Formal Asymptotic Expansion for the Damped Amsler Equation

I construct a formal Poincaré‑type asymptotic expansion for oscillatory solutions of the damped Amsler equation \(t u''+(1+α)u'=sin u\) with \(α∈(-12,0]\) that approach \(u=π\) as \(t→∞\). The method of variation of constants together with systematic averaging reduces the problem to a hierarchy of linear equations. The leading behaviour is a power‑law decay with a phase shift proportional to \(τ-2α\) (\(τ=2√t\)). All higher corrections are obtained explicitly; the first three blocks of Fourier harmonics are given in closed form. The two constants \(C_1>0,\;C_2\) parametrise the family of solutions and can be extracted from initial data by a shooting procedure.