A Julia set maps how a particular nonlinear dynamical system behaves.
The links below are to Julia sets of various iterators.
Using Newton's Method on z^7 - 1
Using Newton's Method on z^7 + z / 1000 - 1
Using Newton's Method on z^6 - 1
Newton's method on three slightly assymmetrical roots
Newton's method on very asymmetric roots
Newton's method on seven slightly asymmetrical roots
Zoom-in of previous picture
Zoom-in of previous zoom-in
Newton's method on z^7 + i*z/100000 - 1
Julia set of z^3 - i*z + 1/4
Julia set of z*sin(z) + 1/5
Julia set of z * (1 + (z - exp(1.4*i)) / 2)
previous picture zoomed-in (This is a much bigger version of the little blue picture at the top of the screen.)
The link below goes to a picture of a tiny fraction of the Mandelbrot set, which maps the properties of Julia set of the iterator z -> z^2 + c.
Zooming In on the Mandelbrot Set
I produced these images back in the day using a Windows application that I wrote in VB6(!).