= Van der Pol equation
{c}
{title2=$\ddot x-\mu(1-x^2)\dot x+x=0$}
The Van der Pol equation is the nonlinear oscillator
$$
\ddot x-\mu(1-x^2)\dot x+x=0,\qquad \mu>0,
$$
in dimensionless variables. Its damping is negative at small amplitude and positive when $|x|>1$. For the quadratic oscillator energy $E=(\dot x^2+x^2)/2$,
$$
\dot E=\mu(1-x^2)\dot x^2.
$$
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 $\langle\dot E\rangle\approx\mu(A^2/2-A^4/8)$, 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>.
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