The Simulation and Analysis of a Single and Double Inverted Pendulum with a Vertically-Driven Pivot

The inverted pendulum is a simple system in which both stable and unstable configurations are easily observed. The upward inverted state is unstable, though it has long been known that a simple rigid pendulum can be stabilized against small disturbances in its inverted state by oscillating its pivot in an up-and-down motion. Numerical simulations of the inverted pendulum are employed to investigate the stable and unstable domains of the system with respect to the excitation amplitudes and frequencies. The associated basins of attractions, extracted by interpolated cell mapping, are fractal.