out is used and only written if the 'F', 'T' or 'V' keys are pressed during a run. The 'F' key will output the Attractor/Lyapunov window, the 'T' key will output the Trajectory window, and the 'V' key will output the Critical Curve window. 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: Turns off calculation and display of phase portrait and bifurcation diagrams.
-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 [ , r , r ]: Specifies the real value to be used as the range over which the horizontal values vary. First value is for the Attractor/Lyapunov window. Second (optional) value is for the Trajectory window. Third (optional) value is for the Critical Curve window. Values are a comma-separated list.
-t r: Specifies the real value to be used as the radius within which periodicity is decided (e.g. the distance between the current and a previous iterate is less than r).
-q: Turns on attractor display (only useful when also detecting periodic orbits).
-z r: Specifies the real absolute value to be used as the threshold within which points are determined to be on the critical curve (e.g. the determinant of the Jacobian at that point is less than abs(r)).

NOTES

During display, endo allows interaction through either the keyboard or the mouse. In most of the windows, pressing the left 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. During 3-D display, pressing the middle mouse button will display the angle values through which the viewpoint will be rotated. Releasing the middle mouse button will rotate the viewpoint. Pressing the right mouse button will cause the viewpoint to be translated as the mouse is moved while the button is down.

The color map adjustment window allows you to alter the color mapping function by use of the mouse or keyboard. In this window, pressing the left mouse button selects the point at which the line segments of the map will intersect. By holding the button down and moving the mouse you can drag this point around. Releasing the mouse button selects the point. Leaving the window aborts the adjustment. In this window, the middle mouse button selects a linear mapping and the right mouse button displays a histogram of color usage.

No mouse interaction is possible in the help or information windows.

Use of the keys

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indicates:

           (0) Zoom out, quadrupling size of selected window.
           (1) Use 1-D model (only appropriate for Chichilnisky model 1).
           (2) Select one of several styles of bifurcation diagrams.
           (3) Toggle 3-D display.
           (5) Pan left by half the width.
           (6) Pan right by half the width.
           (7) Pan down by half the height.
           (8) Pan up by half the height.
           (9) Zoom in to center quarter of selected window.
           (<) Halve dwell value.
           (>) Double dwell value.
           ([) Halve settle value.
           (]) Double settle value.
           ({) Halve delta value.
           (}) Double delta value.
           (-) Halve cdelta value.
           (+) Double cdelta value.
           (,) Positions the viewpoint on the y-axis.
           (.) Positions the viewpoint on the z-axis.
           (/) Positions the viewpoint on the x-axis.
           (|) Toggles display of coordinate axes.
           (A) Displays the next iterate of the selected critical arc.
           (a) Toggle the attractor calculation and display.
           (b or B) Toggle the display of the rate of attraction window.
           (c) Clear the trajectory popup window
           (C) Toggles calculation and display of the critical curves
           (d) Go down to the next window in the zoom sequence.
           (D) Flush the drawing buffers.
           (E or e) Recalculate the indices into the color wheel using a different method
           (f) Toggles periodic orbit detection
           (F) Save current attract velocity window to ouput file
           (G) Recalculate and redisplay the iterates of the critical curve
           (g) Calculate and display the iterates of the selected critical arc
           (H) Toggles calculation and display of a histogram of trajectory values
           (h or ?) Display brief help message
           (I) Toggle display of a parameter and map information window.
           (i) Toggle random initial condition selection.
           (j) Toggle clearing of bifurcation diagram at each new vertical parameter value.
           (k) Translate the viewpoint half the distance toward the view center.
           (K) Translate the viewpoint double the distance from the view center.
           (l or L) Toggle Lyapunov exponent calculation and display.
           (m) Increment the map index, selecting the next available map to iterate.
           (M) Decrement the map index, selecting the previous map to iterate.
           (n) Toggle negative/positive Lyapunov exponent color maps.
           (o) Translate the viewpoint toward the already computed average of points.
           (O) Translate the viewpoint away from the already computed average of points.
           (P or p) Toggle the point orbit calculation and display.
           (r) Redraw the window using previously calculated rates of attraction.
           (R) Redraw the window using the newly set dwell and/or settle values.
           (s) Spin the color wheel.
           (S) Display the menu of available maps for selection.
           (T) Save current point trajectory window to ouput file.
           (t) Select alternate critical arc (if any).
           (u) Go up to the window just prior to the most recent zoom.
           (U) Go all the way up to the original window.
           (V) Save current critical curve window to ouput file
           (W) Use next color map.
           (w) Use previous color map.
           (x) Clear the rate of attraction window
           (X) Delete this frame from the sequence
           (z) Clear the critical curve window
           (Z) Single step display of each iterate of the critical curve
           (Q or q) quit

In addition, several control keys are supported.

<ctrl>-S will pause computation, <ctrl>-Q will resume computation

<ctrl>-C will toggle drawing the trajectory in the critical curve window.

<ctrl>-A toggles between adjusting the sea level and sky level (minimum and maximum bounds for histographic data)

<ctrl>-B toggles a color bar display. This bar indicates the color indices used for histographic images.

<ctrl>-E recalulates colors for basins of attraction

<ctrl>-F sets the sea or sky level to 1.0

<ctrl>-G sets the sea or sky level to 0.0

<ctrl>-H decrements the sea or sky level

<ctrl>-J increments the sea or sky level

<ctrl>-K decrements the histogram algorithm index

<ctrl>-L increments the histogram algorithm index

<ctrl>-M halves the number of iterates of the critical curve to be displayed

<ctrl>-N doubles the number of iterates of the critical curve to be displayed

<ctrl>-T toggles display of the trajectory in the pre-critical window

<ctrl>-P toggles display of the trajectory in the basin window

<ctrl>-V toggles display of the color mapping adjustment window

<ctrl>-W reads a colormap from the file specified by the Mapfile resource or, if no such resource exists, $HOME/.endomap . The format of this file is 3 integers per line, each in the range [0, 256] indicating red, green and blue values.

<ctrl>-X decreases the number of divisions in the multiple rainbow palette

<ctrl>-Y increases the number of divisions in the multiple rainbow palette

AUTHORS

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

and

Dr. Ralph H. Abraham
Mathematics Department
University of California
Santa Cruz, CA 95064
rha@cats.ucsc.edu

ACKNOWLEDGEMENTS

The series of seminars given in 1992 at UCSC led by Dr. Ralph Abraham were the inspiration for the study of iterated endomorphisms of the plane. The 3-D transformation and display routines were ported to X from routines written for CGI by Hiram Clawson. Techniques for calculating the critical arcs and absorbing regions were drawn from Laura Gardini of the Istituto di Scienze economiche, Universita' di Urbino. Studies of various maps of the plane were used including papers by Jon von Neumann, Graciela Chichilnisky, John Guckenheimer, G. Oster, A. Ipaktchi, Wulf Gaertner, Jochen Jungeilges, Ralph Abraham, B. P. Wood, A. J. Lichtenberg, Z. You, E.J. Kostelich, J. A. Yorke, R. Brown, L. Chua, K. Judd, A. I. Mees, K. Aihara, and M. Toyoda.

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Time: 20:44:00 GMT, June 11, 2022