Averaged amplitude equation ($\dot{r}=ϵ⟨\dot{H}/ϵ⟩/H^{\prime }(r)$)

An averaged amplitude equation describes the slow evolution of a periodic orbit's amplitude under a weak perturbation. If unperturbed energy is $H(r)$ with $H^{\prime }(r)\ne 0$, divide the mean energy drift over one fast period by $H^{\prime }(r)$. Simple zeros identify attracting or repelling persistent periodic orbits according to the derivative of this drift; this approximation is local to an appropriate perturbative amplitude range.