Abstract
We show that a simple differential equation modeling a garden-variety damped forced pendulum can exhibit extraordinarily complicated and unstable behavior. While instability and control might at first glance appear contradictory, we can use the pendulum's instability to control it. Such results are vital in robotics: the forced pendulum is a basic subsystem of any robot. Most of the mathematical methods used in this paper were initially developed in celestial mechanics, largely by Poincare. The literature of the field tends to be quite advanced indeed (see [1] and [11]); one object of this paper is to show that computer programs, properly used, can make these advanced topics transparent. All the computer-generated pictures in this paper were produced by the programs Planar Systems and Planar Iterations [6], both written by Ben Hinkle (now at Maple). 1. SOME PARALLELS IN CELESTIAL MECHANICS. When I was a graduate
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