Logistic Orbits. This program will compute the orbits for the logistic equation and display them in a time series plot and in a table.
Comparing Initial Conditions. This program will compute the orbits for two different initial conditions for the logistic equation and display them in a time series plot and in a table. The program also plots the difference between the two orbits. This can be used to see if a function has sensitive dependence on initial conditions.
Bifurcation Diagram. This program will let you plot the bifurcation diagram for the logistic equation and then magnify regions that you select.
Julia Set Programs. There are a number of pretty good Julia Set programs you can choose from. All of these will plot a Julia set for the function f(z) = z^2 + C. You can enter different C values and see how the Julia sets change. You will need these programs to do the exercises from Chapter 24.
- Julia Sets. Enter C values by changing the boxes after "Constant". The first box is the real part of C, and the second box is the imaginary part of C.
- Julia Set on WolframAlpha. This page has a simple interface in which you input the real and imaginary parts of c.
- Julia Set Fractals. (Warning: This program displays Julia sets flipped horizontally. It uses a slightly different definition of the quadratic function and so its Julia sets are mirror images of those in the book and those made by most other programs. To see the image rendered correctly, change the sign on the imaginary part of c. I.e., if you wanted to see the Julia set for c = -0.1 - 0.88i, enter -0.1 for the real part of c and +0.88 for the imaginary part of c.) To change the C value on this program, first click on the "Show Display Controls" button on the lower left of the image. You can then enter in the real and imaginary parts of C. It has several different color options and is quite fast.
Mandelbrot Set Programs.
- Mandelbrot/Julia Set Applet. This program lets you click on the Mandelbrot set and see the corresponding Julia set. Simple to use.
- Julia and Mandelbrot Set Explorer. Click on the Mandelbrot set and you then get a Julia set and zoomed-in image of the Julia set. Scroll down the page and you can then chose to return to the previous page, download the Mandelbrot or Julia set you just generated, or return to the original Mandelbrot set.
- Mandelbrot Maps. As you move your mouse over the Mandelbrot set, the corresponding Julia sets are rendered on the right of the screen. The Julia sets are drawn very quickly. This program is also great for successively zooming in on the Mandelbrot set or a Julia set. The zoom controls are on the top of each image. You can drag the images if you need to re-center them.