Exploration #5

Question: What happens to Julia sets as c moves from the large main cardioid into one of the smaller bulbs attached to the cardioid?

Answer: Two things happen. For c-values inside the cardioid, all orbits are attracted to an attracting fixed point. When c crosses into an adjacent smaller bulb (called a primary bulb), a new attracting cycle is born. At the same time, the filled Julia sets "pinch" together at various junction points in the filled Julia set.

Click on an arrowhead (not stem) to see an animation of the filled Julia sets as c travels along the path shown. You must have a QuickTime player to view the animation. Click here to download a QuickTime player for Macs or PCs. Have patience while these animations are being loaded. They average around 650K per animation.

ImageMap - turn on 

More explanation: The small white dots that suddenly appear in the animation form the points on the attracting cycle. This cycle appears as soon as c enters the small bulb attached to the cardioid. The large white dot that remains becomes a repelling fixed point when this cycle appears.

These changes are called bifurcations. Bifurcation means a splitting apart. Here we see that an attracting cycle of some period splits apart or bifurcates from the fixed point.

Further Exploration:

The Java applet below allows you to make your own movie. Simply click on a point in or near the Mandelbrot set. This gives you your starting c-value. Then click on a second c-value. This determines a path along which c will move. Click Make Movie to compute the resulting animation. Then click Play to view the movie.

Click here to open the Mandelbrot Movie Maker applet.

Please note that your browser must be Java-enabled in order to use this applet. All recent versions of Netsapce or Internet Explorer are Java-enabled.

Devaney home page

Dynamical Systems and Technology Project Home Page