Absolute-value damping amplitude law
= Absolute-value damping amplitude law
{title2=$A(T)=[1+2T/(3\pi)]^{-1}$}
For $x''+\varepsilon|x|x'+x=0$ and unit initial amplitude, averaging uses $\langle|\cos\psi|\sin^2\psi\rangle=2/(3\pi)$. Thus $A_T=-2A^2/(3\pi)$ and the displayed algebraic envelope follows. The leading phase has no first-order drift. The dissipation is amplitude dependent, so an exponential envelope with a constant decay rate would give the wrong long-time scale.