Preface to the CD-ROM

The CD-ROM supplied in the back of the book is intended as an enhancement to the book. Its main function is to animate the graphics in chapters 4 through 7 with 12 movies. It also contains some useful software. This companion CD-ROM may be regarded as a *canned: piece of the World Wide Web. It has an index which may be accessed by any WWW browser, like Netscape Navigator, or Internet Explore. The CD also connects seamlessly with the Web, if your computer has Internet access.


The movies for chapters 4, 5, 6, and 7 are computer graphic animations, created by extensive computations with ENDO, an X-Windows software package for research on discrete dynamical systems in two dimensions created by Ronald Joe Record. These movies provide the best opportunity to understand the role of critical curves in the bifurcations presented in these chapters.

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.


Besides the twelve movies, each in two formats, the CD-ROM also contains additional material: MAPLE and ENDO.

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.


There are two methods for accessing the CD-ROM.

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,

Then all contents of the CD-ROM are displayed for your choice. This is particularly convenient for the MAPLE script files. Also, if you happen to be connected to the World Wide Web, you may click on some links to external servers.

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.

Because this CD-ROM is a hybrid CD, the file structure looks like Windows to Windows, looks like Macintosh to Macintosh, and looks like UNIX to UNIX.


The 12 movies are each provided in two formats on the CD- ROM: AVI and QuickTime. Both are 320x240x8 video with 22kHz by 16 bit sound. On a Macintosh you must use the QuickTime versions. Under Windows you would choose the AVI version, unless you have QuickTime for Windows on your system, in which case you have a choice. QuickTime for Windows is available from Apple over the Internet, and our CD-ROM has a link to Apple to help you obtain a copy. In any case, you may play the movies through the web browser, as described above in the preferred Method #1. On the other hand, with the fall-back Method #2, the QuickTime movies may be played with the Movie Player included in the Macintosh operating system, while the AVI movies may be played with the MediaPlayer which is part of the Windows operating system.

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.


Every hardware/software platform plays CD-ROMs differently, and we cannot anticipate all of the potential problems. We have tested our CD-ROM on several machines of each sort P Windows, Macintosh, and UNIX. All functions have been robust and correct except the movie service function.

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 seems to work.

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.


Ralph H. Abraham created the computational dynamics program at the University of California at Santa Cruz.

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.

Revised 24 September 1996 by Ralph Abraham, <>