Chaotic pendulum: The complete attractor

A commercial chaotic pendulum is modified to study nonlinear dynamics, including the determination of Poincare sections, fractal. dimensions, and Lyapunov exponents. The apparatus is driven by a simple oscillating mechanism powered by a 200 pulse per revolution stepper motor running at constant angular velocity. A computer interface generates the uniform pulse train needed to run the stepper. motor and, with each pulse, read a rotary encoder attached to the pendulum axle. Ten million readings from overnight runs of 50 000 drive cycles were smoothed and differentiated to obtain the pendulum angle theta and the angular velocity omega at each pulse of the drive. A plot of the 50 000 (theta,omega) phase points corresponding to one phase of the drive system produces a single Poincare section. Thus, 200 Poincare. sections are experimentally available, one at each step of the drive. Viewed separately, any one of them strikingly illustrates the fractal geometry of the underlying chaotic attractor. Viewed sequentially in a repeating loop, they demonstrate the stretching and folding of phase point density typical of chaotic dynamics. Results for four pendulum damping conditions are presented and compared. (C) 2003 American Association of Physics Teachers.