**MAT335: Java Applets and Visual Basic Programs **

This page
contains links to several applets and VB programs (specifically written
for this course) that can be used to
illustrate and understand fractals, dynamics and chaos. There are links
to programs
that illustrate the Julia and
Mandelbrot sets on the
MAT 335 Resources web page (at the bottom).

There are various versions of each applet suitable for screen sizes 640 x 480
(VGA), 800 x 600 (SVGA), 1024 x 768 (XVGA), 1280 x 1024 (UVGA), and
1400 x 1050.

Instructions for applets are included in the applet. Instructions for the VB
programs can be viewed in the link provided.

All the software here can be downloaded via the
fractals zip file (8 MB).

**The Chaos Game **

This applet draws fractals using the chaos game.

**Fractal Movie **

This applet plays a movie of an 'evolving' fractal that results when
the IFS defining the fractal changes in time (see Figure 5.36 in the text).

**The Modified Chaos Game **

This applet is a modification of our applet for the standard chaos game, modified by Danny Heap.
Here you can selectively remove certain strings of digits from the random
sequence used to draw
the fractal.
Before running this program read Danny's report.

(Click here for a larger view of the 'fractal' pictured here,
and here
for a colour-coded image indicating the distribution of points. Both images were produced with this
applet.)

**Fractal Pattern **

Visual Basic executable file (called "FractalPattern.exe") that draws fractals via Iterated
Function Systems (IFS) or the Chaos Game. You can input the parameters for the
IFS yourself, or download the following data files to the same folder as the
executable file (these data can also be found in the text on page 295 and
elsewhere):

Cantor
Maze (Figure 5.12), Twin
Christmas Tree (Figure 5.10), Crystal
with 4-fold symmetry (Figure 5.14), Crystal
with 5-fold symmetry (Figure 5.15), Dragon
with 3-fold symmetry (Figure 5.11), Barnsley's
fern (Figure 5.35), von Koch
curve (Figure 3.22), square von
Koch curve, Tree
(Figure 5.16), Twig
(Figure 5.13), Sierpinski
triangle.

For information about how to use this program, go
here.

If you get errors when trying to run the executable file, you may need
some support
programs.

**Graphical Iteration **

Applet for
the graphical iteration of the logistic function and its compositions.

**Bifurcation **

Bifurcations
of the Logistic Equation. Applet demonstrating the changing shape of the
logistic function and its iterates as the parameter *a* changes.
Observe how period doubling bifurcations arise.

**Time Series **

Time
series for Logistic equation. VB executable program (called "Timeseries.exe") for generating time
series and histograms of orbits of the logistic equation ax(1-x) and the
equation ax^2 sin(pi*x). See here
for instructions on how to use this program. If you get errors when trying to
run the program, you may need some support
programs.

**Logistic Movie **

Applet 'movie' of evolving histograms of the logistic equation as
the parameter *a* changes. Observe the period doubling bifurcations and
the histograms of ergodic orbits. You can compute the Feigenbaum constant for
the many 'windows' of the final state diagram.

**Henon Map **

VB executable program (called "TimeSeries2D.exe") for plotting (two dimensional) orbits of the
Henon map. See here
for instructions on how to use this program. If you get errors when trying to
run the program, you may need some support
programs.

**Henon Movie **

Applet 'movie' of the motion of an orbit on the Henon attractor.

**Julia Set Movies **

Applet 'movie' of changing Julia sets.

*This applet was written by Ilona Kowalik*

For a good program that explores the Mandelbrot sets and Julia sets, go to
the XAOS web page (`http://xaos.theory.org`).

*Unless specified otherwise, all Java applets and Visual Basic programs here were written by
Dongmei
Zhang
for the Department of Mathematics,
University of Toronto. *