Chaotic Pendulum

The harmonic oscillator is a paradigm of predictability. The resonance response of a harmonic oscillator can be exploited to make clocks accurate to 1 part in 10. Harmonic oscillations are normally observed in systems operating near equilibrium and governed by linear or “Hooke’s law” restoring forces. Nonetheless, for small enough excursions from equilibrium, systems governed by nonlinear restoring forces can also display harmonic oscillations. A good example is the pendulum. When its motion stays near equilibrium (at θ = 0), the nonlinear restoring torque (proportional to sin θ) behaves like a linear restoring torque (proportional to θ) and the pendulum executes simple harmonic motion. However, when driven strongly, the excursions from equilibrium can grow into regions where the nonlinearity becomes important. The predictable pattern of repeating oscillations can turn into chaos—non-repeating motion characterized by a particular kind of unpredictability.