Chaos in a Pendulum

The damped-and-driven pendulum and its surprisingly complex behavior


The damped-and-driven pendulum is a seemingly simple deterministic (i.e., the equations of motion are known exactly) system with a surprisingly complex behavior. Tiny variations in the damping, forcing amplitude, or forcing frequency can send the pendulum from a periodic into a chaotic regime, then back into a “window” of order, and again into chaos. This hallmark of chaotic systems – extreme sensitivity to initial conditions – eventually leads to loss of long-term predictability.