 Radiation-reaction oscillator pole structure

For $\ddot{x}+x-\epsilon \stackrel{...}{x}=f$ and the $e^{-i\omega t}$ convention, the frequency response is $G(\omega )=[1-{\omega }^2-i\epsilon {\omega }^3{]}^{-1}$. At small nonzero real $\epsilon$, its poles areFor positive $\epsilon$ the slow oscillations decay but the fast mode runs away; for negative $\epsilon$ 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.