Poincaré Section Algorithm

Coloring by time slicing...

The Poincaré section 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 "Poincaré section" 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.

This is a rather long story but it contains some important ideas. The dynamical systems we find interesting all have something in common. They exhibit some sort of recurrence. That means that the trajectory of their state on the map of all possible states hangs around in a limited region of the map. If a system does not have this behavior its moving parts just go over the horizon, never to be seen again.

For certain systems, this hanging around behavior has a convenient definiteness to it. Systems whose time dependence comes from the application of a periodic force will settle into a motion that is also periodic at some multiple of the period of the forcing function. Periodic here just means that the trajectory of the state point passes in the neighborhood of its starting point an interval equal to an integral number of periods of the forcing function.

In all our other discussions of nature's hidden art we have been a little loose in describing attractors. In fact attractors may change as time passes. To make this time dependence explicit we have to include a time dimension in our map of all possible states. In this case a point attractor becomes a line extended in the time dimension, a closed curve becomes a cylinder and a chaotic attractor, for periodically forced systems, becomes a fractal bent into a torous (doughnut looking thing). You can find the details at our online chaos course section on Dynamical Systems.

The Poincaré section algorithm, named for Henri Poincaré who came up with the idea back in the 19th century, in effect slices a chaotic attractor at a particular value of time, revealing its fractal cross section. The image is made up one pixel at a time by integrating the system along a time interval of 2π time units, the period of the forcing function, and illuminating the pixel at the system state point every 2π time units.