| Franics Moon and
Celso Grebogi, Ed Ott, and James Yorke, among others, have done
important and subtle work on this problem. To illustrate the concept, here
we present a simple, slightly idealized, example. |
| Imagine an iron pendulum suspended midway over two magnets. |
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| What happens to the pendulum if we release it from some position? What if
we shake the pendulum support while it swings? Let's see. |
| First, to make the example realistic we include friction. |
| To visualize this problem, we represent the position and speed of the pendulum as a
point in the phase plane. |
| Here are the basins of attraction of the two equilibria of the
damped pendulum. The details of the picture depend
on the amount of friction. |
| Now suppose we shake the pendulum support. Here are the basins of attraction
of the two equilibria of the driven pendulum. As the
driving frequency and amplitude increase, the basin boundaries become fractal. |
|