Quadratically damped oscillator ($x\prime \prime +ϵ|x\prime |x\prime +x=0$)

For initial amplitude one, the method of multiple scales gives $x\simeq A(ϵt)\sin t$ with $A_T=-4A^2/(3\pi )$, hence $A=1/(1+4ϵt/(3\pi ))$. Positive damping causes algebraic decay. Negative damping yields a formal finite-time amplitude pole, but the weak-damping assumption fails before it because $|ϵ|A$ becomes order one. A leading approximation has order-$|ϵ|$ displacement error on fixed slow-time intervals away from that pole.