Van der Pol Relaxation Oscillator

The Van der Pol oscillator has nonlinear damping that extracts energy for small amplitudes and dissipates it for large ones, producing a stable limit cycle. As μ increases, the cycle transitions from sinusoidal (μ≈0) to sawtooth-like relaxation oscillations (μ≫1).

ẍ − μ(1−x²)ẋ + x = 0  |  ẋ=y, ẏ=μ(1−x²)y−x  |  Limit cycle: |x|≈2 for large μ, period T≈(3−2ln2)μ