#
The Quadratic Map

## How to use this applet

In this applet you see on the right the Mandelbrot set (the parameter plane for
the complex function **z**^{2} + c where **c** is a complex
parameter) and, on the left, the corresponding (filled) Julia set for the
particular chosen **c**-value. The default parameter value is **c = -1**.
The filled Julia set is displayed in black; the Julia
set is the boundary of the black region. The colored points are the points
whose orbits escape to infinity, with red points escaping fastest, followed by
orange, yellow, green, blue, indigo, and violet. In the Mandelbrot set, the
black points represent **c**-values in the Mandelbrot set, while colored
points represent **c**-values for which the corresponding orbit of **0**
escapes (with the same coloring scheme as above).

Don't know what I'm talking about??? Visit the Mandelbrot Set Explorer for a
primer on the mathematics behind Mandelbrot and Julia sets.

You can use this applet to:

- Magnify various portions of both the Mandelbrot and Julia sets;
- Choose a different
**c**-value and see the corresponding Julia set;
- Compute the orbit of a particular point in or around the Julia set of
**z**^{2} + c;
- Create an animation of the various Julia sets as
**c** varies along a
selected path in the Mandelbrot set.

Here is how to do these things:
To magnify portions of the Mandelbrot or Julia set, simply select a
rectangular region with your mouse in the appropriate window and click on
** compute** and the selected region will be displayed. You might wish to
increase the number in the **Max. iterations** box below each figure to get
a better image. Alternatively, you can enter the lower left and upper right
points of your desired window manually in the appropriate boxes. Then click
**compute** to see the desired magnification.

To display a Julia set for a different **c**-value, simply click on the
desired **c**-value in the Mandelbrot set window or enter the **c**-value
manually in the window called ** Function parameters**. Then click **
compute**.

To see a particular orbit in or around the filled Julia set, click on the
desired point. You can display the first ten points on the orbit by clicking
the **View orbit** button, or you can see the orbit one-by-one by
successively hitting
the **Iterate** button.

To make a movie of how the Julia sets change, select the starting
**c**-value in the Mandelbrot set and then click **Add current E**.
Then select another **c** value and click again **Add current E**.
You will see a straight line displayed in the Mandelbrot set. If you click
the ** Animate** button, you will then compute nine Julia sets drawn from
along this line and then see the animation in the Juli set window. Click
**animate** to replay this animation. To produce longer animations, simply
choose more points in the Mandelbrot set and don't forget to click **Add
current E** after each choice. Each additional choice adds eight more frames
to your animation. And no, to answer your question: you cannot save this
animation.

This applet was created by Yakov
Shapiro for Professor Robert
Devaney.