Chapter 3 Interactive Run-time Control

3.1 Playing with the title screen and demo

If endo is invoked with no arguments, a title screen is displayed. The full-screen window welcomes. In the center of the screen a graphic is displayed. At the bottom of the screen are instructions for use of the program. The user can choose to allow endo to randomly demonstrate some of its features or continue on to the menu of maps. If the user wants to proceed on, she can press the space bar or click any mouse button. If she decides to watch the demo, several keys provide limited interactive control.

Pressing the 'n' key causes endo to go on to the next demo, 'p' the previous demo. The 'w' key installs another color palette. Pressing 'q' quits the program.

3.2 Selecting a map and view

The menu of maps presents the user with a list of the available maps, the current settings for settle and dwell, and a list of the views. This screen and the title screen are not displayed if the "-T #" option was present. Click the left mouse button to select the desired map. The selected map will be highlighted. Clicking the right and left mouse buttons in the settle or dwell areas of the screen will increase and decrease the values. Clicking the left mouse button in the view areas to the right of the screen will toggle the selection of those views. Pressing the space bar or clicking the right mouse button will bring up the selected views and begin the iterations.

3.3 The information and help windows

The display of an "information" window can be toggled by pressing ''. This window contains the map definition, parameter values currently selected, ranges of the views selected and other data relevant to the map and views. The information window will also display the modes that endo is currently running in. For instance, whether critical curve detection is on, whether periodic orbit detection is on, what level of "Mandelbrot" mode has been selected and what the escape radius is.

The display of a "help" window can be toggled by pressing 'h' or '?'. This window contains a terse description of the interactive run-time control available via the keyboard and mouse. The information and help windows cannot be panned or zoomed and are the only such windows.

3.4 Zooming and panning

The mouse and/or keyboard can be used to zoom in, zoom out and pan in any of the windows. To zoom in with the mouse, position the cursor on the upper left corner of the desired zoom box. Click and hold the left mouse button while rubber-banding a zoom box. When the cursor is on the desired lower right corner, release the left mouse button. The rubber-banded box expands to fill the window.

To zoom out or pan the window, use the keyboard. Pressing the '' key doubles the range of the x and y axes of a window while keeping the center point fixed. This effectively quadruples the area of the plane displayed in a window. Pressing the '' key reverses this by zooming in by a factor of four while keeping the center point fixed.

Panning can be accomplished by pressing the '5', '6 ', '7' and '8' keys to pan left, right, down and up respectively. Each key slides the window half of the respective range, thus positioning the former center at an edge and a former edge point at the center.

3.5 Toggling the views

The several views of the dynamics provided in endo can be displayed individually or in concert with one another. By default the trajectory window is displayed along with either the basins of attraction or the Lyapunov exponents. The display of the trajectory window (also used to display the bifurcation diagrams) can be toggled interactively by pressing the '' key. To toggle between the basin window and the Lyapunov window, press 'L'. The display of the critical curve window is toggled via the 'C' key and that of the precritical curves via the 'N' key.

The trajectory window can be erased after each individual trajectory has been displayed. The 'e' key toggles this feature. The trajectory window can be cleared at any time by pressing 'c'. Periodic trajectory detection is toggled by pressing the 'f' key.

Several views of the bifurcation diagrams are provided. Pressing '2' steps through these views. By default, the bifurcation view is non-erasing and displays the attractor as a parameter value is changed over time. The next view presented (after pressing '2' once) shows the x coordinate of the trajectory on the attractor (y-axis) plotted against a parameter (x-axis). The next view uses the y-coordinate of the trajectory on the attractor and the final view plots both x and y coordinates of the trajectory on the attractor against a parameter axis. All of these views can be displayed in "animation" mode in which the attractor is erased each time the parameter is incremented.

The trajectory can be viewed in other windows as well. In order to display the trajectory in the critical curve window, press <ctrl>-C. To display the trajectory in either the precritical or basin windows, press <ctrl>-T or <ctrl>-P.

3.6 Adjusting values

Several runtime values can be changed interactively. The settle and dwell can be increased or decreased. To double the settle, press ']'. To halve it, press '[ '. To double the dwell, press '>'. To halve it, press '<'. Likewise, the values of or can be halved by pressing '{' or '-' and doubled by pressing ' }' or '+' respectively. Recall, these values set the near-periodic and critical curve detection ranges.

3.7 The color palettes and view enhancements

Included with endo are 16 built-in color palettes and many user-defined palettes to select from. To increment the color palette index, press 'W'. To decrement this index, press 'w '. The user-defined palette is palette index 0 and is defined in the file $HOME/.endomap. Several samples of these files are provided in the source distribution in the directory lib/colormaps. The user is encouraged to play with the color maps provided and create new ones. The author has found that quite often heretofore unseen subtleties of the dynamics are strikingly revealed by use of an alternate color map.

Several view enhancements are also provided. These include 3-D bifurcation diagram display, trajectory histogram calculation and display, color bar indexing, color palette indexing curves and critical arc location and display. The use of 3-D bifurcation diagrams is toggled with the '3' key (if this feature was turned on at compile time). The author, however, found that 3-D enhancements to bifurcation diagrams was not as effective as use of the 'e' key to turn on the erasure of trajectories. In this way, the bifurcation diagram is displayed a slice at a time in a "movie" with time representing the parameter axis.

The 'H' key toggles trajectory histogram calculation. Although this feature requires additional memory and cpu resources, it achieves a pleasing and appropriate coloration of the attractor. The histographic display of an attractor essentially represents a probability measure with colors representing densities. A color bar showing the range used for the histogram can be displayed by pressing <ctrl>-B. The color bar can be displayed in either the lower-left or upper-left corners of the window. Pressing <ctrl>-O toggles display between these two corners.

The algorithm used to calculate the histographic display of the trajectory can be modified by pressing <ctrl>-V. This brings up a window which displays the mapping of histogram values to colors. This mapping can be adjusted by use of the keyboard or mouse. Pressing <ctrl>-[, <ctrl>-], <ctrl>-^, or <ctrl>-_ will move the intersection point left, right, upward or downward. The intersection point can also be moved by "grabbing" and "dragging" it with the mouse. To do this, click the left mouse button and hold it down as you move it to the desired intersection location. In addition to these adjustments, entirely different histogram color mapping functions are available. To step through these, press <ctrl>-K or <ctrl>-L to decrement or increment the color mapping algorithm index.

3.8 Saving to a file

Each of the five main endo windows can be saved to a file. When calculating basins, the basin window is saved to a file by pressing 'F'. The 'F' key is also used to save the Lyapunov exponent window. To save the trajectory window, press 'T'. Pressing 'V' or 'v' saves the critical or precritical curve windows respectively. In each case, the file format is Portable Pixmap (PPM) for color displays or Portable Bitmap (PBM) for monochrome displays. See 2.9 for command-line arguments to specify the output filename.

3.9 Restarting

You may, at any time, return to either the title screen or menu of maps by pressing 'J' or 'S' respectively. In this way, you can re-initialize the view, parameters and map or just view the title demo. If all you want to do is change the map, this can be accomplished by pressing the 'm ' key which increments the map index. Incrementing the map index leaves all other parameters unchanged. Care should be exercised to avoid invalid parameter ranges and initial conditions.