Each of the four chapters 4, 5, 6, and 7 present an exemplary bifurcation sequence. This means that we are given a one-parameter family of maps, and we carefully observe a chaotic attractor as the parameter is varied. Certain special events called bifurcations occur, perhaps very frequently, as the parameter is changed. In each of these chapters, we have singled out just a few of these special events for special attention, we call them stages.
For example, in Chapter 4, there are 12 stages. In the book, monochrome computer graphics are included for each of these stages, along with extensive commentary which tries to explain the (very complicated) images.
In the movies, the stages are embedded in a very large number of in-between images, which are then flashed on the screen like a flip book. Thus, the still-frame black-and-white stage images of the book are embedded in an apparently continuous, uniform, sequence of color-coded images in the movies. The color code is a one-dimensional spectral scale from blue to red, and is shown at the right side of the screen in all of the movies. In the square frames of the movies, the color blue indicates a low relative density of trajectory points in a given small square of the plane, while red indicates a high density.
The 96 computer graphics in chapters 4 through 7 of the book (with four exceptions) have been computed in the mathematical programming language MAPLE by Scott Hotton. For the 92 images that have been made by in this way, the complete programs (they are plain text files) may be read directly from the CD-ROM. Reading one of these files, with the help of a MAPLE programming manual if needed, answers all possible questions about the figures in the book: the size of the domain, the number of points, etc. In addition, the programs are very easily modified and run in the MAPLE environment, to do further research in chaos theory.
The ENDO program, written by Ron Record, was used by him to make all of the frames of the movies on the CD-ROM. It is an easy-to-use research environment which you might use to do frontier research in two-dimensional discrete chaos theory, if you have access to an X-Windows environment. We have included the complete program on the CD-ROM, in an archived and compressed UNIX file. Instructions for its installation are found in the file *index.html: on the CD-ROM.
Finally, the CD-ROM contains (in file *index.html:) a few pointers to relevant websites, for those who have an Internet connection.
Method #1. The first method, which we strongly recommend, makes use of a World Wide Web browser. The one we have used is Netscape Navigator, which is freely available on the Internet. All other browsers should work, but we have not tested them. In this method,
Note: Clicking on a movie in the web browser results in a one-minute wait, while the movie file is copied from the CD-ROM to your hard disk. This is bad, because you have to wait. On the other hand it is good, because the movies play better from the hard disk, unless your equipment is in perfect running order. After the wait, you will see the first frame of the movie in the web browser window. You may then start and stop the movie by clicking anywhere in its frame.
Method #2. This is the fall-back method, and does not require any software other than the Windows FileManager, Macintosh Desktop, or UNIX shell.
These movie players have a simple control panel with run and pause buttons. In addition, you may drag the slider to advance or reverse the movie at slower or faster than normal speeds. You may use either format on a UNIX platform, with appropriate software, such as the freeware *xanim: for X-Windows. On Windows or Macintosh machines, you may also use a World Wide Web Browser to play the movies, as we have explained above.
The movies assume that your computer is capable of playing QuickTime (MOV) or Video for Windows (AVI) movies at 2X speed, that is, at 300 KB per second. If the movies jerk or stick, that probably means that your computer needs a tune-up.
On older versions of Windows and Macintosh operating sys-
tems, the movie players seem to stick inconsistently. As a
work-around, try moving the slider back and forth to loosen things up.
Some older systems display a warning message upon first inserting
the CD-ROM in its drive, but
Here are some tricks to improve Macintosh movie performance.
Virtual Memory: Typically, this is set on, and to about 1MB more
than the actual RAM. For example, with actual RAM 16 MB, set
virtual RAM to 17 MB. Cache Memory: This may be reduced to
improve movie playing. MoviePlayer application memory: Increase
the amount of memory devoted to MoviePlayer if you know how.
Ronald Joe Record is a Ph.D. graduate of the computational
dynamics program at the University of California at Santa Cruz,
and now works as a software engineer in Santa Cruz.
ABOUT THE CD-ROM AUTHORS
Ralph H. Abraham created the computational dynamics
program at the University of California at Santa Cruz.
Revised 24 September 1996 by Ralph Abraham, <firstname.lastname@example.org>
Here are some tricks to improve Macintosh movie performance. Virtual Memory: Typically, this is set on, and to about 1MB more than the actual RAM. For example, with actual RAM 16 MB, set virtual RAM to 17 MB. Cache Memory: This may be reduced to improve movie playing. MoviePlayer application memory: Increase the amount of memory devoted to MoviePlayer if you know how.
Ronald Joe Record is a Ph.D. graduate of the computational dynamics program at the University of California at Santa Cruz, and now works as a software engineer in Santa Cruz.