Chapter 4 Putting It All Together

4.1 Locating Bifurcations

Often the first task in analyzing an endomorphism (or any map) is locating critical parameter values at which bifurcations occur. The use of endo can usually make this task much easier. After having established the domain of iteration and applying the classical dynamical systems tools such as calculation of the eigenvalues at fixed points, endo can be employed to construct a graphical display of the Lyapunov exponents and corresponding bifurcations diagrams.

Use the menu of maps, command-line options, or interactive run-time control to select the map and Lyapunov exponent mode. To view bifurcation diagrams, also select the bifurcation diagram mode. Endo will now construct a graphical display of the Lyapunov exponents. To locate critical bifurcation values, zoom in on an area of the graphical display which exhibits a sudden shift in color (e.g. shades of blue next to shades of green or red). Use the "information" window to determine the parameter values being displayed. The "information" window can be toggled by use of the "I" key. Continue zooming until a small enough window is displaying the transition between negative and positive Lyapunov exponents. Use values from these parameter intervals to calculate and display attractors, basins of attraction, critical and pre-critical curves, and absorbing areas.

4.2 Displaying Bifurcation Diagrams

If you selected the bifurcation diagram mode as well, an additional window displaying bifurcation diagrams is displayed. As the Lyapunov exponents are being calculated, bifurcation diagrams are simultaneously displayed. Several bifurcation diagram display modes are available. By default, the bifurcation diagram window displays an attractor in the selected region of the plane for each parameter value. Alternate modes are selected by pressing the '2' key. The alternate modes display the x-coordinate, y-coordinate, or x and y coordinates of the trajectory on the attractor as the y-coordinate in the bifurcation diagram and the parameter value as the x-coordinate in the bifurcation diagram. If endo was compiled with the 3-D option, then an additional three dimensional mode is available. This mode is selected with the '3' key and displays the bifurcation diagram with one axis representing the parameter value and the other two representing the x and y coordinates of the attractor for that parameter value.

Each of the bifurcation diagram modes has an "animation" mode that can be toggled with the 'e' key. In animation mode, the bifurcation diagram window is cleared when a new parameter value is selected. In this mode the window displays a "movie" of the attractor as it changes with the changing parameter values. Zooming in and out of the bifurcation diagram window is also possible.

4.3 Locating an Absorbing Area

The procedure developed by Christian Mira for determining an absorbing area (described in Chapter 5) is implemented as an algorithm in endo. To display the boundary of an absorbing area with endo, select the display of the critical curves window. It is only necessary to iterate the map for a single iteration in order to calculate the critical curve an its iterate . To do this, set the settle to 0 and the dwell to 1 (e.g. use the command-line switches "-S 0 -n 1 -P 1"). After calculating and displaying the critical curves, endo will highlight the critical arc (if any) to be iterated by displaying it in white. If there is no intersection between and , alternate domains of iteration may be tried.

After the critical curves are calculated and critical arc highlighted, increase the dwell by pressing the '>' key the desired number of times (each time doubles the dwell count). Press 'g' to iterate the critical arc. Each iteration of the critical arc will be displayed in a different color. Increase or decrease the dwell as desired, erase the critical curves window and redisplay critical curves and iterates of the critical arc. If an additional arc was detected, that arc can be selected by pressing 't'.

4.4 Making a Movie

Once an "interesting" regime of parameter values is located and bifurcation diagrams viewed, it is often desirable to generate a sequence of "frames" for use in generating a computer animation of the bifurcations. This is simplified through the use of endo's "demo" mode. When given the "-0" command-line option, endo exits after completing the specified number of iterations and optionally writing the contents of a window to a file. To generate a sequence of files representing frames in an animated bifurcation sequence, use the "-0 -o filename -A parameter-value" command-line switches in conjunction with the shell scripting capabilities of a UNIX shell. An example for use with the Bourne shell (/bin/sh on most systems) is as follows :

for APARM in 90 80 70 60 50 40 30 20 10 00

do

PIC=pics/demo1_$APARM

[ -f $PIC.gif ] && continue

endo -0 -7 -9 1 -8 0 -T 28 -S 512 -n 512 -r 100 -A -1.$APARM -B -0.42 -x -3 , -2 -y -3 , -2 -w 6 , 4 -h 6 , 4 -W 128 , 320 -H 128 , 240 -o $PIC.ppm , 2 $*

done