Two-time expansion of a weakly damped linear oscillator ($x=e^{-ϵt}\sin (\sqrt{1-{ϵ}^2}t)/\sqrt{1-{ϵ}^2}$)

The fixed-time regular perturbation is $(1-ϵt)\sin t+O({ϵ}^2)$. Retaining the exponential decay and the first accumulated phase correction gives $e^{-ϵt}[\sin t-{ϵ}^{2}t\cos t/2]+O({ϵ}^2)$ on $ϵt=O(1)$. When ${ϵ}^{2}t=O(1)$ its phase series is unordered. For positive damping the absolute error is nevertheless exponentially small on that later scale, so phase failure and failure in absolute displacement must be distinguished.