Chaos in the motion of an inverted pendulum: An undergraduate laboratory experiment

A suitable experiment for undergraduates is described which illustrates chaotic motion, i.e., a driven inverted pendulum. By varying the driving frequency, one finds a transition from regular (periodic) motion to chaotic motion. A sonar coupled to a microcomputer is used to measure the position of the pendulum as a function of time, and fast Fourier transforms are used to give the power spectrum for the motion. The chaos is detected by following the evolution of the power spectrum, as the driving frequency is varied, from a line spectrum to a continuous one; the former corresponds to regular motion and the latter to chaotic motion. A second characteristic nonlinear phenomenon, subharmonic response, is also illustrated with the apparatus.