= Two-time expansion of a weakly damped linear oscillator
{title2=$x=e^{-\epsilon t}\sin(\sqrt{1-\epsilon^2}\,t)/\sqrt{1-\epsilon^2}$}
The fixed-time <regular perturbation> is $(1-\epsilon t)\sin t+O(\epsilon^2)$. Retaining the exponential decay and the first accumulated phase correction gives $e^{-\epsilon t}[\sin t-\epsilon^2t\cos t/2]+O(\epsilon^2)$ on $\epsilon t=O(1)$. When $\epsilon^2t=O(1)$ 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.
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