Velocity-switched driven oscillator ($\ddot{x}+2k\dot{x}+{\omega }^{2}x=a{1}_{\{\dot{x}>0\}}$)

This damped harmonic oscillator receives a constant force only while moving toward increasing $x$. For $0<k<\omega$, a positive maximum $X$ evolves without forcing to a minimum $-rX$, then with forcing to the next maximum $r^{2}X+(1+r)a/{\omega }^2$, where $r=e^{-k\pi /\sqrt{{\omega }^2-k^2}}$. Its unique periodic maximum is $a/[{\omega }^2(1-r)]$ and its period is $2\pi /\sqrt{{\omega }^2-k^2}$. Position and velocity are continuous at switches, but acceleration has jumps. The slope $r^2<1$ of this turning-point return map shows attraction of nearby turning-point amplitudes.