= Radiation-reaction oscillator pole structure
For $\ddot x+x-\varepsilon\dddot x=f$ and the $e^{-i\omega t}$ convention, the frequency response is $G(\omega)=[1-\omega^2-i\varepsilon\omega^3]^{-1}$. At small nonzero real $\varepsilon$, its poles are
$$
\omega_\pm=\pm1-\frac{i\varepsilon}{2}+O(\varepsilon^2),\qquad
\omega_3=\frac i\varepsilon+i\varepsilon+O(\varepsilon^3).
$$
For positive $\varepsilon$ the slow oscillations decay but the fast mode runs away; for negative $\varepsilon$ the fast mode decays but the slow oscillations grow. Thus no nonzero sign makes all modes stable. A causal initial-value response and a bounded response free of runaways impose different boundary conditions.
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