Consider the following equation for the position of an oscillator $x(t)$, $$\ddot x(t) + f(t) \dot x + g(x(t)) + h(t)= 0\,.$$ We will assume we live in a basin of attraction of this equation, where it admits quasi-periodic solutions. My goal is to compute the period of the oscillations as a function of time.
For a 'simple' example, take $f(t) = t^{-n}$, $g = (1 -\gamma x^2)x$, and $h = t^{-m}\sin(\Omega t)$ where $m$ and $n$ are positive integers, and $\gamma$ is a positive real number. This is an example of a driven Duffing oscillator with nonlinear damping, but none of the techniques I could find (say, in this text) seem to generalize. Any suggestions are appreciated!