As part of the Dynamical Systems and Technology Project, we have developed several JAVA Applets for use in exploring the topics of chaos and fractals. These applets are designed to accompany the four booklets in the series A Toolkit of Dynamics Activities, published by Key Curriculum Press.

Important Notice: Security has been greatly enhanced recently on the web for java applets like those on this page. Because of this, you may have problems accessing these applets. If so, you can remedy this as follows (for Mac users --- a similar procedure should work for PC users, but I don't know the details).

Here is how to correct problems with access to these applets:

• Go to system preferences in the Apple menu
• Click on java
• Click on see the security tab
• Click on edit site list
• Add " http://math.bu.edu/DYSYS/applets " to this list, or add the address of specific applets
• Click OK and exit securely

Applets associated with the book Fractals include:

The chaos game. Yes, this is a game. Try to beat the computer by hitting specific targets via the moves of an iterated function system. This game allows students to understand the construction of the Sierpinski triangle via the chaos game.

Fractalina. This applet allows you to set up the vertices, compression ratios, and rotations associated to an iterated function system and then compute and view the resulting fractal. On some browsers, apparently the numbers that you enter in this program show up as white-on-white, so the numbers do not appear on the screen. If this is the case, you can just highlight the appropriate window to see the numbers. Someday I'll figure out how to fix this....

Fractanimate. This applet allows you to string together a collection of fractal images generated by Fractalina into a movie. We encourage you to become quite familiar with Fractalina before trying to use this applet.

Applets associated with the book Chaos include:

Linear Web. This applet allows you to see the results of iteration of linear functions of the form Ax + B in three different ways: as a list of the orbit, as a web diagram (graphical iteration), and as a time series graph.

Nonlinear Web. This applet allows you to see the results of iteration of nonlinear functions such as the quadratic function x² + c, the logistic function kx(1 - x), and many others. You can iterate step-by-step or view the entire web diagram.

Target Practice. This applet allows you to practice graphical iteration in the setting of a game. We give you a target (a particular "itinerary"); your job is to find an orbit graphically that has this itinerary.

Cycle Practice. This applet is a more advanced version of Target Practice. Your job in this game is to find a cycle (not just an orbit) with a given itinerary. Be sure to master Target Practice first before attempting this game.

Orbit Diagram. This applet allows you to draw the orbit diagram (or bifurcation diagram) for a variety of families of functions. You may also magnify various portions of these complicated sets.

Applets associated with the book The Mandelbrot and Julia Sets include the following. These applets are also part of the Mandelbrot Set Explorer, an interactive tutorial associated with this book.

The Quadratic Map Applet. This applet allows you to view simultaneously the Mandelbrot set, a Julia set, and the fate of any orbit in or around the Julia set. You can also use this program to make animations of various Julia sets derived from moving along different paths in the Mandelbrot set.

The following applets are somewhat older and some do not run on Macs using OS X (and some other machines).

The Mandelbrot Set Iterator. This applet allows you to choose a c-value in the Mandelbrot set and then view the corresponding fate of the orbit of 0.

The Mandelbrot/Julia Set Applet. (Old Version.) This applet allows you to view simultaneously the Mandelbrot set, a Julia set, and the fate of any orbit in or around the Julia set.

The Mandelbrot Movie Maker. This applet allows you to choose a path in the Mandelbrot set and then compute and view the corresponding Julia sets along this path in movie format.

Applets for computing the parameter planes and Julia sets for other complex functions. These applets are really a research tool for me and my students, but we place them here so other people can see the interesting parameter planes and Julia sets that arise from iteration of certain non-quadratic complex functions in the complex plane.

Rational Maps. These applets deal primarily with rational functions that are singular perturbations of polynomials. These maps are of the form of the form zⁿ + C + A / z^d for very special values of C, usually C = 0.

The complex cosine function.

The complex sine function.

The complex exponential function.

More applets to come soon. Maybe...

Applets associated with the book Iteration include:

Towers of Hanoi. This applet allows you to play the classic game called the Towers of Hanoi.

The Function Iterator. This applet allows you to iterate a number of different functions, including the quadratic functions x² + c, the logistic family cx(1 - x) the doubling function 2x mod 1, linear functions of the form cx + 1 as well as several other families. You can also display the results as an orbit list, a time series, or a histogram.

For comments and suggestions write to Robert L. Devaney at bob@bu.edu