Julia Sets and the Mandelbrot Set

Newton's Method Basins of Attraction

Newton's method is designed to find the roots of equations. Science is filled with examples of new problems suggested by techiques developed to solve other problems. Arthur Cayley recognized that if we already know the roots of a function, Newton's method suggests another problem:

which initial guesses iterate to which roots.

That is,

what are the basins of attraction of the roots?

We shall see this problem leads to fractals (no surprise at that), and also a simple experiment making an optical Sierpinski gasket using Christmas tree ornaments.

Introducing the problem: finding the basins of attraction of the roots of z2 - 1.

How much harder can it be to find the basins of attraction of the roots of z3 - 1?

Here we illustrate how the basins change as the roots move.

The crinlky edges of the z3 - 1 basins illustrate fractal basin boundaries and the corresponding loss of predictability.

The stirring together of basins on smaller and smaller scales illustrates the Wada property from topology.

Here is an exmple of complicated basins of attraction in a mechanical system.

Here is an exmple of complicated basins of attraction in a magnetic system.

Here is an exmple of complicated basins of attraction in a optical system.

Return to Complex Newton's Method.