Changing the power in the equation

Some setup

import chaoseverywhere as chaos
from mayavi import mlab

An example with a power of 4

The Mandelbrot set is defined by its famous equation \(z_{n+1}=z_n^2+c\). But one could rightfully ask : why \(2\) exactly ? So let’s fulfill our curious needs and see what happens if we take \(z_{n+1}=z_n^4+c\) as our equation.

Construction

First, we define the function of the Mandelbrot set \(f(z,c):\mathbb{C}\times\mathbb{C}\longrightarrow\mathbb{C}\) as above.

def transform(z,c):
    return(z**4 + c)

Then, we use the Mandelbrot_disp class in this package to create a basis for the set, and then we use mayavi to display our work.

mandel = chaos.Mandelbrot_disp(1.5,0,2.5, precision=600).mandel_transform(FUN=transform)
mlab.figure(size=(800, 800))
mlab.surf(mandel, colormap='hot', warp_scale='auto', vmax=1.5)
mlab.view(elevation=18)
mlab.close()

We can clearly see some king of duplicates into three of Mandelbrot sets displaying themselves around the main bulb and allowing for each of them a third of the space to grow.

And for the other powers ?

For a power of \(2\), we see one big ramification extending over the real axis. For a power of \(4\) we see three of them. So we can conjecture that for any positive integer \(p\) as the power, we will see \(p-1\) main ramifications around the main bulb. Let’s see if that checks out for the first hundred powers.

One can reproduce this animation using the code below.

chaos.Mandelbrot_disp(0,0,2,t_max=150, precision=500).anim_puiss_mandel(remove=True)