Van der Pol amplitude evolution
= Van der Pol amplitude evolution
{c}
{title2=$A_T=A/2-A^3/8$}
For $x''+\varepsilon(x^2-1)x'+x=0$, the <averaged amplitude for position-dependent damping> has the displayed logistic-type equation. With $A(0)=1$, its positive solution is $A(T)=2/\sqrt{1+3e^{-T}}$. Small amplitudes grow and large ones decay toward the stable value two; the leading oscillation is $A(\varepsilon t)\cos t$.