Absolute-value damping amplitude law ($A(T)=[1+2T/(3\pi ){]}^{-1}$)

For $x^{\prime \prime }+\epsilon |x|x^{\prime }+x=0$ and unit initial amplitude, averaging uses $⟨|\cos \psi |{\sin }^2\psi ⟩=2/(3\pi )$. 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.