Slow oscillator damping by a relaxing auxiliary variable ($A(\tau )=\exp [-3\tau /2-3(1-e^{-\tau })]$)

For $x^{\prime \prime }+3ϵy{x}^{\prime }+x=2(y^{\prime }{)}^2$ and $y^{\prime }=ϵ(1+x-y)$ with leading oscillation $x=A(\tau )\sin t$, $\tau =ϵt$, averaging the auxiliary equation gives $Y^{\prime }=1-Y$. Cancellation of resonant forcing in the oscillator gives $2A^{\prime }+3YA=0$. With $Y(0)=3$ and $A(0)=1$, $Y=1+2e^{-\tau }$ and the displayed amplitude follows. Bounded oscillatory corrections repair the instantaneous auxiliary equation; secular terms in a fixed-time expansion show why the slow scale is needed.