= Averaged oscillator damping by a power of velocity
{title2=$R_T=-C_nR^n,\quad C_n=\langle\sin^{n+1}t\rangle$}
For $y''+\epsilon(y')^n+y=0$, the <method of multiple scales> gives no first-order phase drift. If $n$ is odd, the <amplitude> obeys the displayed damping law with $C_n=2^{-(n+1)}\binom{n+1}{(n+1)/2}$. If $n$ is even, the average is zero and the amplitude is unchanged on the first slow time $T=\epsilon t$. The odd case dissipates <mechanical energy>, whereas the even case is reversible and its force is not damping of both velocity signs.
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