Spectrum Intercept Algorithm
Coloring according to position...

The spectrum intercept 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 "spectrum intercept" 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 spectrum intercept 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 for a fixed time as the system approaches its attractor. At the end of that time the value of the state variables are used to determine which color is picked from a pre-prepared spectrum. The algorithm then assigns that color to the pixel from which the integration started and moves on to the next pixel representing another set of initial conditions.

This algorithm is an abreviated form of the basin of attraction algorithm. It eliminates the time required to integrate all the way to the attractor and to identify that you have arrived. The images generated contain less information about the basins of attraction but are still an interesting example of nature's art.

This is the spectrum used in our examples of the spectrum intercept algorithm. If for example, at the end of the chosen integration time the position of the moving part of the system lies anywhere to the left of the dark blue boundary the starting pixel in the domain of the state map gets the dark blue color. If the position falls between the right edge of the dark blue and the left edge of the dark red, the appropriate color is picked from the spectrum. If the position falls anywhere to the right of the left edge of the dark red, the assigned color will be dark red.