 Energy balance for the weakly perturbed double-well oscillator

Forthe unperturbed Hamiltonian function $H=y^2/2-x^2/2+x^4/4$ satisfies the exact drift identity $\dot{H}=\epsilon (1-\alpha x^2)y^2$. The perturbation adds energy where $\alpha x^2<1$ and removes it where $\alpha x^2>1$, for positive $\epsilon$.

On a regular unperturbed periodic orbit of energy $h$, let $x_1<x_2$ be its turning points and write $y=\pm \sqrt{2h+x^2-x^4/2}$. Since $dt=dx/y$, both halves contribute the same integral:A simple zero of this energy increment selects a nearby perturbed periodic orbit by the energy balance method. The approximation is local to regular closed levels; at $h=0$, the homoclinic balance for the weakly perturbed double-well oscillator uses a convergent drift integral despite the infinite traversal time.