Unlike a simple pendulum that slows down, the Van der Pol oscillator sustains its own oscillation: it draws energy from a source at small amplitude and dissipates it at large amplitude, settling on a unique limit cycle independent of starting conditions.
The equation ẍ − μ(1−x²)ẋ + x = 0 has a nonlinear damping term. For μ = 0 it is a harmonic oscillator. As μ grows, the limit cycle becomes increasingly non-sinusoidal — the 'relaxation oscillation' typical of cardiac pacemakers and electronic circuits.
Watch the phase portrait — position vs velocity — spiral onto the limit cycle from any initial condition. Increase μ to strengthen nonlinearity and observe the waveform becoming sawtooth-like. With external forcing enabled, find the Arnold tongues of frequency-locking.
Balthasar van der Pol discovered this oscillator studying vacuum-tube radio circuits in the 1920s. He also noticed that under certain forcing conditions the circuit made 'irregular noise' — one of the first experimental observations of deterministic chaos, decades before the word existed.