= Solution
Eliminating $y$ gives
$$
\ddot x+x-F=\mu(1-x^2)\dot x.
$$
For $\mu\ll1$, take $x=F+A\cos(t+\delta)$ over one fast period. The oscillator energy satisfies
$$
\dot E=\mu(1-x^2)\dot x^2,
$$
and averaging gives
$$
\langle\dot E\rangle
=\frac{\mu A^2}{2}\left(1-F^2-\frac{A^2}{4}\right).
$$
Thus, if $F^2<1$, amplitudes approach
$$
A=2\sqrt{1-F^2},
$$
producing a stable weakly nonlinear limit cycle. If $F^2>1$, every nonzero small amplitude loses energy and trajectories approach the stable equilibrium.
Back to article page
