Duffing equation

This is an equation for a damped non-linear oscillator, so it doesn't obey energy conservation. Multiply it by $dot{omega}$ one can get: $$ frac{d}{dt}left(frac{dot{omega}^2}{2} + k^2omega^2 - frac{omega^4}{2}right) = -2kdot{omega}^2 $$ One usually deals with this equation using Van der Pol method, although it depends on the context. You can find the references in this article, particularly the one by Dykman (note however that it contains some typos).