Order reduction of a radiation-reaction oscillator

For $|\epsilon \omega |\ll 1$, use the zeroth-order equation $\ddot{x}=f-x$ inside the small third-derivative term, givingFor positive $\epsilon$ this reduced equation has a stable retarded response with no high-frequency runaway. Its response $G_{\mathrm{red}}=(1-i\epsilon \omega )/(1-{\omega }^2-i\epsilon \omega )$ agrees with the unreduced oscillator to first order at fixed nonresonant frequency. Order reduction deliberately removes the singular fast mode outside the assumed low-frequency regime; it is an approximation rather than an exact alternative boundary condition for the third-order equation.