The last two posts have discussed Chaotic Systems and Fractals, and the Mandelbrot Set. As stated in the second of these posts, the Mandelbrot Set is a specialization of the Multibrot Set. These sets are created by iterating zn+1 = znp + c where z, p, and c are complex numbers. c is the coordinate, in the complex plane, of the point being calculated. Points inside the set are bounded as n approaches infinity, and points outside the set are unbounded (tend to infinity) as n approaches infinity. More information is provided in the post on the Mandelbrot Set and in the ChaosExplorer program’s wiki on GitHub.
For the Mandelbrot Set, p = 2. This post will look at other values of p, including both real and complex values, and also the effects of varying the real and imaginary portions of Z0.
The first menu item in the context menu of MultibrotPanel is a submenu containing menu items that set p to integer values between 2 and 10. The next four images show the resulting images for the values 2, 3, 4, and 10.
Note that the number of secondary bulbs (the second largest bulbs) in each image is equal to p – 1 where p is the integer power in the iterated equation.
The Animate Real Powers menu item changes the power value in steps of 0.01 from 1.0 to 10.0. Here is a video showing the display when you select this menu item:
The Animate Imaginary Powers menu item changes the imaginary portion of power in steps of 0.01i from -1.0i to +1.0i. Here is a video showing the display when you select this menu item starting from the display of Power = 3:
The Animate Z0 Real menu item changes the real portion of z0 from -1.0 to +1.0 in increments of 0.01. Here is a video of the result with Power = 6:
And finally, the Animate z0 Imaginary menu item changes the imaginary portion of z0 from -1.0i to +1.0i in increments of 0.01i. Here is a video of the result with Power = 3:
This ends the exploration of the Mandelbrot and Multibrot Sets. In the next post, we will look at Julia Sets related to the Mandelbrot Set.