Newton’s Basins of Attraction

This picture is a Newton pattern given by the dynamical system

zn+1 = zn- g(zn)/g'(zn),

g(z) = z5 + 2z3 + 3z3 - 4z3 - 5z - 6,

plotted on a small rectangular neighborhood of the point (3.077956, 2.8834923).

Advanced Method for Newton Patterns: If we can find the roots of a given polynomial prior to the fractal plotting, we can choose a color for each initial value z0 (which is a pixel in the canvas for a Julia pattern) depending upon which root the corresponding sequence zn converges to. In plotting the picture above, the computer is programmed to find all five roots of g through the method called Müller's algorithm and then color an initial value red, for instance, if the sequence converges to the first root. This causes the canvas to be separated into the five regions that are painted red, orange, skyblue, blue, and green, called the basins of attraction of the roots of g.

The basins of attraction are intricately interlaced and separated by a nonempty set of points called the basin boundary. It is difficult to visualize the boundary as it happens to be a totally disconnected dusty Julia set often called Cantor dust in topology. We can generally find beautiful Newton patterns near the Julia set but ironically it causes the Newton's root-finding algorithm to behave unpredictably because of chaos associated with the Julia set. It is one of the few shortcomings of the otherwise extremely powerful and widely applicable algorithm, and it explains why I use stable and “automatic” Müller's algorithm instead in finding the roots prior to the fractal plotting.