Position-dependent damping
= Position-dependent damping
{title2=$\dot E=-\epsilon f(x)\dot x^2$}
In $x\prime\prime+\epsilon f(x)x\prime+x=0$, the damping coefficient depends on the displacement. The oscillator energy $E=(x\prime^2+x^2)/2$ satisfies $E\prime=-\epsilon f(x)x\prime^2$. 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.