Van der Pol equation ($\ddot{x}-\mu (1-x^2)\dot{x}+x=0$)

The Van der Pol equation is the nonlinear oscillatorin dimensionless variables. Its damping is negative at small amplitude and positive when $|x|>1$. For the quadratic oscillator energy $E=({\dot{x}}^2+x^2)/2$,Thus small oscillations gain energy while sufficiently large excursions dissipate it. This balances growth and saturation, producing the familiar attracting limit cycle of this system.

When $\mu =0$, the equation reduces to the simple harmonic oscillator. For small positive $\mu$, averaging $x\approx A\cos t$ gives $⟨\dot{E}⟩\approx \mu (A^2/2-A^4/8)$, predicting a stable amplitude near two. This amplitude is an approximation in the weakly nonlinear regime, not an exact sinusoidal solution. The equation is a standard model for self-sustained oscillation and, for large $\mu$, relaxation oscillations.