Basin of Attraction Algorithm
Coloring according to attractor...

The basin of attraction algorithm is a technique for displaying certain aspects of system behavior under integration. An algorithm is just a piece of computer code designed to achieve a desired result. The "basin of attraction" part of the name refers to the decision making process by which a particular color is assigned to each pixel in the image being generated.

Mathematical systems have the property that, depending on their initial conditions, they tend to seek out a certain set of states called an attractor. In a map of the possible system states the attractor may be a single point, a collection of points, a curve, a collection of curves or a region with an intricate fractal boundary. An attractor that has a fractal boundary is called a chaotic (or strange) attractor. In fact it is often the case that multiple attractors exist, each "owning" its own set of initial conditions.

The basin of attraction algorithm takes the location of each pixel in a certain domain of the map of possible system states and assigns it a color. Taking the pixel's location as initial conditions, the algorithm itegrates the mathematical expression for the system until the systems settles on an attractor. The algorithm then assigns a color to that attractor and moves on to the next pixel representing another set of initial conditions and integrates from that start to an attractor. If it is the same attractor as before, the pixel gets the same color. If not, a new color is assigned to the point leading to that attractor.

In our examples of this algorithm the number of attractors ranges from 2 to 5. Every point in the map of possible states will end up belonging to one and only one attractor and the collection of all the points belonging to a particular attractor is its basin of attraction. The surprising issue is how thoroughly mixed these basins are in the map of possible states. Even attractors consisting of a single point may have a fractal basin of attraction.