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1995-07-25
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595 lines
EEEENNNNDDDDOOOO((((LLLLOOOOCCCCAAAALLLL)))) UUUUNNNNIIIIXXXX SSSSyyyysssstttteeeemmmm VVVV EEEENNNNDDDDOOOO((((LLLLOOOOCCCCAAAALLLL))))
NNNNAAAAMMMMEEEE
endo - display an iterated endomorphism of the plane
SSSSYYYYNNNNOOOOPPPPSSSSIIIISSSS
_e_n_d_o [-^+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 ]
DDDDEEEESSSSCCCCRRRRIIIIPPPPTTTTIIIIOOOONNNN
_e_n_d_o generates and graphically displays an iterated
endomorphism of the plane. Endomorphisms are _i_n_t_o 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.
OOOOPPPPTTTTIIIIOOOONNNNSSSS
-^ 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
Page 1 (printed 3/9/94)
EEEENNNNDDDDOOOO((((LLLLOOOOCCCCAAAALLLL)))) UUUUNNNNIIIIXXXX SSSSyyyysssstttteeeemmmm VVVV EEEENNNNDDDDOOOO((((LLLLOOOOCCCCAAAALLLL))))
-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.
Page 2 (printed 3/9/94)
EEEENNNNDDDDOOOO((((LLLLOOOOCCCCAAAALLLL)))) UUUUNNNNIIIIXXXX SSSSyyyysssstttteeeemmmm VVVV EEEENNNNDDDDOOOO((((LLLLOOOOCCCCAAAALLLL))))
-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
Page 3 (printed 3/9/94)
EEEENNNNDDDDOOOO((((LLLLOOOOCCCCAAAALLLL)))) UUUUNNNNIIIIXXXX SSSSyyyysssstttteeeemmmm VVVV EEEENNNNDDDDOOOO((((LLLLOOOOCCCCAAAALLLL))))
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 _f_i_l_e_n_a_m_e
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.
Page 4 (printed 3/9/94)
EEEENNNNDDDDOOOO((((LLLLOOOOCCCCAAAALLLL)))) UUUUNNNNIIIIXXXX SSSSyyyysssstttteeeemmmm VVVV EEEENNNNDDDDOOOO((((LLLLOOOOCCCCAAAALLLL))))
-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)).
NNNNOOOOTTTTEEEESSSS
During display, _e_n_d_o 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
Page 5 (printed 3/9/94)
EEEENNNNDDDDOOOO((((LLLLOOOOCCCCAAAALLLL)))) UUUUNNNNIIIIXXXX SSSSyyyysssstttteeeemmmm VVVV EEEENNNNDDDDOOOO((((LLLLOOOOCCCCAAAALLLL))))
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
_0_1_2_3_5_6_7_8_9<>[]{}-
+^%,./|_a_A_b_B_c_C_d_D_e_E_f_F_g_G_h_H_i_I_j_J_k_K_l_L_m_M_n_N_o_O_p_P_q_Q_r_R_s_S_t_T_u_U_V_w_W_x_X_y_Y_z_Z?
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.
Page 6 (printed 3/9/94)
EEEENNNNDDDDOOOO((((LLLLOOOOCCCCAAAALLLL)))) UUUUNNNNIIIIXXXX SSSSyyyysssstttteeeemmmm VVVV EEEENNNNDDDDOOOO((((LLLLOOOOCCCCAAAALLLL))))
(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
Page 7 (printed 3/9/94)
EEEENNNNDDDDOOOO((((LLLLOOOOCCCCAAAALLLL)))) UUUUNNNNIIIIXXXX SSSSyyyysssstttteeeemmmm VVVV EEEENNNNDDDDOOOO((((LLLLOOOOCCCCAAAALLLL))))
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.
Page 8 (printed 3/9/94)
EEEENNNNDDDDOOOO((((LLLLOOOOCCCCAAAALLLL)))) UUUUNNNNIIIIXXXX SSSSyyyysssstttteeeemmmm VVVV EEEENNNNDDDDOOOO((((LLLLOOOOCCCCAAAALLLL))))
<ctrl>-X decreases the number of divisions in the multiple
rainbow palette
<ctrl>-Y increases the number of divisions in the multiple
rainbow palette
AAAAUUUUTTTTHHHHOOOORRRRSSSS
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
AAAACCCCKKKKNNNNOOOOWWWWLLLLEEEEDDDDGGGGEEEEMMMMEEEENNNNTTTTSSSS
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.
Page 9 (printed 3/9/94)