Position-dependent damping ($\dot{E}=-ϵf(x){\dot{x}}^2$)

In $x\prime \prime +ϵf(x)x\prime +x=0$, the damping coefficient depends on the displacement. The oscillator energy $E=(x{\prime }^2+x^2)/2$ satisfies $E\prime =-ϵf(x)x{\prime }^2$. Positive $f$ dissipates energy and negative $f$ supplies it; a changing sign can select a stable amplitude. The averaged amplitude for position-dependent damping measures the net effect over an oscillation.