Types of Fractals

Strange Attractors

WHAT ARE THEY?

Strange attractors are the third type of formula fractals, besides Julia and Mandelbrot sets. Making a strange attractor is very easy. Starting with some original point on a plane or in space, you calculate every next point using a formula and the coordinates of the current point. The formulas are usually expressed in one of the following three ways:

1. Formula for the complex plane:
new z = f(z)   where z is the complex number of the current point and f is some function
2. Formulas for the normal Cartesian plane:
new x = f(x, y)
new y = g(x, y)
where (x, y) are the coordinates of the current point and f and g are some functions
3. Formulas for 3D strangle attractors:
new x = f(x, y, z)
new y = g(x, y, z)
new z = h(x, y, z)
where (x, y, z) are the coordinates of the current point and f, g, and h are some functions

For example, the Henon Attractor is a very famous fractal:

It uses the formulas:
new x = 1 + y – 1.4x2
new y = 0.3x

The above fractal is an example of a quadratic attractor, where the sum of powers on every term is not greater than 2. Quadratic attractors are the most common strange attractors. Their general formulas are:

new x = ax2 + bxy + cy2 + dx + ey + f
new y = gx2 + hxy + iy2 + jx + ky + l

The coefficients a – l are constants which determine the fractal. Note that some of these coefficients can be 0, and that not all of them create fractals. However, the ones that do create an endless variety of fractals:

It isn’t surprising that after seeing such pictures created by simple formulas mathematicians called them "strange" attractors!

USES?

The only method strange attractors use is changing numbers using formulas. This turns out to be very useful in studying nature, where we look at the ways things like population, weather, and chemical reactions change. Scientists found many fractal patterns in these natural changes. In fact, strange attractors like the Rossler Attractor and the Lorenz Attractor were discovered while studying natural, not mathematical phenomena.