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.




The Chaos Game

Run Applet
640 x 480
800 x 600
1024 x 768
1280 x 1024
1400 x 1050
This applet draws fractals using the chaos game.


Fractal Movie

Run Applet
640 x 480
800 x 600
1024 x 768
1280 x 1024
1400 x 1050
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

Run Applet
640 x 480
800 x 600
1024 x 768
1280 x 1024
1400 x 1050
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

Download
Visual Basic executable file 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):




Graphical Iteration

Run Applet
640 x 480
800 x 600
1024 x 768
1280 x 1024
1400 x 1050
Applet for the graphical iteration of the logistic function and its compositions.


Bifurcation

Run Applet
640 x 480
800 x 600
1024 x 768
1280 x 1024
1400 x 1050
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

Download
Time series for Logistic equation. VB executable program 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

Run Applet
640 x 480
800 x 600
1024 x 768
1280 x 1024
1400 x 1050
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

Download
VB executable program 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

Run Applet
640 x 480
800 x 600
1024 x 768
1280 x 1024
1400 x 1050
Applet 'movie' of the motion of an orbit on the Henon attractor.


Julia Set Movies

Run Applet
640 x 480
800 x 600
1024 x 768
1280 x 1024
1400 x 1050
Applet 'movie' of changing Julia sets.
(This is a preliminay version of the applet.)

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.


Back to the MAT335 homepage