(z^3+1)/(1+cz^2)

The Fractal dropdown menu in the ChaosExplorer program contains 4 menu items:

  • z^p + c
  • (z^3+1)/(1+cz^2)
  • c*e^z, and
  • z^3-z^2+z+c

Each of these menu items displays the fractal (z0 = 0.0 + 0.0i, varying c over the complex plane). Selecting the first menu item displays the Mandelbrot Set; selecting any of the others displays the fractal for the indicated formula. This post specifically shows the fractal and Julia Sets for the second menu item, whose formula is:

zn+1 = (zn3 + 1) / (1 + czn2)

Here is the fractal display. This is for the region -10.0 ≤ x ≤ 10.0, -10.0 ≤ y ≤ 10.0:

Fractal2

And here are 4 Julia Set displays for points in this fractal:

Julia21

Julia22

Julia23

Julia24

As you can see, the fractal and Julia Sets are very different from the Mandelbrot/Multibrot fractals and Julia Sets.

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