Fractal Basins of Attraction in a Magnetic Pendulum

If you've seen the toy in my office then you know what these are about. If not, have a look here. But here's the basic idea...

Imagine a magnet tied to the end of a piece of string. Hold the string fixed at the other end, and let the magnet swing around. Now hold this magnetic pendulum over a table on which three other magnets are glued at the corners of an equilateral triangle. Give the pendulum a swing, and watch what it does: it follows a chaotic trajectory for a while, but eventually (because there's friction) it will stop above one of the magnets on the table. If you paint the initial position of the pendulum according to the color of the magnetic that the pendulum eventually stops at, and you do this for all possible initial positions, then you get a map of the "basins of attraction" for the system.

The images below were computed, one pixel at a time, by doing this with a numerical simulation of the pendulum. For a medium-resolution image, say 512x512 pixels, this takes about 3 hours on a fast computer. Different images are obtained for different values of the physical parameters, like the coefficient of friction, strengths of the magnets, etc.

Image gallery

Click on image to view the larger version.

Zoom sequences

Click on an image to view a sequence of enlargements (up to a magnification factor of 10 million) illustrating fine detail at all scales.

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